Icps-1997-4.ps

%!ps-adobe-2.0 %%creator: dvipsk 5.55a copyright 1986, 1994 radical eye software %%title: fthese.dvi %%pages: 133 %%pageorder: ascend %%boundingbox: 0 0 596 842 %%endcomments %dvipscommandline: dvips -f fthese.dvi %dvipsparameters: dpi=300, compressed, comments removed %dvipssource: tex output 1997.02.25:1621 %%beginprocset: texc.pro /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley

false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end %%endprocset %%beginprocset: special.pro texdict begin /sdict 200 dict n sdict begin /@specialdefaults{/hs 612 n /vs 792 n /ho 0 n /vo 0 n /hsc 1 n /vsc 1 n /ang 0 n /clip 0 n /rwiseen false n /rhiseen false n /letter{}n /note{}n /a4{}n /legal{}n}b /@scaleunit 100 n /@hscale{@scaleunit div /hsc x}b /@vscale{@scaleunit div /vsc x}b /@hsize{/hs x /clip 1 n}b /@vsize{/vs x /clip 1 n}b /@clip{ /clip 2 n}b /@hoffset{/ho x}b /@voffset{/vo x}b /@angle{/ang x}b /@rwi{ 10 div /rwi x /rwiseen true n}b /@rhi{10 div /rhi x /rhiseen true n}b /@llx{/llx x}b /@lly{/lly x}b /@urx{/urx x}b /@ury{/ury x}b /magscale true def end /@macsetup{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}n /note{}n /legal{} n /od{txpose 1 0 mtx defaultmatrix dtransform s atan/pa x newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 f/|______courier fnt invertflag{paintblack}if}n /txpose{pxs pys scale ppr aload pop por{noflips{pop s neg s tr pop 1 -1 scale}if xflip yflip and{pop s neg s tr 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg tr}if xflip yflip not and{pop s neg s tr pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 tr}if yflip xflip not and{ppr 1 get neg ppr 0 get neg tr}if}{noflips{tr pop pop 270 rotate 1 -1 scale}if xflip yflip and{tr pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg tr}if xflip yflip not and{tr pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 tr}if yflip xflip not and{tr pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 s tr}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy tr .96 dup scale neg s neg s tr}if}n /cp {pop pop showpage pm restore}n end}if}if}n /normalscale{resolution 72 div vresolution 72 div neg scale magscale{dvimag dup scale}if 0 setgray} n /psfts{s 65781.76 div n}n /starttexfig{/psf$savedstate save n userdict maxlength dict begin /magscale false def normalscale currentpoint tr /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy x /psf$cx x /psf$sx psf$x psf$urx psf$llx sub div n /psf$sy psf$y psf$ury psf$lly sub div n psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub tr /showpage{}n /erasepage{}n /copypage{}n /p 3 def @macsetup}n /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll s lineto s lineto s lineto closepath clip newpath moveto}n /endtexfig{end psf$savedstate restore}n /@beginspecial{sdict begin /specialsave save n gsave normalscale currentpoint tr @specialdefaults count /ocount x /dcount countdictstack n}n /@setspecial {clip 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo tr hsc vsc scale ang rotate rwiseen{rwi urx llx sub div rhiseen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg tr }{rhiseen{rhi ury lly sub div dup scale llx neg lly neg tr}if}ifelse clip 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}n /erasepage{}n /copypage{}n newpath }n /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{

end}repeat grestore specialsave restore end}n /@defspecial{sdict begin} n /@fedspecial{end}b /li{lineto}b /rl{rlineto}b /rc{rcurveto}b /np{ /savex currentpoint /savey x n 1 setlinecap newpath}n /st{stroke savex savey moveto}n /fil{fill savex savey moveto}n /ellipse{/endangle x /startangle x /yrad x /xrad x /savematrix matrix currentmatrix n tr xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}n end %%endprocset texdict begin 39158280 55380996 1000 300 300 (fthese.dvi) @start /fa 7 110 df<90380fe02090387018603801c00439030003e000061301000e13 004814605a15201278127000f01400a80070142012781238a26c14407e000614806ceb01 003801c00638007018eb0fe01b1e7d9c21>67 d80 d<13201370a313b8a3ea011ca2ea 031eea020ea2487eea07ffea040738080380a2001813c01301123838fc07f815157f9419 >97 d101 d<eaff80ea1c00b3eaff8009157f940d>105 d<eaffc0001cc7fcac1480a3130114005b5bb5fc11157f9415>108 d<00feeb0fe0001e140000171317a338138027a23811c047a33810e087a2eb7107a3133a a2131ca2123839fe083fe01b157f941f>i e /fb 4 121 df<913807f00491383ffc0c91 38fc0e1c903803e00390390f8001b8d91e0013f84913005b491470485a12035b00071560 485aa248c8fc1600123ea35aa4127800f8903803fff05c9138001f00151ea21278a25d12 7c123ca26c5c001f14f86c13013907c003b83903f01e103800fff8d93fc0c7fc262b78a9 2d>71 d<133eebe118380180bc380300fc481378120e5aa2003c13f05aa338f001e0a438 e003c014c3a3eb0786ea600fea701b3830338c38186188380f80f0181a7a991e>97 d<ebf00f39019c30c039031cc0e090381f80600006eb0070a2011e1378a2485a1200a349 13f0a315e0ebf001a215c0ec03801201ec0700140e6d5a3803cc30ebc3c001c0c7fca248 5aa448c8fca35aeafff0a21d2680991e>112 d<ebf03e39031c618038040cc1390c0f83 c00018eb87800010eb0300003090c7fca2ea601e1200a35ba45b1403a21406ea60f838f0 f00cebb80800e15b38631860383c07c01a1a7d991b>120 d e /fc 6 113 df<13fe3803ffc0000f13f0381f01f8383e007c48133c0078131e5a140f7ea212 70c7fca2141ea3143c147814f8eb01f0eb03e0eb07c0eb0f80eb1f00133e5b5b485aea03 c0485a380f000f121e5a007fb5fcb6fc7e18257da41f>50 d<137f3801ffc0000713e038 0f81f0381f0078123e003c133c5a141e127012f0b512fea300f0c7fca21278a36c131e7e 001f133e380fc0fc3803fff86c13f038003f80171a7d991f>101 d<eb01fceb07ff011f1380eb3f0f133c903878070091c7fca5387ffffeb5fca2d80078c7 fcb3a2383ffff0487f6c5b19257fa41f>i<ea7f8012ff127f1207a8eb83f0eb8ffcebbf feebfc1febf00f9038e0078013c0a21380af397ff87ff839fffcfffc397ff87ff81e2580 a41f>104 d<137013f8a3137090c7fca7ea7ff812ff127fea0078b3a2b512f8a315267b a51f>i<387f83e038ff9ffc387fbffe3807fc1f9038f007809038e003c013c090388001 e0a2ec00f0a8ec01e013c0ec03c0ebe0079038f00f809038f81f00ebbffeeb9ff8eb87e0 0180c7fcaaea7ff8487e6c5a1c2780991f>112 d e /fd 6 121 df<ea0ec012311260ea418012c1a213a0ea47c012390b097e8810>97 d<1208a21200a41270129812b01230a21260126412681270060f7d8e0b>105 d<12381218a35a13c0ea3360ea3440ea7800127e12631320eac340eac1800b0e7e8d10> 107 d<ea73c7389c6880389830c0383861801230a2eb63103860c320ebc1c014097d8819 >109 d<ea7380ea9c40ea9860ea30c0a3ea3188ea6190ea60e00d097d8812>i<ea1cf0ea 23181310ea0600a3ea4610eace20ea73c00d097f8810>120 d e /fe 6 108 df<126012f0a212701210a21220a21240a2040a7d830a>59 d<14c0a21301a21303130514e01308131813101320a213401380a23801fff0eb00701202 5aa25a121838fe03fe17177f961a>65 d<3807fff83800e00e14071403a2ea01c01407a2 140e3803803cebfff0a2eb803c3807001c140ea3000e131ca214381470381c01e0b51200 18177f961b>i<120313801300c7fca6121c12241246a25a120c5aa31231a21232a2121c 09177f960c>105 d<1318133813101300a6ea01c0ea0220ea0430a2ea08601200a313c0 a4ea0180a4ea630012e312c612780d1d80960e>i<121f1206a45aa4ea181c1366138eea 190cea3200123c123fea3180ea60c013c4a3eac0c813700f177e9612>i e /ff 5 108 df<eb78603801c4e0ea0302380601c0120e121c123c383803801278a338 f00700a31408eb0e101270131e38302620ea18c6380f03c015157e941a>97 d<ea03c0ea3f801203a348c7fca4120ea45a137cea1d86ea1e03003c1380123814c0a212 70a438e00780a3eb0f00130eea601ca2ea3030ea18e0ea0f8012237ea215>i<13e0a212 01ea00c01300a9121e1223ea4380a21283ea8700a21207120ea35aa3ea38201340127013 801230ea3100121e0b227ea111>105 d<147014f0a214601400a9130feb3180eb41c013

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-21 b(ees)18 b(p)q(our)g(l'exploitation)h(de)f(ces)g(mac)o(hines)h(son) o(t)e(de)h(deux)h(natures)7 b(:)17 b(la)i(pa-)75 2542 y(rall)o(\023)-21 b(elisation)23 b(automatique)e(de)h(progammes)f(s)o (\023)-21 b(equen)o(tiels)22 b(et)g(l'utilisation)i(de)e(mo)q(d)o(\022) -21 b(eles)22 b(de)g(pro-)p eop %%page: 6 8 6 7 bop 75 269 a fq(6)1051 b fl(chapitre)16 b(1.)29 b(intr)o(oduction) 75 397 y fq(grammation)16 b(adapt)o(\023)-21 b(es)15 b(aux)i(mac)o(hines)h(parall)o(\022)-21 b(eles.)17 b(la)g(premi)o(\022) -21 b(ere)17 b(v)o(oie)g(est)e(\023)-21 b(economiquemen)o(t)17 b(tr)o(\022)-21 b(es)75 453 y(prometteuse,)13 b(\023)-21 b(etan)o(t)14 b(donn)o(\023)-21 b(e)14 b(l'immense)i(quan)o(tit)o(\023) -21 b(e)15 b(d'applications)h(d)o(\023)-21 b(ej\022)e(a)13 b(\023)-21 b(ecrites)15 b(dans)g(les)h(mo)q(d)o(\022)-21 b(eles)75 510 y(de)13 b(programmation)e(s)o(\023)-21 b(equen)o(tiels.)12 b(malheureusemen)o(t,)h(si)f(on)g(sait)g(aujourd'h) o(ui)h(extraire)f(automati-)75 566 y(quemen)o(t)i(du)f(parall)o(\023) -21 b(elisme)15 b(des)f(nids)g(de)g(b)q(oucles,)g(les)g(structures)f (it)o(\023)-21 b(erativ)o(es)13 b(dynamiques)h(\(comme)75 623 y(la)i(structure)e fk(while)p fq([24)o(]\))g(p)q(osen)o(t)i (toujours)e(de)i(s)o(\023)-21 b(erieux)15 b(probl)o(\022)-21 b(emes.)14 b(les)i(limitations)h(actuelles)f(de)75 679 y(ces)d(compilateurs)h(parall)o(\023)-21 b(eliseurs)15 b(fon)o(t)d(qu'ils)i(ne)g(p)q(euv)o(en)o(t)f(pr)o(\023)-21 b(etendre)13 b(p)q(our)g(l'instan)o(t,)f(^)-21 b(etre)12 b(une)i(so-)75 736 y(lution)e(unique)g(satisfaisan)o(te)e(p)q(our)h (les)g(utilisateurs.)g(concernan)o(t)f(les)i(mo)q(d)o(\022)-21 b(eles)10 b(de)h(programmation,)75 792 y(de)k(nom)o(breuses)h(prop)q (ositions)f(on)o(t)f(\023)-21 b(et)o(\023)g(e)13 b(faites,)h(souv)o(en) o(t)h(inspir)o(\023)-21 b(ees)16 b(de)f(l'arc)o(hitecture)h(sous-jacen) o(te)75 849 y(de)g(la)f(mac)o(hine.)h(resp)q(ectan)o(t)f(la)h (classi\014cation)g(de)g(flynn[19],)e(les)i(langages)f(les)h(plus)g(r)o (\023)-21 b(epandus)15 b(se)75 905 y(scinden)o(t)e(en)g(deux)f(cat)o (\023)-21 b(egories)7 b(:)k(les)i(langages)f(\022)-23 b(a)12 b fj(p)n(ar)n(al)r(l)o(\023)-22 b(elisme)12 b(de)i(c)n(ontr^)-23 b(ole)p fq(,)11 b(calquan)o(t)h(l'arc)o(hitecture)75 962 y(mimd-dm,)e(et)i(les)h(langages)e(\022)-23 b(a)12 b fj(p)n(ar)n(al)r(l)o(\023)-22 b(elisme)12 b(de)h(donn)o(\023)-22 b(e)n(es)10 b fq(utilis)o(\023)-21 b(es)13 b(initialemen)o(t)h(sur)e (des)g(mac)o(hines)75 1018 y(simd.)146 1074 y(les)17 b(programmes)g(\022)-23 b(a)17 b(parall)o(\023)-21 b(elisme)18 b(de)g(con)o(tr^)-23 b(ole)17 b(son)o(t)g(constitu)o(\023)-21 b(es)17 b(de)g(plusieurs)j(pro)q(cessus)d(s)o(\023)-21 b(e-)75 1131 y(quen)o(tiels)19 b(s'ex)o(\023)-21 b(ecutan)o(t)16 b(libremen)o(t)j(sur)f(les)g(di\013)o(\023)-21 b(eren)o(ts)17 b(pro)q(cesseurs,)h(les)g(sync)o(hronisations)h(n)o(\023)-21 b(eces-)75 1187 y(saires)17 b(en)o(tre)g(pro)q(cessus)h(se)f(faisan)o (t)g(par)e(\023)-21 b(ec)o(hanges)17 b(explicites)i(de)f(message.)e (c'est)g(aujourd'h)o(ui)i(le)75 1244 y(mo)q(d)o(\022)-21 b(ele)17 b(le)i(plus)f(emplo)o(y)o(\023)-21 b(e,)17 b(notammen)o(t)f (\022)-23 b(a)17 b(tra)o(v)o(ers)f(les)i(biblioth)o(\022)-21 b(eques)19 b(pvm[20)o(])e(et)g(mpi[42)o(].)g(la)75 1300 y(raison)e(de)h(ce)g(succ)o(\022)-21 b(es)15 b(tien)o(t)g(certainemen)o (t)h(\022)-23 b(a)15 b(leur)h(disp)q(onibil)q(i)q(t)o(\023)-21 b(e)17 b(sur)e(tous)g(les)h(t)o(yp)q(es)f(de)h(mac)o(hines,)75 1357 y(y)f(compris)h(sur)f(les)g(r)o(\023)-21 b(eseaux)15 b(de)g(stations)g(de)g(tra)o(v)m(ail,)g(ce)h(qui)g(a)f(fait)g(d'elles)h (un)g(standard)e fj(de)j(facto)p fq(.)146 1413 y(les)c(programmes)e (\022)-23 b(a)12 b(parall)o(\023)-21 b(elisme)14 b(de)f(donn)o(\023)-21 b(ees,)12 b(ou)g(data-parall)o(\022)-21 b(eles,)13 b(extraien)o(t)f(le)

h(parall)o(\023)-21 b(elisme)75 1470 y(de)16 b(la)g(structure)g(des)g (donn)o(\023)-21 b(ees,)15 b(qui)i(son)o(t)e(m)o(\023)-21 b(emoris)o(\023)g(ees)14 b(dans)i(des)g fj(variables)g(p)n(ar)n(al)r(l) o(\022)-22 b(eles)p fq(.)14 b(l'applica-)75 1526 y(tion)19 b(d'une)g(instruction)h(p)q(ortan)o(t)e(sur)g(une)i(v)m(ariable)g (parall)o(\022)-21 b(ele,)19 b(a)f(p)q(our)h(e\013et)f(d'ex)o(\023)-21 b(ecuter)18 b(en)i(pa-)75 1583 y(rall)o(\022)-21 b(ele)16 b(l'instruction)h(sur)e(c)o(haque)g(\023)-21 b(el)o(\023)g(emen)o(t)14 b(de)i(la)f(v)m(ariable.)i(le)f(programme)e(appara)-5 b(^)-18 b(\020t)15 b(comme)g(un)75 1639 y(seul)21 b(et)e(m)o(^)-21 b(eme)19 b(co)q(de)h(s)o(\023)-21 b(equen)o(tiel)21 b(agissan)o(t)e (sur)g(des)h(v)m(ariables)i(d'un)e(t)o(yp)q(e)f(nouv)o(eau.)h(ce)g(mo)q (d)o(\022)-21 b(ele)75 1695 y(de)17 b(programmation,)e(bien)j(que)f(tr) o(\022)-21 b(es)16 b(adapt)o(\023)-21 b(e)15 b(aux)i(structures)f(de)h (donn)o(\023)-21 b(ees)17 b(r)o(\023)-21 b(eguli)o(\022)g(eres,)15 b(couvre)i(de)75 1752 y(larges)j(domaines)g(d'applications)h([26)o(].)e (les)i(comm)o(unications)f(in)o(ter-pro)q(cesseurs)h(requises)f(son)o (t)75 1808 y(cac)o(h)o(\023)-21 b(ees)13 b(\022)-23 b(a)13 b(l'utilisateur,)h(et)f(la)h(t^)-23 b(ac)o(he)12 b(de)i(r)o(\023)-21 b(epartir)12 b(les)h(\023)-21 b(el)o(\023)g(emen)o(ts)12 b(des)h(v)m(ariables)i(parall)o(\022)-21 b(eles)14 b(ainsi)g(que)75 1865 y(la)j(g)o(\023)-21 b(en)o(\023)g(eration)15 b(des)h(comm)o (unications)i(n)o(\023)-21 b(ecessaires)16 b(incom)o(b)q(e)i(au)e (compilateur.)h(il)h(faut)e(ici)i(citer)f(le)75 1921 y(high)d(p)o(erformance)f(f)l(ortran)f(\(hpf\))h([27)o(])g(qui)h(a)f(p) q(our)h(ob)s(jectif)f(de)h(s'imp)q(oser)g(comme)f(le)h(standard)75 1978 y(en)j(mati)o(\022)-21 b(ere)16 b(de)h(langage)g(data-parall)o (\022)-21 b(ele,)17 b(car)g(d)o(\023)-21 b(e\014ni)17 b(lors)g(d'un)g(forum)f(rassem)o(blan)o(t)h(utilisateurs,)75 2034 y(c)o(herc)o(heurs)k(et)g(constructeurs.)f(cep)q(endan)o(t,)h(la)g (norme)g(d)o(\023)-21 b(e\014nie)21 b(n'a)f(pas)h(p)q(ermis)g(de)h (mettre)e(au)75 2091 y(p)q(oin)o(t)15 b(\022)-23 b(a)15 b(ce)g(jour,)f(des)h(compilateurs)g(don)o(t)g(les)g(p)q(erformances)g (con)o(v)m(aincraien)o(t)h(les)f(utilisateurs.)h(le)75 2147 y(fait)g(que)h(les)g(comm)o(unications)g(soien)o(t)f(masqu)o(\023) -21 b(ees)16 b(au)g(programmeur)f(par)h(exemple,)h(rend)g(di\016cile)75 2204 y(la)h(g)o(\023)-21 b(en)o(\023)g(eration)17 b(d'un)h(co)q(de)h (e\016cace)f(en)h(terme)f(de)g(comm)o(unications.)h(cette)e(lacune)j (est)e(aussi)g(un)75 2260 y(argumen)o(t)c(qui)i(fait)g(p)q(enc)o(her)g (le)g(c)o(hoix)f(des)h(utilisateurs)h(v)o(ers)d(pvm.)146 2373 y(p)o(ourtan)o(t,)e(ce)k(mo)q(d)o(\022)-21 b(ele)14 b(p)q(oss)o(\022)-21 b(ede)15 b(\022)-23 b(a)14 b(nos)h(y)o(eux)g(des)g (qualit)o(\023)-21 b(es)15 b(in)o(t)o(\023)-21 b(eressan)o(tes)13 b(par)i(rapp)q(ort)f(au)h(mo-)75 2429 y(d)o(\022)-21 b(ele)16 b(\022)-23 b(a)15 b(parall)o(\023)-21 b(elisme)17 b(de)g(con)o(tr^)-23 b(ole,)15 b(notammen)o(t)g(en)h(terme)f(de)i (clart)o(\023)-21 b(e)15 b(du)h(co)q(de,)g(de)g(p)q(ortabilit)o(\023) -21 b(e,)16 b(et)75 2486 y(d'abstraction)f(par)f(rapp)q(ort)h(au)g(mo)q (d)o(\022)-21 b(ele)15 b(d'ex)o(\023)-21 b(ecution.)15 b(le)h(fait)f(que)g(ce)h(mo)q(d)o(\022)-21 b(ele)15 b(soit)g(sync)o (hrone)h(et)75 2542 y(d)o(\023)-21 b(eterministe)16 b(facilite)h(aussi) e(b)q(eaucoup)i(la)f(compr)o(\023)-21 b(ehension)15 b(du)h(comp)q (ortemen)o(t)f(des)h(programmes)p eop %%page: 7 9 7 8 bop 1824 269 a fq(7)74 397 y(\023)-21 b(ecrits.)19 b(il)h(subsiste)h(toutefois)e(des)h(questions)g(d)o(\023)-21 b(elicates)20 b(concernan)o(t)f(la)h(ma)-5 b(^)-18 b(\020trise)20 b(du)g(mo)q(d)o(\022)-21 b(ele)20 b(de)75 453 y(programmation,)15 b(la)i(transformation)e(et)h(la)h(preuv)o(e)g(des)g(programmes.)e(le)i (formalisme)f fn(pei)h fq(pro-)75 510 y(p)q(os)o(\023)-21

b(e)11 b(par)g([56)o(])g(ten)o(te)g(d'app)q(orter)f(des)i(solutions)g (\022)-23 b(a)11 b(ces)g(probl)o(\022)-21 b(emes.)11 b(nous)g(mon)o(trerons)g(qu'il)h(p)q(oss)o(\022)-21 b(ede)75 566 y(l'essen)o(tiel)14 b(des)f(caract)o(\023)-21 b(eristiques)11 b(du)i(mo)q(d)o(\022)-21 b(ele)13 b(data-parall)o(\022)-21 b(ele,)12 b(et)g(p)q(eut)g(^)-21 b(etre)11 b(utilis)o(\023)-21 b(e)14 b(p)q(our)e(mo)q(d)o(\023)-21 b(eliser)75 623 y(les)14 b(programmes)e(\023)-21 b(ecrits)12 b(dans)i(des)g(langages)f (data-parall)o(\022)-21 b(eles.)14 b(a)f(c^)-23 b(ot)o(\023)i(e)12 b(d'autres)h(tra)o(v)m(aux)g(sur)h(la)f(s)o(\023)-21 b(e-)75 679 y(man)o(tique)13 b(et)f(la)g(preuv)o(e)h(des)f(programmes)g (data-parall)o(\022)-21 b(eles)12 b(\()p fj(cf.)g fq([5)o(])g(par)g (exemple\),)h fn(pei)f fq(fournit)g(un)75 736 y(cadre)i(formel)f(p)q (our)h(l'expression)h(de)f(ces)g(programmes,)e(dans)i(la)g(lign)o(\023) -21 b(ee)14 b(des)g(langages)f(d')o(\023)-21 b(equations)75 792 y(r)o(\023)g(ecurren)o(tes)10 b(comme)h fn(alpha)p fq([39])g(ou)g fn(cr)m(yst)m(al)p fq([12)o(].)f(gr^)-23 b(ace)11 b(\022)-23 b(a)11 b(ce)h(supp)q(ort)f(formel,)g(d)o(\023)-21 b(e\014nissan)o(t)11 b(clai-)75 849 y(remen)o(t)j(la)h(s)o(\023)-21 b(eman)o(tique)14 b(des)h(op)o(\023)-21 b(erations,)13 b(il)j(devien)o(t)g(p)q(ossible)g(de)f(raisonner)g(sur)g(les)g (programmes.)75 905 y(une)h(application)i(in)o(t)o(\023)-21 b(eressan)o(te)15 b(est)g(par)h(exemple)h(un)f(con)o(tr^)-23 b(oleur)16 b(de)g(t)o(yp)q(e)g(v)o(\023)-21 b(eri\014an)o(t)15 b(que)i(les)f(op)o(\023)-21 b(e-)75 962 y(rations)18 b(son)o(t)g(appliqu)o(\023)-21 b(ees)20 b(sur)f(des)f(domaines)i(coh)o (\023)-21 b(eren)o(ts.)17 b(mais)h(l'app)q(ort)h(le)g(plus)h (signi\014catif)g(de)75 1018 y fn(pei)15 b fq(est)g(certainemen)o(t)h (la)f(p)q(ossiblit)o(\023)-21 b(e)16 b(de)g(transformer)e(les)i (programmes,)e(sur)h(la)g(base)g(d'un)h(calcul)75 1074 y(de)g(ra\016nemen)o(t)g(d)o(\023)-21 b(e\014ni)16 b(\022)-23 b(a)16 b(partir)g(de)g(la)g(s)o(\023)-21 b(eman)o(tique)16 b(des)g(op)o(\023)-21 b(erations)15 b(du)h(langage.)g(ce)f(pro)q(c)o (\023)-21 b(ed)o(\023)g(e)15 b(de)75 1131 y(ra\016nemen)o(t)e(p)q(eut)i (servir)f(en)o(tre)g(autres,)f(\022)-23 b(a)13 b(mon)o(trer)g(l')o (\023)-21 b(equiv)m(alence)16 b(de)e(di\013)o(\023)-21 b(eren)o(ts)13 b(programmes,)g(ou)75 1187 y(d)o(\023)-21 b(eriv)o(er)11 b(des)h(solutions)g(di\013)o(\023)-21 b(eren)o(tes)10 b(\022)-23 b(a)12 b(partir)f(d'un)g(programme)g (initial.)i(ces)e(asp)q(ects)h(nous)f(sem)o(blen)o(t)74 1244 y(^)-21 b(etre)17 b(des)h(caract)o(\023)-21 b(eristiques)17 b(in)o(t)o(\023)-21 b(eressan)o(tes,)17 b(v)o(oire)h(indisp)q(ensables) j(p)q(our)d(le)h(g)o(\023)-21 b(enie)18 b(logiciel)i(dans)e(le)75 1300 y(domaine)e(de)f(l'algorithmique)i(parall)o(\022)-21 b(ele.)16 b(quan)o(t)f(au)g(placemen)o(t)h(des)g(donn)o(\023)-21 b(ees,)14 b(il)j(est)e(explicite)i(en)75 1357 y fn(pei)g fq(et)h(fait)f(partie)h(in)o(t)o(\023)-21 b(egran)o(te)16 b(du)i(programme.)e(les)i(transformations)f(que)h(l'on)f(p)q(eut)i (e\013ectuer)75 1413 y(sur)13 b(un)g(programme)f(mo)q(di\014en)o(t)i (par)e(cons)o(\023)-21 b(equen)o(t)13 b(ce)g(placemen)o(t)h(des)f(donn) o(\023)-21 b(ees)7 b(;)13 b(cette)f(mo)q(di\014cation)75 1470 y(du)20 b(placemen)o(t)g(constitue)g(d'ailleurs)h(souv)o(en)o(t)e (une)h(motiv)m(ation)g(\022)-23 b(a)19 b(part)g(en)o(ti)o(\022)-21 b(ere)19 b(p)q(our)g(ten)o(ter)g(de)75 1526 y(trouv)o(er)14 b(un)i(programme)e(plus)i(e\016cace)g(apr)o(\022)-21 b(es)14 b(transformation.)146 1583 y(les)h(p)q(ossibilit)o(\023)-21 b(es)17 b(qu'app)q(orte)d fn(pei)h fq(concernen)o(t)g(donc)h(pleinemen) o(t)g(le)g(probl)o(\022)-21 b(eme)15 b(de)g(l'exploita-)75 1639 y(tion)g(e\016cace)g(des)g(mac)o(hines)h(parall)o(\022)-21 b(eles)14 b(\023)-21 b(ev)o(o)q(qu)o(\023)g(e)13 b(pr)o(\023)-21

b(ec)o(\023)g(edemmen)o(t.)13 b(l'ob)s(jectif)i(est)g(de)g(faire)g (pro\014ter)75 1695 y(l'utilisateur)22 b(de)f(ce)g(formalisme,)g(et)g (de)g(l'assister)f(\022)-23 b(a)21 b(tra)o(v)o(ers)e(des)i(outils,)g (qui)h(\022)-23 b(a)20 b(terme)g(doiv)o(en)o(t)75 1752 y(automatiser)d(le)h(plus)g(de)g(c)o(hoses)g(p)q(ossibles.)h(dans)e (cette)g(th)o(\022)-21 b(ese,)16 b(nous)i(essaierons)g(de)g(mon)o(trer) e(ce)75 1808 y(que)j(p)q(ourrait)e(^)-21 b(etre)17 b(un)i(en)o (vironnemen)o(t)g(logiciel)i(de)d(conception)i(de)e(programmes)g (parall)o(\022)-21 b(eles)18 b(re-)75 1865 y(p)q(osan)o(t)c(sur)h(le)h (formalisme)f fn(pei)p fq(.)f(cet)h(en)o(vironnmen)o(t)g(doit)g (inclure)i(les)e(\023)-21 b(etap)q(es)14 b(de)h(conception)h(des)75 1921 y(programmes,)e(de)h(transformation,)f(et)h(en\014n)h(celle)h(de)f (la)f(g)o(\023)-21 b(en)o(\023)g(eration)13 b(de)j(co)q(de)g (data-parall)o(\022)-21 b(ele.)146 2034 y(le)19 b(premier)h(c)o (hapitre)f(exp)q(ose)g(les)h(princip)q(es)h(fondamen)o(taux)e(du)g (formalisme,)g(ainsi)h(que)f(les)75 2091 y(asp)q(ects)g(syn)o(taxiques) g(\022)-23 b(a)19 b(tra)o(v)o(ers)e(quelques)k(exemples)f(simples.)g (suit)g(dans)f(un)g(second)h(c)o(hapitre,)75 2147 y(une)e(d)o(\023)-21 b(e\014nition)18 b(des)f(r)o(\022)-21 b(egles)16 b(et)h(du)h(calcul)g (de)g(ra\016nemen)o(t)e(tels)i(qu'ils)g(son)o(t)e(d)o(\023)-21 b(e\014nis)18 b(en)f fn(pei)p fq(.)g(nous)75 2204 y(mon)o(trerons)h (aussi)h(dans)g(ce)g(c)o(hapitre)h(les)f(protot)o(yp)q(es)f(d'outils)i (qui)f(p)q(euv)o(en)o(t)g(^)-21 b(etre)17 b(asso)q(ci)o(\023)-21 b(es)19 b(\022)-23 b(a)18 b(la)75 2260 y(construction)d(des)h (programmes)e(et)h(\022)-23 b(a)15 b(leur)h(ra\016nemen)o(t.)146 2316 y(le)i(troisi)o(\022)-21 b(eme)17 b(c)o(hapitre)h(illustre)h (l'utilisation)h(du)e(calcul)h(de)f(ra\016nemen)o(t)g(p)q(our)g (e\013ectuer)f(des)75 2373 y(transformations)e(plus)i(globales)g(sur)f (un)h(programme.)e(il)i(s'agit)e(en)i(fait)f(de)h(l'application)h(de)e (plu-)75 2429 y(sieurs)h(r)o(\022)-21 b(egles)16 b(de)h(ra\016nemen)o (t)f(selon)i(un)f(sc)o(h)o(\023)-21 b(ema)15 b(se)i(pr)o(\023)-21 b(esen)o(tan)o(t)15 b(fr)o(\023)-21 b(equemmen)o(t.)15 b(nous)h(quali\014ons)75 2486 y(ces)f(transformations)f(de)i fj(str)n(at)o(\023)-22 b(e)n(gies)14 b fq(et)h(nous)g(les)h(emplo)o(y)o (ons)f(sur)g(un)h(exemple.)146 2542 y(le)k(quatri)o(\022)-21 b(eme)18 b(c)o(hapitre)i(sera)f(consacr)o(\023)-21 b(e)19 b(\022)-23 b(a)19 b(l'outil)h(de)g(traduction)g(de)g fn(pei)f fq(v)o(ers)g(hpf.)g(nous)p eop %%page: 8 10 8 9 bop 75 269 a fq(8)1051 b fl(chapitre)16 b(1.)29 b(intr)o(oduction) 75 397 y fq(exp)q(osons)16 b(ici)g(les)g(m)o(\023)-21 b(ecanismes)15 b(p)q(ermettan)o(t)g(de)g(passer)h(d'un)e(\023)-21 b(enonc)o(\023)g(e)13 b(\023)-21 b(equationnel)16 b fn(pei)f fq(\022)-23 b(a)15 b(un)h(pro-)75 453 y(gramme)d(imp)o(\023)-21 b(eratif)13 b(data-parall)o(\022)-21 b(ele,)14 b(et)f(les)i (restrictions)f(imp)q(os)o(\023)-21 b(ees)13 b(sur)h(ces)f(\023)-21 b(enonc)o(\023)g(es)12 b(p)q(our)i(p)q(ouv)o(oir)75 510 y(les)i(traduire.)g(nous)g(mon)o(trons)e(ensuite)j(ce)f(qui)g(a)f(\023) -21 b(et)o(\023)g(e)13 b(implan)o(t)o(\023)-21 b(e)16 b(dans)g(l'outil)h(actuel.)f(en\014n,)f(nous)75 566 y(prop)q(osons)i (quelques)i(pistes)g(p)q(our)f(am)o(\023)-21 b(eliorer)17 b(ce)h(traducteur)f(dans)h(le)g(but)g(de)g(pro\014ter)f(de)h(la)g(s)o (\023)-21 b(e-)75 623 y(man)o(tique)19 b(particuli)o(\022)-21 b(ere)20 b(d'hpf.)e(le)h(langage)f(hpf)h(p)q(oss)o(\022)-21 b(ede)18 b(la)h(particularit)o(\023)-21 b(e)19 b(par)g(rapp)q(ort)f (aux)75 679 y(autres)c(langages)g(data-parall)o(\022)-21 b(eles,)15 b(d')o(^)-21 b(etre)13 b(insensible,)k(du)e(p)q(oin)o(t)g (de)g(vue)f(de)h(la)g(correction)g(du)g(pro-)75 736 y(gramme,)d(au)i (placemen)o(t)g(des)f(donn)o(\023)-21 b(ees.)13 b(ceci)h(p)q(ermet)g (d')o(\023)-21 b(ecrire)13 b(des)h(algorithmes)g(don)o(t)f(le)h(texte)f

(est)75 792 y(pro)q(c)o(he)g(du)f(programme)f(s)o(\023)-21 b(equen)o(tiel.)13 b(cette)f(caract)o(\023)-21 b(eristique)12 b(para)-5 b(^)-18 b(\020t)12 b(in)o(t)o(\023)-21 b(eressan)o(te)11 b(p)q(our)h(la)h(v)o(\023)-21 b(eri\014ca-)75 849 y(tion)13 b(in)o(tuitiv)o(e)i(de)e(la)h(correction)f(du)g(programme,)f(par)h (simple)h(comparaison)f(a)o(v)o(ec)g(les)g(algorithmes)75 905 y(s)o(\023)-21 b(equen)o(tiels)16 b(existan)o(ts.)146 962 y(en)f(conclusion,)i(nous)f(repla\030)-20 b(cons)17 b fn(pei)e fq(dans)h(le)g(con)o(texte)f(des)h(tra)o(v)m(aux)f (comparables,)h(et)f(nous)75 1018 y(exp)q(osons)22 b(les)h (particularit)o(\023)-21 b(es)22 b(de)g(ce)g(langage.)g(nous)g (soulignons)h(l'in)o(t)o(\023)-21 b(er)o(^)g(et)20 b(de)i(ce)h(t)o(yp)q (e)f(de)g(for-)75 1074 y(malisme,)f(en)h(rapp)q(elan)o(t)f(les)h(tec)o (hniques)g(de)g(transformations)d(d)o(\023)-21 b(ev)o(elopp)o(\023)g (ees)21 b(dans)g(le)h(troisi)o(\022)-21 b(eme)75 1131 y(c)o(hapitre.)14 b(nous)g(insistons)h(sur)g(la)f(n)o(\023)-21 b(ecessit)o(\023)g(e)13 b(de)i(disp)q(oser)g(d'outils)g(p)q(our)f (manipuler)i(un)f(tel)f(forma-)75 1187 y(lisme,)k(et)f(nous)h(a)o(v)m (an\030)-20 b(cons)17 b(des)h(id)o(\023)-21 b(ees)17 b(de)h(fonctionnalit)o(\023)-21 b(es)18 b(suppl)o(\023)-21 b(emen)o(taires)18 b(que)g(nous)f(p)q(ourrions)75 1244 y(in)o(t)o(\023)-21 b(egrer)14 b(\022)-23 b(a)15 b(ces)h(outils.)f (nous)g(parlons)g(\023)-21 b(egalemen)o(t)14 b(des)i(ob)s(jectifs)f (vis)o(\023)-21 b(es)14 b(a)o(v)o(ec)h(la)h(traduction)f(v)o(ers)g(le) 75 1300 y(langage)g(hpf,)g(et)g(nous)g(concluons)h(sur)f(les)h(am)o (\023)-21 b(eliorations)15 b(p)q(ossibles)i(de)f(cette)f(traduction.)p eop %%page: 9 11 9 10 bop 1824 269 a fq(9)75 708 y fi(chapitre)32 b(2)75 916 y fp(d)m(\023)-56 b(e\014nition)39 b(du)f(formalisme)f fh(pei)146 1138 y fq(depuis)16 b(quelques)h(ann)o(\023)-21 b(ees,)15 b(un)h(certain)g(nom)o(bre)f(de)h(tra)o(v)m(aux)f(sur)h (l'analyse)g(statique)g(des)g(pro-)75 1195 y(grammes)c(on)o(t)g(ab)q (outi)i(\022)-23 b(a)12 b(des)h(tec)o(hniques)h(de)g(parall)o(\023)-21 b(elisation)14 b(automatique)7 b(:)12 b(rec)o(herc)o(he)i(de)f(b)q (oucles)75 1251 y(parall)o(\022)-21 b(eles,)21 b(distribution)i(des)e (calculs)i(et)e(des)g(donn)o(\023)-21 b(ees,)21 b fj(etc)p fq(.)f(les)i(tec)o(hniques)g(de)f(r)o(\023)-21 b(e)o(\023)g(ecriture)20 b(de)75 1308 y(nids)d(de)g(b)q(oucles,)h(en)f(vue)g(de)g(leur)g(parall) o(\023)-21 b(elisation,)17 b(rep)q(osen)o(t)g(sur)f(des)h (manipulations)h(de)f(nature)75 1364 y(g)o(\023)-21 b(eom)o(\023)g (etrique)9 b(du)j(domaine)f(d'it)o(\023)-21 b(eration)11 b(ou)g(des)g(domaines)h(d'indices)h(des)e(tableaux,)g(sous)g(des)g(h)o (yp)q(o-)75 1421 y(th)o(\022)-21 b(eses)11 b(fortes,)g(comme)h(la)h (lin)o(\023)-21 b(earit)o(\023)g(e)12 b(des)g(fonctions)h(de)f(d)o (\023)-21 b(ep)q(endance)14 b(et)e(des)g(b)q(ornes)h(d'it)o(\023)-21 b(eration,)11 b(par)75 1477 y(exemple.)h(des)f(consid)o(\023)-21 b(erations)12 b(du)f(m)o(^)-21 b(eme)11 b(t)o(yp)q(e)g(se)h(retrouv)o (en)o(t)e(en)i(programmation)e(data-parall)o(\022)-21 b(ele,)75 1534 y(dans)14 b(les)f(\023)-21 b(etudes)14 b(sur)f(l'alignemen)o(t)i(et)f(la)g(distribution)h(des)f(donn)o(\023) -21 b(ees,)13 b(ou)h(des)g(comm)o(unications)h(in-)75 1590 y(duites.)j(ces)g(remarques)f(mon)o(tren)o(t)f(l'imp)q(ortance)j (d'une)f(certaine)g(forme)f(de)h(g)o(\023)-21 b(eom)o(\023)g(etrie)16 b(dans)h(la)75 1646 y(s)o(\023)-21 b(eman)o(tique)18 b(des)i(programmes)e(parall)o(\022)-21 b(eles,)19 b(et)g(l'in)o(t)o (\023)-21 b(er)o(^)g(et)17 b(de)j(p)q(ouv)o(oir)f(manipuler)i(les)e (indices)i(des)75 1703 y(tableaux)14 b(de)h(mani)o(\022)-21 b(ere)13 b(abstraite,)h(au)g(m)o(^)-21 b(eme)13 b(titre)h(que)g(les)h (d)o(\023)-21 b(e\014nitions)15 b(des)f(calculs,)h(en)g(expriman)o(t)75

1759 y(le)20 b(placemen)o(t)g(des)g(v)m(aleurs)h(dans)e(un)h fj(r)o(\023)-22 b(ef)o(\023)g(er)n(entiel)18 b fq(qui)j(p)q(ermette)e (de)h(distinguer)h(c)o(hacune)f(d'elles)75 1816 y(par)h(des)h(co)q (ordonn)o(\023)-21 b(ees)21 b(discr)o(\022)-21 b(etes.)21 b(la)h(transformation)e(du)i(placemen)o(t)g(des)g(v)m(aleurs)h(p)q(eut) f(alors)75 1872 y(p)q(ermettre)15 b(de)h(d)o(\023)-21 b(e\014nir)15 b(des)h(calculs)h(e\016caces)e(sur)g(une)h(arc)o (hitecture)g(parall)o(\022)-21 b(ele)15 b(donn)o(\023)-21 b(ee.)146 1929 y(c'est)11 b(ainsi)i(que)f(l'activit)o(\023)-21 b(e)12 b(de)h(programmation)e(p)q(eut)h(fondamen)o(talemen)o(t)g(rep)q (oser)h(sur)f(un)g(tr)o(\022)-21 b(es)75 1985 y(p)q(etit)14 b(nom)o(bre)f(de)h(primitiv)o(es)h(n)o(\023)-21 b(ecessaires)13 b(\022)-23 b(a)14 b(l')o(\023)-21 b(ecriture)13 b(des)h(programmes.)e (nous)h(en)h(mettons)f(trois)75 2042 y(en)i(\023)-21 b(evidence)8 b(:)143 2120 y({)23 b(l'application)17 b(de)e(fonctions)h (sur)f(des)g(v)m(aleurs,)h(p)q(our)g(d)o(\023)-21 b(e\014nir)15 b(les)h(calculs,)143 2207 y({)23 b(le)15 b(d)o(\023)-21 b(eplacemen)o(t)15 b(de)h(v)m(aleurs)f(dans)g(le)h(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel,)14 b(p)q(our)h(exprimer)g(les)h(d)o(\023)-21 b(ep)q(endances,)15 b(et)g(par)189 2264 y(cons)o(\023)-21 b(equen)o(t)14 b(les)h(\023)-21 b(ev)o(en)o(tuelles)16 b(comm)o(unications,)143 2351 y({)23 b(le)11 b(c)o(hangemen)o(t)g(de)g (r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel,)10 b(p)q(our)h(transformer)f (la)h(structure)g(des)g(donn)o(\023)-21 b(ees)10 b(et)h(des)g(calculs.) 75 2429 y(ces)17 b(id)o(\023)-21 b(ees)18 b(tr)o(\022)-21 b(es)16 b(simples)j(son)o(t)d(le)i(fondemen)o(t)g(du)g(formalisme)f fn(pei)g fq(que)h(nous)g(allons)g(main)o(tenan)o(t)75 2486 y(rapp)q(eler.)g(la)e(d)o(\023)-21 b(e\014nition)18 b(originale)g(de)f fn(pei)g fq(se)g(trouv)o(e)f(dans)h(la)g(th)o(\022) -21 b(ese)16 b(d'eric)h(violard)h([56)o(])e(et)h(les)75 2542 y(articles)f([58)o(],[59)n(])f(a\016nen)o(t)g(les)h(id)o(\023)-21 b(ees)15 b(pr)o(\023)-21 b(esen)o(t)o(\023)g(ees)14 b(dans)h(sa)g(th)o (\022)-21 b(ese.)p eop %%page: 10 12 10 11 bop 75 269 a fq(10)587 b fl(chapitre)16 b(2.)45 b(d)1091 257 y(\023)1085 269 y(efinition)16 b(du)f(f)o(ormalisme)h fn(pei)75 397 y ft(2.1)66 b(in)n(tro)r(duction)25 b(\022)-33 b(a)21 b(la)i(programmation)g fs(pei)75 500 y fw(2.1.1)55 b(les)18 b(c)n(hamps)h(de)f(donn)o(\023)-27 b(ees)146 586 y fq(de)18 b(mani)o(\022)-21 b(ere)19 b(g)o(\023)-21 b(en)o(\023)g(erale,)17 b(consid)o(\023)-21 b(erons)19 b(qu'un)g(probl)o(\022)-21 b(eme)19 b(est)g(sp)o(\023)-21 b(eci\014)o(\023)g(e)18 b(par)h(une)g(relation)h(en)o(tre)75 643 y(des)14 b fj(multi-ensembles)e fq(de)i(v)m(aleurs,)g(app)q(el)o (\023)-21 b(es)14 b(ses)g fj(entr)o(\023)-22 b(e)n(es)12 b fq(et)i(ses)f fj(sorties)p fq(.)27 b(en)13 b fn(pei)p fq(,)g(les)i(v)m(aleurs)f(d'un)75 699 y(m)o(ulti-ensem)o(ble)j(son)o(t) e(plac)o(\023)-21 b(ees)15 b(dans)g(un)h(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel)14 b(p)q(our)i(former)e(un)i fj(champ)h(de)f(donn)o(\023)-22 b(e)n(es)1710 683 y fg(1)1727 699 y fq(.)146 756 y(evidemmen)o(t,)15 b(p)q(our)h(un)g(m)o(ulti-ensem) o(ble)i(donn)o(\023)-21 b(e,)15 b(il)i(y)e(a)h(une)g(in\014nit)o(\023) -21 b(e)16 b(de)g(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiels)15 b(en)o(visa-)75 812 y(geables,)c(et)g(p)q(our)g(un)h(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel)10 b(c)o(hoisi,)i(plusieurs)h(placemen)o(ts)e (p)q(ossibles.)i(un)e(placemen)o(t)h(p)q(ossible)75 869 y(p)q(our)18 b(le)g(m)o(ulti-ensem)o(ble)i(de)e(v)m(aleurs)h fm(v)26 b fq(=)18 b ff(f)p fq(1)p fm(;)8 b fq(2)p fm(;)g ff(\000)p fq(3)p fm(;)f fq(1)p ff(g)p fq(,)14 b(est)k(celui)h(qui)g (consiste)f(\022)-23 b(a)17 b(asso)q(cier)h(des)75 925

y(indices)h(de)e fe(z)-13 b fq(,)13 b ff(f)p fq(0)p fm(;)8 b fq(1)p fm(;)g fq(2)p fm(;)g fq(3)o ff(g)14 b fq(par)i(exemple,)i (\022)-23 b(a)16 b(c)o(haque)g(\023)-21 b(el)o(\023)g(emen)o(t)15 b(de)i fm(v)10 b fq(,)16 b(comme)g(le)i(mon)o(tre)e(la)g(\014gure)75 982 y(2.1\(a\).)9 b(d)o(\023)-21 b(e\014nissons)11 b(ce)g(placemen)o(t) h(comme)f(forman)o(t)f(le)i(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)11 b fk(v)p fq(.)f(ce)h(m)o(ulti-ensem)o(ble)75 1038 y(p)q(eut)g(\023)-21 b(egalemen)o(t)10 b(^)-21 b(etre)11 b(plac)o(\023)-21 b(e)12 b(dans)g(un)g(domaine)g(d'indices)i(\()p fm(i;)8 b(j)s fq(\))h(de)j fe(z)1309 1022 y fg(2)1324 1038 y fq(,)g(tels)g(que)g(\(0)g ff(\024)h fm(i;)8 b(j)14 b ff(\024)f fq(1\))e(par)75 1095 y(exemple)18 b(\(\014gure)e (2.1\(b\)\).)f(on)i(forme)f(ainsi)i(un)f(autre)g(c)o(hamp)f(de)h(donn)o (\023)-21 b(ees)17 b fk(m)p fq(.)f(ces)g(deux)i(c)o(hamps)75 1151 y(exprimen)o(t)k(di\013)o(\023)-21 b(eremmen)o(t)20 b(le)i(m)o(^)-21 b(eme)20 b(m)o(ulti-ensem)o(ble)j(de)e(v)m(aleurs,)h (et)f(nous)g(disons)g(qu'ils)h(son)o(t)74 1208 y fj(\023)-22 b(e)n(quivalents)17 b fq(\(p)q(our)i(plus)h(de)f(pr)o(\023)-21 b(ecisions)19 b(sur)g(cette)f(notion)h(d')o(\023)-21 b(equiv)m(alence,)20 b(on)f(se)g(rep)q(ortera)f(\022)-23 b(a)18 b(la)75 1264 y(section)e(2.3.4\).)d(f)l(ormellemen)o(t,)i(il)i (existe)e(une)h(bijection)h(p)q(ermettan)o(t)e(de)g(passer)g(du)h (premier)g(pla-)488 1375 y 14917438 8055411 526254 39995310 18418892 49796792 starttexfig 488 1375 a %%begindocument: figures/chbase.eps /tgifdict 3 dict def tgifdict begin /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def end tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 0 setgray

72 0 mul 72 11.00 mul translate 72 128 div 100 mul 100 div dup neg scale gsave % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -38.438 -73.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg}

{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -59.906 -100.125 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath

/showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi)

@start /fa 1 51 df<ea03f0ea0c1cea100700201380384003c0a2008013e012f0eaf8 01a3ea2003120014c0a2eb07801400130e5b13185b5b5b485a90c7fc000213205a5a0018 1360481340383fffc05ab5fc13217ea018>50 d e end texdict begin 1 0 bop 73 203 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -65.438 -82.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{

rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % polygon/closed-spline 0 setgray newpath 80 80 moveto 64 128 lineto 80 144 lineto 128 160 lineto 144 144 lineto 176 128 lineto 176 96 lineto 176 64 lineto

144 64 lineto 128 80 lineto 96 64 lineto closepath gsave grestore gsave stroke grestore % box 0 setgray gsave 1.415 setmiterlimit newpath 64 224 moveto 192 224 lineto 192 272 lineto 64 272 lineto closepath stroke 1 setmiterlimit grestore % box 0 setgray gsave 1.415 setmiterlimit newpath 368 192 moveto 464 192 lineto 464 288 lineto 368 288 lineto closepath stroke 1 setmiterlimit grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -74.438 -163.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0

0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat

countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -26.109 -91.125 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2

chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 52 df<ea03f8ea0c1eea100f38200780004013c0127813031307123800 001380a214005b130c1338ea03f0ea001c130feb0780a2eb03c0a214e01220127012f8a2 00f013c01240eb0780122038100f00ea0c1cea03f013227ea018>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 73 203 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -50.906 -163.125 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath

/showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi)

@start /fa 1 51 df<ea03f0ea0c1cea100700201380384003c0a2008013e012f0eaf8 01a3ea2003120014c0a2eb07801400130e5b13185b5b5b485a90c7fc000213205a5a0018 1360481340383fffc05ab5fc13217ea018>50 d e end texdict begin 1 0 bop 73 203 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -26.109 -163.125 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{

rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 52 df<ea03f8ea0c1eea100f38200780004013c0127813031307123800 001380a214005b130c1338ea03f0ea001c130feb0780a2eb03c0a214e01220127012f8a2 00f013c01240eb0780122038100f00ea0c1cea03f013227ea018>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 73 203 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def

0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -20.438 -163.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false}

ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % poly/open-spline 0 setgray gsave newpath 96 224 moveto 96 272 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 128 224 moveto 128 272 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 160 224 moveto 160 272 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 96 112 moveto 64 -48 atan dup cos 8 mul exch sin 8 mul rmoveto 64.00 154.67 58.67 192.00 80 224 curveto

stroke grestore gsave newpath 96 112 8 3 48 -64 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 96 144 moveto 48 0 atan dup cos 8 mul exch sin 8 mul rmoveto 96.00 176.00 101.33 202.67 112 224 curveto stroke grestore gsave newpath 96 144 8 3 0 -48 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 144 128 moveto 96 0 atan dup cos 8 mul exch sin 8 mul rmoveto 144 224 lineto stroke grestore gsave newpath 144 128 8 3 0 -96 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 160 96 moveto 64 32 atan dup cos 8 mul exch sin 8 mul rmoveto 181.33 138.67 186.67 181.33 176 224 curveto stroke grestore gsave newpath 160 96 8 3 -32 -64 tgifarrowtip closepath fill grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def

userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 123.562 -73.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for

65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 129.094 -145.125 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn

begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 51 df<ea03f0ea0c1cea100700201380384003c0a2008013e012f0eaf8 01a3ea2003120014c0a2eb07801400130e5b13185b5b5b485a90c7fc000213205a5a0018 1360481340383fffc05ab5fc13217ea018>50 d e end texdict begin 1 0 bop 73 203 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat

countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 96.562 -82.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2

chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % polygon/closed-spline 0 setgray newpath 368 80 moveto 352 128 lineto 368 144 lineto 416 160 lineto 432 144 lineto 464 128 lineto 464 96 lineto 464 64 lineto 432 64 lineto 416 80 lineto 384 64 lineto closepath gsave grestore gsave stroke grestore % eps

end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 135.891 -91.125 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul

add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 52 df<ea03f8ea0c1eea100f38200780004013c0127813031307123800 001380a214005b130c1338ea03f0ea001c130feb0780a2eb03c0a214e01220127012f8a2 00f013c01240eb0780122038100f00ea0c1cea03f013227ea018>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 73 203 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % poly/open-spline 0 setgray gsave newpath 384 112 moveto 48 -32 atan dup cos 8 mul exch sin 8 mul rmoveto 362.67 144.00 362.67 170.67 384 192 curveto stroke grestore gsave newpath 384 112 8 3 32 -48 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 400 144 moveto 48 48 atan dup cos 8 mul exch sin 8 mul rmoveto 448 192 lineto stroke grestore

gsave newpath 400 144 8 3 -48 -48 tgifarrowtip closepath fill grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 105.562 -145.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg}

{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 102.094 -100.125 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath

/showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi)

@start /fa 1 51 df<ea03f0ea0c1cea100700201380384003c0a2008013e012f0eaf8 01a3ea2003120014c0a2eb07801400130e5b13185b5b5b485a90c7fc000213205a5a0018 1360481340383fffc05ab5fc13217ea018>50 d e end texdict begin 1 0 bop 73 203 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 117.891 -172.125 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{

rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 52 df<ea03f8ea0c1eea100f38200780004013c0127813031307123800 001380a214005b130c1338ea03f0ea001c130feb0780a2eb03c0a214e01220127012f8a2 00f013c01240eb0780122038100f00ea0c1cea03f013227ea018>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 73 203 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def

0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 123.562 -172.125 translate 1.312 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false}

ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 1 50 df<13801203120f12f31203b3a9ea07c0eafffe0f217ca018>49 d e end texdict begin 1 0 bop 73 203 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % poly/open-spline 0 setgray gsave newpath 368 240 moveto 464 240 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 416 192 moveto 416 288 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 448 96 moveto 80 48 atan dup cos 8 mul exch sin 8 mul rmoveto 480.00 149.33 496.00 184.00 496.00 200.00 curveto 496.00 216.00 485.33 234.67 464 256 curveto stroke grestore gsave newpath 448 96 8 3 -48 -80 tgifarrowtip closepath fill grestore

% poly/open-spline 0 setgray gsave newpath 416 128 moveto 64 -80 atan dup cos 8 mul exch sin 8 mul rmoveto 362.67 170.67 346.67 213.33 368 256 curveto stroke grestore gsave newpath 416 128 8 3 80 -64 tgifarrowtip closepath fill grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -95.906 -130.142 translate 1.266 1.074 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id

ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 2 127 df<13e0a21201ea00c01300a9121e1223ea4380a21283ea8700a212 07120ea35aa3ea38201340127013801230ea3100121e0b227ea111>105 d<14c0a31460b512f0a2c712e0eb0180eb03001302140a77a318>126 d e end texdict begin 1 0 bop 65 192 a fa(~)73 203 y(i)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def

/legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 50.625 -129.174 translate 1.237 1.085 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7

getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 2 127 df<147014f0a214601400a9130feb3180eb41c01381a2ea0101a238 000380a4eb0700a4130ea45ba45ba3ea7070eaf0605bea6380003ec7fc142c81a114> 106 d<14c0a31460b512f0a2c712e0eb0180eb03001302140a77a318>126 d e end texdict begin 1 0 bop 69 192 a fa(~)73 203 y(j)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % poly/open-spline 0 setgray gsave newpath 16 224 moveto 0 32 atan dup cos 8 mul 48 exch sub exch sin 8 mul 224 exch sub lineto stroke grestore gsave newpath 48 224 8 3 32 0 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 304 224 moveto 32 0 atan dup cos 8 mul 304 exch sub exch sin 8 mul 256 exch sub lineto stroke grestore gsave newpath 304 256 8 3 0 32 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray

gsave newpath 304 224 moveto 0 32 atan dup cos 8 mul 336 exch sub exch sin 8 mul 224 exch sub lineto stroke grestore gsave newpath 336 224 8 3 32 0 tgifarrowtip closepath fill grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 66.094 -103.142 translate 1.266 1.074 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp

x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14143.dvi) @start /fa 2 127 df<13e0a21201ea00c01300a9121e1223ea4380a21283ea8700a212 07120ea35aa3ea38201340127013801230ea3100121e0b227ea111>105 d<14c0a31460b512f0a2c712e0eb0180eb03001302140a77a318>126 d e end texdict begin 1 0 bop 65 192 a fa(~)73 203 y(i)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath

1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -33.648 -200.250 translate 1.074 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley

scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14890.dvi) @start /fa 1 98 df<ebf180380389c038070780ea0e03121c123c383807001278a3ea f00ea31410eb1c20a2ea703ceb5c40ea308c380f078014147e9318>97 d e /fb 2 42 df<13401380ea01005a12061204120c5aa212381230a212701260a412e0 ac1260a412701230a212381218a27e120412067e7eea008013400a2e7ba112>40 d<7e12407e12307e1208120c7ea212077ea213801201a413c0ac1380a412031300a25a12 06a25a120812185a12205a5a0a2e7ea112>i e end texdict begin 1 0 bop 71 199 a fb(\()p fa(a)p fb(\))p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 123.750 -200.250 translate 1.125 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo

setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa14890.dvi) @start /fa 1 99 df<ea0780123fea0700a4120ea45aa213f0ea1d0cea3a0e123cea38 0fa21270a4eae01ea3131c133c1338ea607013e0ea31c0ea1f0010207e9f14>98 d e /fb 2 42 df<13401380ea01005a12061204120c5aa212381230a212701260a412e0 ac1260a412701230a212381218a27e120412067e7eea008013400a2e7ba112>40 d<7e12407e12307e1208120c7ea212077ea213801201a413c0ac1380a412031300a25a12 06a25a120812185a12205a5a0a2e7ea112>i e end texdict begin 1 0 bop 71 199 a fb(\()p fa(b)p fb(\))p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore

count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin grestore tgifsavedpage restore end %matchingcreationdate: fri dec 13 16:41:36 1996 %%enddocument endtexfig 268 2029 a fn(fig.)f fq(2.1)g(-)30 b fj(deux)17 b(plac)n(ements)e (di\013)o(\023)-22 b(er)n(ents)15 b(d'un)i(multi-ensemble)e(de)h (valeurs)75 2133 y fq(cemen)o(t)h(au)h(second.)f(en)h(l'o)q(ccurrence,) g(cette)g(bijection)g(est)g fm(\033)r fq(\()p fm(i)p fq(\))c(=)j(\()p fm(i)g(mod)g fq(2)p fm(;)8 b(i)16 b(div)i fq(2\).)e(en)i fn(pei)75 2190 y fq(nous)d(exprimons)h(cette)f(relation) h(en)o(tre)f(les)h(deux)g(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees)15 b(par)f(l')o(\023)-21 b(equation)15 b(suiv)m(an)o(te)9 b(:)620 2345 y fk(m)15 b fq(=)g fk(align)7 b(::)g(v)485 2401 y fq(a)o(v)o(ec)50 b fk(align)23 b(=)15 b fm(\025)p fk(\(i\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p ff(\024)p fk(3\).\(i)22 b(mod)h(2,i/2\))p 75 2500 709 2 v 124 2542 a fd(1)p fc(:)9 b fd(a)k(la)h(fa\030)-17 b(con)13 b(des)g fb(table)n(aux)d fd(dans)k(les)g(langages)h(imp)o(\023)-18 b(eratifs)p eop %%page: 11 13 11 12 bop 75 269 a fl(2.1.)29 b(intr)o(oduction)570 257 y(\022)562 269 y(a)16 b(la)f(pr)o(ogramma)l(tion)h fn(pei)577 b fq(11)146 397 y(l'application)25 b(d'une)g(fonction)g(comme)f fk(align)f fq(sur)h(un)h(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)24 b(est)g(app)q(el)o(\023)-21 b(ee)75 453 y(une)22 b fj(op)o(\023)-22 b(er)n(ation)p fq(,)21 b(et)h(rend)f(comme)h(r)o (\023)-21 b(esultat)20 b(un)i(autre)f(c)o(hamp)g(de)h(donn)o(\023)-21 b(ees.)21 b(l')o(\023)-21 b(equation)21 b(pr)o(\023)-21 b(ec)o(\023)g(e-)75 510 y(den)o(te)16 b(exprime)g(l')o fj(\023)-22 b(e)n(galit)o(\023)g(e)15 b fq(en)o(tre)h(les)g(c)o(hamps)f (de)h(part)f(et)h(d'autre)f(de)h(l')o(\023)-21 b(equation,)14 b(c'est-\022)-23 b(a-dire)16 b fk(m)g fq(et)75 566 y fk(\(align)7 b(::)g(v\))p fq(.)14 b(ceci)i(signi\014e)g(qu'ils)g(on)o (t)e(les)i(m)o(^)-21 b(emes)14 b(v)m(aleurs,)h(et)g(que)g(ces)g(v)m (aleurs)h(son)o(t)e(plac)o(\023)-21 b(ees)15 b(de)75 623 y(la)h(m)o(^)-21 b(eme)14 b(mani)o(\022)-21 b(ere)15 b(sur)g(le)h(m)o(^)-21 b(eme)14 b(r)o(\023)-21 b(ef)o(\023)g(eren)o (tiel.)75 807 y fw(2.1.2)55 b(les)18 b(op)o(\023)-27 b(erations)18 b(et)g(l'op)o(\023)-27 b(erateur)18 b(de)g(sup)r(erp)r (osition)75 895 y fo(les)f(op)o(\023)-25 b(erations)146 983 y fq(nous)15 b(distinguons)h(trois)f(sortes)f(d'op)o(\023)-21 b(erations)14 b(en)i fn(pei)p fq(.)143 1081 y({)23 b(celle)16 b(que)f(nous)g(v)o(enons)f(de)h(v)o(oir)g(mo)q(di\014e)h(le)f(r)o(\023) -21 b(ef)o(\023)g(eren)o(tiel)14 b(dans)h(lequel)h(son)o(t)e(repr)o (\023)-21 b(esen)o(t)o(\023)g(ees)13 b(les)189 1138 y(v)m(aleurs)20 b(et)e(nous)i(l'app)q(elons)g fj(changement)f(de)h(b)n(ase)p fq(.)e(l'application)j(d'un)e(c)o(hangemen)o(t)g(de)189 1194 y(base)c(est)g(d)o(\023)-21 b(enot)o(\023)g(ee)14 b(par)g(le)i(sym)o(b)q(ole)9 b fk(::)143 1293 y fq({)23 b(une)18 b(autre)f(op)o(\023)-21 b(eration)16 b(mo)q(di\014e)j(le)f (placemen)o(t)h(des)f(v)m(aleurs)g(dans)g(ce)g(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel.)16 b(elle)j(p)q(er-)189 1349

y(met,)13 b(par)h(exemple,)i(d'exprimer)e(les)h(comm)o(unications)g (dans)g(une)g(mac)o(hine)g(parall)o(\022)-21 b(ele.)15 b(nous)189 1406 y(l'app)q(elons)j(op)o(\023)-21 b(eration)17 b(de)g fj(r)n(outage)p fq(,)h(et)f(son)g(application)i(est)e(d)o(\023) -21 b(enot)o(\023)g(ee)16 b(par)h(le)h(sym)o(b)q(ole)g fa(c)1815 1389 y fg(2)1834 1406 y fq(.)189 1462 y(la)c(\014gure)g(2.2)f (mon)o(tre)g(le)i(placemen)o(t)f(des)g(v)m(aleurs)h(du)g(c)o(hamp)f(de) g(donn)o(\023)-21 b(ees)14 b fk(w)f fq(d)o(\023)-21 b(e\014ni)15 b(par)f fk(w)f fq(=)189 1518 y fk(v)i fa(c)g fk(shift)o fq(,)g(a)o(v)o(ec)g fk(shift)p fq(=)p fm(\025)p fk(\(i\).\(\(i-1\))22 b(mod)h(4\))p fq(.)p 726 1704 2 75 v 801 1704 v 876 1704 v 650 1628 304 2 v 650 1703 2 75 v 951 1703 v 650 1705 304 2 v 764 1685 a(2)-98 b(1)108 b(-3)764 1798 y fk(v)914 1685 y fq(1)p 1251 1704 2 75 v 1326 1704 v 1401 1704 v 1175 1628 304 2 v 1175 1703 2 75 v 1476 1703 v 1175 1705 304 2 v 483 1630 94 2 v 535 1629 a fz(-)1214 1685 y fq(1)52 b(1)g(2)33 b(-3)1289 1798 y fk(w)513 1675 y fm(~)520 1685 y(i)350 1946 y fn(fig.)15 b fq(2.2)f(-)31 b fj(l)n(e)15 b(champ)i fk(w)f fj(d)o(\023)-22 b(e\014ni)16 b(p)n(ar)g(un)g(r)n(outage)h(des)f(valeurs)g(de)h fk(v)189 2075 y fq(noter)i(que)h(la)f(fonction)h(in)o(terv)o(enan)o(t)f(dans)h (une)g(op)o(\023)-21 b(eration)19 b(de)h(routage)e(n'est)h(pas)h(n)o (\023)-21 b(eces-)189 2131 y(sairemen)o(t)19 b(injectiv)o(e.)i(ceci)f (p)q(ermet)f(d'exprimer)i(des)e(op)o(\023)-21 b(erations)19 b(de)h(t)o(yp)q(e)f(di\013usion)9 b(:)19 b(par)189 2188 y(exemple,)i(l')o(\023)-21 b(equation)20 b fk(w)g fq(=)g fk(v)15 b fa(c)h fk(spread)o fq(,)k(a)o(v)o(ec)g fk(spread)p fq(=)p fm(\025)p fk(\(i\))p ff(j)p fk(\(0<=i<4\).\()o(0\))d fq(di\013use)k(la)189 2244 y(v)m(aleur)c(plac)o(\023)-21 b(ee)16 b(au)f(p)q(oin)o(t)i(d'indice)h(0)d(de)h fk(v)g fq(p)q(our)g(former)f(le)i(c)o(hamp)f fk(w)p fq(.)f(ceci)i(est)e (illustr)o(\023)-21 b(e)17 b(par)189 2301 y(la)e(\014gure)h(2.3.)143 2399 y({)23 b(la)17 b(troisi)o(\022)-21 b(eme)17 b(op)o(\023)-21 b(eration)16 b(est)h(celle)h(qui)g(c)o(hange)g(les)f(v)m(aleurs,)h(app) q(el)o(\023)-21 b(ee)18 b(op)o(\023)-21 b(eration)16 b fj(fonction-)189 2456 y(nel)r(le)p fq(.)21 b(elle)j(p)q(ermet)e (d'appliquer)i(un)f(m)o(^)-21 b(eme)22 b(calcul)i(aux)e(v)m(aleurs)i (d'un)f(c)o(hamp)f(de)h(don-)p 75 2500 709 2 v 124 2542 a fd(2)p fc(:)9 b fd(l'op)o(\023)-18 b(eration)15 b(est)e(su\016x)o (\023)-18 b(ee)13 b(car)g(la)h(comp)q(osition)i(s'applique)f(\022)-19 b(a)13 b(droite)h(\()p fb(cf.)e fd(section)i(2.3.1\))p eop %%page: 12 14 12 13 bop 75 269 a fq(12)587 b fl(chapitre)16 b(2.)45 b(d)1091 257 y(\023)1085 269 y(efinition)16 b(du)f(f)o(ormalisme)h fn(pei)1195 592 y fz(?)1194 458 y fy($)p 1194 592 2 52 v 933 458 180 2 v -691 w(')p 669 592 2 52 v 753 458 180 2 v 1270 592 a fz(?)1269 432 y fy($)p 1269 592 2 79 v 970 432 217 2 v -766 w(')p 669 592 2 79 v 753 432 217 2 v 1345 592 a fz(?)1344 421 y fy($)p 1344 592 2 90 v 1008 421 255 2 v -841 w(')p 669 592 2 90 v 753 421 255 2 v 1420 592 a fz(?)1419 406 y fy($)p 1419 592 2 105 v 1045 406 292 2 v -916 w(')p 669 592 2 105 v 753 406 292 2 v 1232 667 2 75 v 1307 667 v 1382 667 v 1156 591 304 2 v 1156 666 2 75 v 1458 666 v 1156 668 304 2 v 464 593 94 2 v 516 592 a fz(-)p 707 667 2 75 v 782 667 v 857 667 v 631 591 304 2 v 631 666 2 75 v 933 666 v 631 668 304 2 v 1270 649 a fq(1)-98 b(1)-323 b(1)-117 b(-3)-94 b(2)c(1)745 761 y fk(v)501 b(w)1345 649 y fq(1)52 b(1)495 638 y fm(~)502 649 y(i)238 909 y fn(fig.)15 b fq(2.3)f(-)31

b fj(l)n(e)15 b(champ)i fk(w)f fj(d)o(\023)-22 b(e\014ni)16 b(p)n(ar)g(un)g(r)n(outage)h(\(di\013usion\))e(des)h(valeurs)g(de)h fk(v)189 1065 y fq(n)o(\023)-21 b(ees.)15 b(l'application)k(de)e(l'op)o (\023)-21 b(eration)16 b(fonctionnelle)j(est)d(d)o(\023)-21 b(enot)o(\023)g(ee)15 b(par)i(le)g(sym)o(b)q(ole)h fa(b)1729 1048 y fg(3)1747 1065 y fq(.)e(p)o(ar)189 1121 y(exemple,)f fk(w)f fq(=)g fk(inc)h fa(b)g fk(v)p fq(,)f(a)o(v)o(ec)f fk(inc)p fq(=)p fm(\025)p fk(\(a\).\(a+3\))o fq(,)g(est)h(un)h(c)o (hamp)f(don)o(t)f(c)o(haque)i(v)m(aleur)g(est)189 1177 y(calcul)o(\023)-21 b(ee)16 b(\022)-23 b(a)15 b(partir)g(de)g(la)h(v)m (aleur)g(de)g fk(v)f fq(plac)o(\023)-21 b(ee)15 b(au)g(m)o(^)-21 b(eme)15 b(p)q(oin)o(t)g(\(\014gure)g(2.4\).)p 1147 1379 2 75 v 1222 1379 v 1297 1379 v 1072 1303 304 2 v 1072 1378 2 75 v 1373 1378 v 1072 1380 304 2 v 623 1379 2 75 v 698 1379 v 773 1379 v 547 1303 304 2 v 547 1378 2 75 v 848 1378 v 547 1380 304 2 v 1186 1473 a fk(w)661 1360 y fq(2)-98 b(1)108 b(-3)56 b(1)661 1473 y fk(v)1186 1360 y fq(5)c(0)g(4)-248 b(4)292 1621 y fn(fig.)15 b fq(2.4)f(-)31 b fj(l)n(e)15 b(champ)i fk(w)f fj(d)o(\023)-22 b(e\014ni)16 b(p)n(ar)g(une)g(op)o(\023)-22 b(er)n(ation)17 b(fonctionnel)r(le)d(sur)j fk(v)146 1809 y fq(on)c(d)o(\023)-21 b(e\014nit)13 b(\023)-21 b(egalemen)o(t)12 b(l'in)o(v)o(erse)i(de)f (l'op)o(\023)-21 b(eration)12 b(de)i(routage)e(et)h(de)g(l'op)o(\023) -21 b(eration)13 b(fonctionnelle,)75 1865 y(quelle)h(que)e(soit)g(la)g (fonction,)g(qui)g(son)o(t)g(resp)q(ectiv)o(emen)o(t)g(la)g fj(r)o(\023)-22 b(e)n(duction)14 b(g)o(\023)-22 b(e)n(om)o(\023)g (etrique)12 b fq(et)f fj(la)j(r)o(\023)-22 b(e)n(duction)75 1922 y(fonctionnel)r(le)10 b fq([59)o(].)h(nous)h(n'utiliserons)h(que)f (la)g(r)o(\023)-21 b(eduction)12 b(g)o(\023)-21 b(eom)o(\023)g(etrique) 10 b(dans)h(cette)h(pr)o(\023)-21 b(esen)o(tation.)75 1978 y(la)23 b(r)o(\023)-21 b(eduction)22 b(g)o(\023)-21 b(eom)o(\023)g(etrique)21 b(d'un)i(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)22 b fk(x)h fq(par)f(une)h(fonction)g fk(g)f fq(\(d)o(\023)-21 b(enot)o(\023)g(ee)21 b(par)75 2034 y fk(g)15 b fq(;)8 b fa(b)15 b fk(x)p fq(\),)h(place)j(en)e(tout)g(p)q(oin)o(t)h(de)g(co)q (ordonn)o(\023)-21 b(ees)16 b fm(z)r fq(,)i(une)g(s)o(\023)-21 b(equence)17 b(des)h(v)m(aleurs)g(de)g fk(x)f fq(situ)o(\023)-21 b(ees)17 b(en)75 2091 y(tous)f(les)g(p)q(oin)o(ts)h fm(y)r fq(,)e fm(y)h ff(2)f fm(dom)p fq(\()p fk(g)p fq(\),)f(tels)i(que)h fk(g)p fq(\()p fm(y)r fq(\))c(=)h fm(z)r fq(.)i(p)o(ar)f(exemple,)i(le) g(c)o(hamp)f fk(y)g fq(d)o(\023)-21 b(e\014ni)17 b(par)e fk(y)h fq(=)75 2147 y fk(merge)e fq(;)8 b fa(b)15 b fk(x)d fq(a)o(v)o(ec)g fk(merge)p fq(=)p fm(\025)p fk(\(i\))p ff(j)p fk(\(0<=i<3\).\(0\))d fq(a)j(toutes)g(ses)h(v)m(aleurs)g(plac)o (\023)-21 b(ees)12 b(au)h(p)q(oin)o(t)g(d'indice)75 2204 y(0,)i(dans)h(une)g(s)o(\023)-21 b(equence)16 b(d'ordre)f(quelconque.)i (la)f(\014gure)f(2.5)g(mon)o(tre)g(une)h(d)o(\023)-21 b(e\014nition)17 b(p)q(ossible)g(du)75 2260 y(c)o(hamp)e fk(y)p fq(.)146 2325 y(nous)k(disons)h(que)h(cette)e(op)o(\023)-21 b(eration,)18 b(est)i(l'in)o(v)o(erse)g(de)g(l'op)o(\023)-21 b(eration)19 b(g)o(\023)-21 b(eom)o(\023)g(etrique)18 b(car)h(quand)75 2381 y(la)h(fonction)h(est)e(bijectiv)o(e,)i(la)g(r)o (\023)-21 b(eduction)20 b(est)e(\023)-21 b(egale)20 b(\022)-23 b(a)19 b(l'op)o(\023)-21 b(eration)20 b(g)o(\023)-21 b(eom)o(\023)g(etrique)18 b(d)o(\023)-21 b(e\014nie)21 b(par)e(la)75 2437 y(fonction)d(in)o(v)o(erse.)p 75 2500 709 2 v 124 2542 a fd(3)p fc(:)9 b fd(l'op)o(\023)-18 b(eration)15 b(est)e(pr)o(\023)-18 b(e\014x)o(\023)g(ee)14 b(car)f(la)g(comp)q(osition)j(s'applique)g(\022)-19 b(a)13 b(gauc)o(he)g(\()p fb(cf.)f fd(section)j(2.3.1\))p eop

%%page: 13 15 13 14 bop 75 269 a fl(2.1.)29 b(intr)o(oduction)570 257 y(\022)562 269 y(a)16 b(la)f(pr)o(ogramma)l(tion)h fn(pei)577 b fq(13)1214 619 y fz(?)1213 409 y fy($)p 1213 619 2 127 v 944 409 187 2 v -710 w(')p 669 619 2 127 v 753 409 191 2 v 1214 619 a fz(?)1213 409 y fy($)p 1213 619 2 127 v 981 409 150 2 v -635 w(')p 744 619 2 127 v 828 409 154 2 v 1213 610 2 7 v 1213 619 a(\001)1213 406 y($)p 1213 604 2 116 v 1019 406 112 2 v -560 w(')p 819 604 2 116 v 903 406 116 2 v 819 610 2 7 v 819 619 a(\002)p 1156 618 304 2 v 1156 693 2 75 v 1458 693 v 1156 695 304 2 v 464 619 94 2 v 516 618 a fz(-)p 707 694 2 75 v 782 694 v 631 618 229 2 v 631 693 2 75 v 858 693 v 631 695 229 2 v 1251 694 2 75 v 1344 694 v 495 664 a fm(~)502 675 y(i)152 b fq(4)52 b(7)g(5)745 787 y fk(x)501 b(y)1158 675 y fd(7;5;)o(4)194 935 y fn(fig.)15 b fq(2.5)f(-)31 b fj(l)n(e)15 b(champ)i fk(y)f fj(d)o(\023)-22 b(e\014ni)16 b(p)n(ar)g(une)g(r)o(\023)-22 b(e)n(duction)17 b(g)o(\023)-22 b(e)n(om)o(\023)g(etrique)16 b(des)g(valeurs)g(de)h fk(x)75 1076 y fo(op)o(\023)-25 b(erateur)17 b(de)h(sup)q(erp)q(osition)146 1165 y fq(en)g(plus)i(de)f(ces)g(trois)f(op)o(\023)-21 b(erations)17 b(et)i(de)g(leurs)g(in)o(v)o(erses,)g(on)f(d)o(\023)-21 b(e\014nit)19 b(en)g fn(pei)p fq(,)f(un)h(op)o(\023)-21 b(erateur)75 1221 y(in)o(terne)23 b(de)g(sup)q(erp)q(osition,)g(not)o (\023)-21 b(e)21 b fk(/&/)8 b fq(.)22 b(la)g(sup)q(erp)q(osition)i(de)f (plusieurs)h(c)o(hamps)e(de)h(donn)o(\023)-21 b(ees)75 1278 y(p)q(ermet)13 b(de)h(d)o(\023)-21 b(e\014nir)13 b(un)h(autre)e(c)o(hamp)h(de)h(donn)o(\023)-21 b(ees,)12 b(don)o(t)h(les)h(v)m(aleurs)g(son)o(t)e(des)i(s)o(\023)-21 b(equences.)12 b(chaque)75 1334 y(s)o(\023)-21 b(equence)15 b(est)g(la)h(concat)o(\023)-21 b(enation)14 b(des)h(v)m(aleurs)h(des)g (c)o(hamps)f(qui)h(son)o(t)e(plac)o(\023)-21 b(ees)15 b(aux)g(m)o(^)-21 b(emes)15 b(indices)75 1391 y(dans)g(le)h(r)o(\023) -21 b(ef)o(\023)g(eren)o(tiel.)13 b(ainsi,)j(la)g(\014gure)f(2.6)f(mon) o(tre)g(le)i(c)o(hamp)f(r)o(\023)-21 b(esultan)o(t)13 b(de)j(la)f(sup)q(erp)q(osition)h(de)g fk(x)75 1447 y fq(et)f fk(w)p fq(.)p 866 1625 v 941 1625 v 715 1549 304 2 v 715 1624 2 75 v 1017 1624 v 715 1626 304 2 v 791 1625 2 75 v 360 1625 v 435 1625 v 284 1549 229 2 v 284 1624 2 75 v 511 1624 v 284 1626 229 2 v 1316 1625 2 75 v 1429 1625 v 1541 1625 v 1184 1549 454 2 v 1184 1624 2 75 v 1636 1624 v 1184 1626 454 2 v 867 1700 a fk(w)538 b(z)323 1606 y fq(4)52 b(7)34 b(5)398 1719 y fk(x)755 1606 y fq(4)52 b(5)g(0)33 b(4)221 b(4;)8 b(4)64 b(7;)8 b(5)42 b(5;)8 b(0)67 b(4)513 1867 y fn(fig.)15 b fq(2.6)g(-)30 b fj(l)n(e)15 b(champ)j fk(z)e fj(d)o(\023)-22 b(e\014ni)15 b(p)n(ar)i fk(z)f fj(=)g fk(x/&/)7 b(w)75 2058 y fw(2.1.3)55 b(les)18 b(programmes)e fv(pei)146 2147 y fq(un)21 b fj(\023)-22 b(enonc)o(\023)g(e)22 b fn(pei)g fq(est)h(un)g(ensem)o(ble)g(d')o(\023)-21 b(equations)22 b(p)q(ermettan)o(t)g(de)h(d)o(\023)-21 b(e\014nir)23 b(des)g(c)o(hamps)f(de)75 2204 y(donn)o(\023)-21 b(ees)18 b(en)g(sortie)g(en)g(fonction)g(des)g(c)o(hamps)g(de)g(donn)o(\023)-21 b(ees)18 b(en)g(en)o(tr)o(\023)-21 b(ee.)17 b(lorsqu'ils)i(satisfon)o (t)e(aux)74 2260 y(\023)-21 b(equations,)12 b(l'ensem)o(ble)j(de)f(ces) g(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees,)13 b(et)g(les)h(c)o(hamps)g (in)o(term)o(\023)-21 b(ediaires)14 b(p)q(ouv)m(an)o(t)g(ap-)75 2316 y(para)-5 b(^)-18 b(\020tre)15 b(dans)g(les)g(\023)-21 b(equations,)14 b(constituen)o(t)h(une)g fj(solution)g fq(du)h(syst)o(\022)-21 b(eme)13 b(d')o(\023)-21 b(equations.)14

b(les)i(c)o(hamps)75 2373 y(de)k(donn)o(\023)-21 b(ees)20 b(en)g(en)o(tr)o(\023)-21 b(ee)19 b(et)h(en)h(sortie)e(son)o(t)h(pr)o (\023)-21 b(ecis)o(\023)g(es)19 b(dans)h(l'en)o(t)o(^)-21 b(ete)19 b(de)h(l')o(\023)-21 b(enonc)o(\023)g(e.)19 b(les)h(fonctions)75 2429 y(apparaissan)o(t)c(dans)i(les)f(op)o(\023) -21 b(erations,)16 b(d)o(\023)-21 b(e\014nissen)o(t)17 b(le)h(con)o(texte)f(de)g(l')o(\023)-21 b(enonc)o(\023)g(e,)16 b(sous)h(la)g(forme)g(de)g fm(\025)p fq(-)75 2486 y(expressions.)f (dans)e(la)i(th)o(\023)-21 b(eorie)15 b fn(pei)p fq(,)f(certains)h (\023)-21 b(enonc)o(\023)g(es)14 b(seulemen)o(t)i(son)o(t)e(des)i (programmes.)p eop %%page: 14 16 14 15 bop 75 269 a fq(14)587 b fl(chapitre)16 b(2.)45 b(d)1091 257 y(\023)1085 269 y(efinition)16 b(du)f(f)o(ormalisme)h fn(pei)75 397 y fo(d)o(\023)-25 b(e\014nition)19 b(1)31 b fq(\(programme)13 b fn(pei)p fq(\))75 453 y(un)18 b(\023)-21 b(enonc)o(\023)g(e)18 b fn(pei)h fq(est)g(un)h fj(pr)n(o)n(gr)n(amme)f fq(si,)g(p)q(our)g(tout)g(jeu)g(de)h(c)o(hamps)f(de)g(donn)o(\023)-21 b(ees)19 b(en)h(en)o(tr)o(\023)-21 b(ee,)17 b(il)75 510 y(existe)i(au)g(plus)h(un)g(ensem)o(ble)g(de)f(c)o(hamps)g(de)g(donn)o (\023)-21 b(ees)19 b(en)g(sortie)g(d)o(\023)-21 b(e\014nissan)o(t)19 b(une)g(solution)h(du)75 566 y(syst)o(\022)-21 b(eme)14 b(d')o(\023)-21 b(equations.)146 679 y(cette)14 b(d)o(\023)-21 b(e\014nition)17 b(a)e(p)q(our)g(cons)o(\023)-21 b(equence)16 b(de)g(ne)g(pas)f(consid)o(\023)-21 b(erer)15 b(comme)g(des)h (programmes,)e(les)74 736 y(\023)-21 b(enonc)o(\023)g(es)14 b(a)o(y)o(an)o(t)g(des)h(r)o(\023)-21 b(esultats)14 b(non-d)o(\023)-21 b(eterministes.)146 849 y(nous)15 b(p)q(ouv)o(ons)g(d'ores)f(et)h(d)o (\023)-21 b(ej\022)e(a)14 b(illustrer)j(\022)-23 b(a)15 b(l'aide)h(de)g(quelques)g(exemples)g(simples,)g(les)g(primi-)75 905 y(tiv)o(es)f(du)h(langage)8 b(:)75 1018 y fo(exemple)17 b(1)f fj(a)n(ddition)g(d'une)g(matric)n(e)g(et)h(de)f(sa)g(tr)n(ansp)n (os)o(\023)-22 b(e)n(e)75 1131 y fk(matsum)23 b(:)h(a)15 b ff(7!)g fk(c)75 1143 y fx(8)75 1181 y(>)75 1193 y(<)75 1268 y(>)75 1280 y(:)137 1185 y fk(a)g fq(=)g fk(matrix)7 b(::)g(a)137 1242 y(t)15 b fq(=)g fk(a)g fa(c)h fk(transp)137 1298 y(c)f fq(=)g fk(add)g fa(b)h fk(\(a)23 b(/&/)8 b(t\))96 1361 y(matrix)40 b fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p fm(<)p fk(n)7 b(&)h(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(i,j\))96 1418 y(transp)40 b fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p fm(<)p fk(n)7 b(&)h(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(j,i\))96 1474 y(add)112 b fq(=)42 b fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+b\))143 1590 y fq({)23 b(la)14 b(premi)o(\022)-21 b(ere)14 b(\023)-21 b(equation)14 b(est)g(une)h(pr)o(\023)-21 b(econdition)15 b(sur)f(le)i(c)o(hamp)e(de)h(donn)o(\023)-21 b(ees)14 b(en)h(en)o(tr)o(\023)-21 b(ee)13 b fk(a)p fq(.)h(elle)189 1646 y(exprime)19 b(que)f fk(a)h fq(n'est)e(pas)i(mo)q(di\014)o(\023) -21 b(e)18 b(par)g(l'application)i(du)f(c)o(hangemen)o(t)f(de)h(base)f fk(matrix)p fq(.)189 1703 y(ce)d(c)o(hangemen)o(t)g(de)h(base)g(est)f (l'iden)o(tit)o(\023)-21 b(e)16 b(sur)f(le)h(domaine)g(carr)o(\023)-21 b(e)14 b(\(0)p ff(\024)p fm(i;)8 b(j)s(<)o(n)p fq(\))f(:)15 b(comme)g(nous)189 1759 y(le)20 b(v)o(errons)e(par)h(la)h(suite,)f (ceci)h(signi\014e)h(que)f(le)g(domaine)g(discret)f(sur)h(lequel)h(on)e (place)h(les)189 1815 y(v)m(aleurs)c(de)f fk(a)g fq(est)g(inclus)i (dans)f(ce)f(carr)o(\023)-21 b(e.)143 1906 y({)23 b(la)e(deuxi)o(\022) -21 b(eme)20 b(\023)-21 b(equation)20 b(d)o(\023)-21 b(e\014nit)21 b(le)g(c)o(hamp)g fk(t)f fq(par)h(une)g(op)o(\023)-21 b(eration)20 b(de)h(routage)f(appliqu)o(\023)-21 b(ee)189 1962 y(sur)16 b fk(a)p fq(.)f(de)h(mani)o(\022)-21 b(ere)16

b(g)o(\023)-21 b(en)o(\023)g(erale,)15 b(un)h(routage)g(exprime)h(des)f (d)o(\023)-21 b(ep)q(endances)17 b(sur)f(le)h(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel)7 b(:)189 2019 y(le)18 b(c)o(hamp)h fk(x)c fa(c)g fk(g)j fq(est)g(tel)g(que)h(ses)f(v)m(aleurs)h(de)f(co)q (ordonn)o(\023)-21 b(ees)18 b fm(z)h ff(2)f fm(dom)p fq(\()p fk(g)p fq(\))f(son)o(t)g(issues)i(des)189 2075 y(p)q(oin)o(ts)e(de)h(co)q(ordonn)o(\023)-21 b(ees)16 b fk(g)p fq(\()p fm(z)r fq(\))h(de)h fk(x)p fq(.)e(en)i(l'o)q (ccurrence)g(sur)f(l'exemple,)h(les)g(v)m(aleurs)h(plac)o(\023)-21 b(ees)189 2132 y(aux)14 b(p)q(oin)o(ts)h(de)g(co)q(ordonn)o(\023)-21 b(ees)14 b(\()p fm(i;)8 b(j)s fq(\))13 b(dans)h fk(t)h fq(son)o(t)f(celles)i(plac)o(\023)-21 b(ees)15 b(en)g(\()p fm(j;)8 b(i)p fq(\))k(dans)j(le)h(c)o(hamp)e fk(a)p fq(.)143 2222 y({)23 b(la)13 b(troisi)o(\022)-21 b(eme)11 b(\023)-21 b(equation)13 b(d)o(\023)-21 b(e\014nit)13 b(le)h(c)o(hamp)f(de)g(donn) o(\023)-21 b(ees)13 b(en)g(sortie)g fk(c)p fq(.)f(la)h(partie)g(droite) g(d)o(\023)-21 b(e\014nit)189 2279 y(la)18 b(sup)q(erp)q(osition)i(de)f fk(a)f fq(et)g fk(t)p fq(,)g fj(i.e.)g fq(l'ensem)o(ble)h(des)g (couples)g fk(\(a)p fq(;)p fk(b\))p fq(,)e(a)o(v)o(ec)h fk(a)g ff(2)g fk(a)h fq(et)f fk(b)g ff(2)g fk(t)p fq(.)189 2335 y(l'op)o(\023)-21 b(eration)10 b(fonctionnelle)k(d)o(\023)-21 b(e\014nie)13 b(ici)g(consiste)f(en)g(l'application)i(de)e(la)g (fonction)g fk(add)g fq(\022)-23 b(a)11 b(tous)189 2391 y(les)k(\023)-21 b(el)o(\023)g(emen)o(ts)13 b(de)j(ce)g(c)o(hamp.)75 2486 y fa(\006)p eop %%page: 15 17 15 16 bop 75 269 a fl(2.1.)29 b(intr)o(oduction)570 257 y(\022)562 269 y(a)16 b(la)f(pr)o(ogramma)l(tion)h fn(pei)577 b fq(15)75 397 y fo(exemple)17 b(2)f fj(somme)g(de)g fm(n)h fj(nombr)n(es)75 515 y fq(p)o(our)g(calculer)i(la)f(somme)f(de)h fm(n)g fq(nom)o(bres)f(plac)o(\023)-21 b(es)17 b(initialemen)o(t)j(sur) d(le)h(c)o(hamp)g fk(a)p fq(,)f(nous)h(p)q(ouv)o(ons)75 571 y(amener)e(l'ensem)o(ble)h(des)f(v)m(aleurs)h(en)f(un)g(p)q(oin)o (t)g(unique)i(et)d(y)h(former)f(la)h(s)o(\023)-21 b(equence)16 b(de)g(ces)g(v)m(aleurs.)75 628 y(on)f(applique)h(ensuite)g(une)f(op)o (\023)-21 b(eration)13 b(fonctionnelle)k(p)q(our)e(sommer)f(cette)g(s)o (\023)-21 b(equence.)14 b(la)h(mani)o(\022)-21 b(ere)75 684 y(la)18 b(plus)g(g)o(\023)-21 b(en)o(\023)g(erale)16 b(et)h(la)h(plus)g(simple)h(d'exprimer)f(ce)f(d)o(\023)-21 b(eplacemen)o(t)18 b(est)f(d'utiliser)i(l'op)o(\023)-21 b(eration)16 b(de)75 740 y(r)o(\023)-21 b(eduction)12 b(g)o(\023)-21 b(eom)o(\023)g(etrique)6 b(:)12 b(on)g(pr)o(\023)-21 b(ecise)13 b(seulemen)o(t)g(la)g(destination)g(des)g(v)m(aleurs,)g (sans)f(dire)h(commen)o(t)75 797 y(elles)18 b(son)o(t)d(ac)o(hemin)o (\023)-21 b(ees)16 b(v)o(ers)g(cette)f(destination.)i(cette)e (expression)i(traduit)f(en)h(fait)e(la)i(d)o(\023)-21 b(e\014nition)75 853 y(globale)16 b(du)g(scalaire)g fm(t)d fq(=)534 821 y fx(p)578 831 y fw(n)p fx(\000)p fg(1)578 865 y fw(i)p fg(=0)650 853 y fm(a)674 860 y fw(i)688 853 y fq(.)i(le)h(programme)e(s')o(\023)-21 b(ecrit)14 b(simplemen)o(t)1385 837 y fg(4)1412 853 y fq(:)75 971 y fk(redsum)23 b(:)h(a)15 b ff(7!)g fk(t)75 985 y fx(\()133 1029 y fk(a)g fq(=)h fk(vector)7 b(::)g(a)133 1085 y(t)15 b fq(=)h fk(sum)f fa(b)g fk(\(red)23 b fq(;)8 b fa(b)24 b fk(a\))96 1152 y(vector)40 b fq(=)i fm(\025)p fk(\(i\))p ff(j)p fk(\(0<=i
(c)o(hemin)h(suivi)g(par)e(les)h(v)m(aleurs)g(p)q(our)g(atteindre)f(la) h(destina-)75 1491 y(tion.)g(dans)g(ce)g(but,)g(nous)h(utilisons)g(une) g(d)o(\023)-21 b(e\014nition)16 b(r)o(\023)-21 b(ecursiv)o(e)15 b(p)q(our)h(d)o(\023)-21 b(e\014nir)15 b(le)h(c)o(hamp)f fk(r)8 b fq(:)75 1609 y fk(prefixsum)23 b(:)g(a)15 b ff(7!)h fk(r)75 1623 y fx(\()133 1667 y fk(a)f fq(=)h fk(vector)7 b(::)g(a)133 1723 y(r)15 b fq(=)h fk(add)f fa(b)g fk(\(a)24 b(/&/)7 b(\(r)24 b fa(c)g fk(prev\)\))96 1789 y(vector)40 b fq(=)i fm(\025)p fk(\(i\))p ff(j)p fk(\(0<=i
b fy(\()p fu(r)14 b ft(c)g fu(prev)o fy(\))1258 1086 y fu(add)f ft(b)h fu(\(a/&/)6 b(\(r)21 b ft(c)i fu(prev)o(\)\))-717 b fy(15)33 b(21)-192 b(10)-99 b(6)j(3)g(1)751 692 y(1)54 b(3)g(6)36 b(10)d(15)g(21)826 824 y(1)54 b(3)g(6)36 b(10)d(15)751 955 y fs(1)39 b(2;)o(1)30 b(3;)o(3)g(4;)o(6)86 b(6;)o(15)-138 b(5;)o(10)p 1896 1208 2 756 v 75 1210 1822 2 v 305 1287 a fn(fig.)15 b fq(2.7)f(-)31 b fj(l)n(e)15 b(champ)i fk(r)f fj(c)n(ontient)g(les)f(sommes)h(p)n(artiel)r(les)g(du)g(champ)h fk(a)146 1407 y fq(v)o(\023)-21 b(eri\014er)17 b(qu'un)h(c)o(hamp)f (appartien)o(t)g(\022)-23 b(a)18 b(la)f(solution)h(du)g(syst)o(\022)-21 b(eme)17 b(d')o(\023)-21 b(equations)16 b(est)h(la)h(fa\030)-20 b(con)17 b(la)75 1463 y(plus)h(abstraite)f(de)g(consid)o(\023)-21 b(erer)17 b(un)h(programme)e fn(pei)p fq(.)g(on)i(p)q(eut)e(\023)-21 b(evidemmen)o(t)17 b(souhaiter)g(p)q(ouv)o(oir)75 1519 y(calculer)g(une)e(telle)i(solution,)e(c'est-\022)-23 b(a-dire)16 b(asso)q(cier)g(une)f(s)o(\023)-21 b(eman)o(tique)15 b fj(op)o(\023)-22 b(er)n(ationnel)r(le)14 b fq(\022)-23 b(a)15 b(ces)f(\023)-21 b(equa-)75 1576 y(tions.)18 b(nous)g(y)h (reviendrons)g(en)g(d)o(\023)-21 b(etail)18 b(dans)g(le)i(dernier)f(c)o (hapitre.)f(sur)h(cet)f(exemple)i(simple)g(on)75 1632 y(p)q(ourrait)e(naturellemen)o(t)h(d)o(\023)-21 b(eterminer)17 b(le)i(c)o(hamp)f(solution)g(en)g(calculan)o(t)h(p)q(our)f(c)o(haque)f (\023)-21 b(etap)q(e)17 b(de)75 1689 y(la)g(r)o(\023)-21 b(ecurrence,)17 b(la)g(somme)f(de)i(la)f(s)o(\023)-21 b(equence)17 b(de)h(v)m(aleurs)f(obten)o(ue)h(\022)-23 b(a)16 b(cette)g(\023)-21 b(etap)q(e,)16 b(ceci)i(par)f(sup)q(er-)75 1745 y(p)q(osition)i(de)g fk(a)f fq(a)o(v)o(ec)g(ce)h(m)o(^)-21 b(eme)17 b(c)o(hamp)h(don)o(t)g(les)h(v)m(aleurs)g(on)o(t)e(\023)-21 b(et)o(\023)g(e)16 b(d)o(\023)-21 b(ecal)o(\023)g(ees)18 b(sur)g(la)h(droite)f(d'une)75 1802 y(p)q(osition.)e(le)h(c)o(hamp)f (solution)g(est)g(\014nalemen)o(t)g(obten)o(u)g(lorsque)g(le)h(d)o (\023)-21 b(ecalage)15 b(v)o(ers)h(la)g(droite)g(n'est)75 1858 y(plus)g(p)q(ossible,)h(les)f(v)m(aleurs)g(n')o(\023)-21 b(etan)o(t)13 b(plus)j(dans)g(le)g(domaine)f(de)h(la)g(fonction)f fk(prev)p fq(.)146 1917 y(en)f(somme,)f(le)h(programme)f fk(prefixsum)g fq(est)h(une)g(v)o(ersion)g(plus)h(pr)o(\023)-21 b(ecise,)14 b(concernan)o(t)g(les)h(com-)75 1973 y(m)o(unications)h (induites,)g(que)f(le)g(programme)f fk(redsum)p fq(.)f(comme)h(nous)h (le)h(v)o(errons)e(dans)h(le)g(c)o(hapitre)75 2030 y(3,)g(nous)g (dirons)h(que)f(ce)h(programme)e fj(r)n(a\016ne)h fk(redsum)p fq(.)75 2145 y fo(exemple)i(3)f fj(d)o(\023)-22 b(ete)n(ction)16 b(de)g(p)n(alindr)n(omes)75 2260 y fq(un)d(v)o(ecteur)f fm(v)22 b fq(est)12 b(un)h(palindrome)h(si)f fm(v)788 2267 y fw(i)813 2260 y fq(=)g fm(v)888 2267 y fw(n)p fx(\000)p fg(1)p fx(\000)p fw(i)991 2260 y fq(,)f ff(8)p fm(i)g ff(2)h fq([0)8 b fm(:)g(:)g(:)d(n)t ff(\000)t fq(1].)13 b(p)o(our)d(\023)-21 b(ecrire)12 b(le)i(programme)75 2316 y fn(pei)p fq(,)h(qui)h(d)o(\023)-21 b(etecte)14 b(si)i(un)g(v)o(ecteur)f(est)g(un)h(palindrome,)g(nous)f(n'a)o(v)o(ons) g(pas)g(b)q(esoin)h(de)g(plus)g(de)g(pri-)75 2373 y(mitiv)o(es)g(que)h (p)q(our)f(le)g(programme)f fk(redsum)p fq(.)g(dans)g(le)i(programme)e (suiv)m(an)o(t,)h(la)g(v)m(aleur)h(b)q(o)q(ol)o(\023)-21 b(eenne)75 2429 y(du)18 b(c)o(hamp)f fk(r)g fq(plac)o(\023)-21 b(ee)18 b(en)g fm(i)e fq(=)g(0)h(indique)j(si)e(le)g(placemen)o(t)g (des)f(v)m(aleurs)i(du)f(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)17 b fk(a)75 2486 y fq(forme)e(un)g(palindrome.)p eop %%page: 17 19 17 18 bop 75 269 a fl(2.2.)29 b(objets)16 b(du)f(langa)o(ge)1114 b fq(17)146 397 y fk(palind)23 b(:)g(a)15 b ff(7!)h fk(r)146

411 y fx(\()204 455 y fk(t)f fq(=)g fk(equ)g fa(b)h fk(\(a)23 b(/&/)8 b(\(a)23 b fa(c)i fk(opp)o(\)\))204 511 y(r)15 b fq(=)g fk(and)g fa(b)h fk(\(red)23 b fq(;)8 b fa(b)23 b fk(t\))170 589 y(equ)50 b fq(=)g fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a)23 b(=)h(b\))170 646 y(opp)50 b fq(=)g fm(\025)p fk(\(i\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p ff(\024)p fk(n/2\).\(n-1-i\))170 702 y(red)g fq(=)g fm(\025)p fk(\(i\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p ff(\024)p fk(n/2\).\(0\))170 759 y(and)g fq(=)g fk(id)25 b fq(#)10 b fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a)d(&)h(\(and)23 b(b\)\))75 827 y fa(\006)75 1032 y ft(2.2)66 b(ob)t(jets)21 b(du)i(langage)146 1135 y fq(nous)14 b(v)o(enons)g(de)h(v)o(oir)f(in)o (tuitiv)o(emen)o(t)i(quelques)f(programmes)e fn(pei)p fq(.)h(nous)g(d)o(\023)-21 b(etaillons)15 b(main)o(te-)75 1192 y(nan)o(t)g(les)h(notations)e(utilis)o(\023)-21 b(ees,)16 b(et)f(les)h(ob)s(jets)f(du)g(langage)g fn(pei)p fq(.)143 1302 y({)23 b(les)16 b(programmes)7 b(:)15 b(la)i(structure)f (d'un)g(programme)f fn(pei)h fq(est)g(la)h(suiv)m(an)o(te.)f(un)h (programme)189 1359 y fk(p)h fq(p)q(oss)o(\022)-21 b(ede)19 b(en)g(en)o(tr)o(\023)-21 b(ee)18 b(un)i fm(n)p fq(-uplet)g(\()p fk(i)878 1366 y fg(1)896 1359 y fm(;)8 b(:)g(:)g(:)d(;)j fk(i)1022 1366 y fw(n)1044 1359 y fq(\))18 b(de)i(c)o(hamps)f(de)g (donn)o(\023)-21 b(ees,)18 b(et)h(un)h fm(m)p fq(-uplet)189 1415 y(\()p fk(o)231 1422 y fr(1)248 1415 y fm(;)8 b(:)g(:)g(:)t(;)g fk(o)373 1422 y fw(m)404 1415 y fq(\))15 b(de)h(c)o(hamps)f(de)g(donn)o (\023)-21 b(ees)15 b(en)h(sortie.)189 1492 y(les)d(c)o(hamps)g(en)h(en) o(tr)o(\023)-21 b(ee)12 b(et)h(les)g(c)o(hamps)g(en)h(sortie)f(son)o(t) f(li)o(\023)-21 b(es)14 b(par)f(des)f(\023)-21 b(equations)12 b(appartenan)o(t)189 1549 y(au)18 b(syst)o(\022)-21 b(eme)18 b(d')o(\023)-21 b(equations)18 b fm(s)s fq(,)f(\023)-21 b(ev)o(en)o(tuellemen)o(t)19 b(\022)-23 b(a)18 b(tra)o(v)o(ers)g(des)g (\023)-21 b(equations)18 b(p)q(ortan)o(t)g(sur)g(des)189 1605 y(c)o(hamps)d(de)g(donn)o(\023)-21 b(ees)15 b(in)o(term)o(\023)-21 b(ediaires.)189 1738 y fm(p)30 b fk(:)24 b fq(\()p fk(i)338 1745 y fg(1)356 1738 y fm(;)8 b(:)g(:)g(:)d(;)j fk(i)482 1745 y fw(n)503 1738 y fq(\))15 b ff(7!)h fq(\()p fk(o)639 1745 y fg(1)657 1738 y fm(;)8 b(:)g(:)g(:)d(;)j fk(o)783 1745 y fw(m)813 1738 y fq(\))189 1756 y fx(n)241 1803 y fq(s)214 1884 y fj([d)o(\023)-22 b(e\014nitions)14 b(de)j(fonctions])143 2050 y fq({)23 b(les)13 b(\023)-21 b(equations)14 b(en)o(tre)g(c)o(hamps)g(de)g(donn)o(\023)-21 b(ees)7 b(:)14 b(c)o(haque)g(\023)-21 b(equation)13 b(\(non-orien)o(t)o (\023)-21 b(ee\))13 b(du)i(syst)o(\022)-21 b(eme)189 2106 y fm(s)15 b fq(exprime)f(l')o(\023)-21 b(egalit)o(\023)g(e)12 b(des)h(expressions)h(de)f(c)o(hamps)g(de)h(donn)o(\023)-21 b(ees)12 b(de)i(part)e(et)h(d'autre)f(du)i(sym-)189 2163 y(b)q(ole)k(\023)-21 b(egale.)17 b(une)h(expression)h(de)g(c)o(hamp)f (de)g(donn)o(\023)-21 b(ees)18 b(est)f(soit)h(celle)i(d'un)f(c)o(hamp)f (simple,)189 2219 y(soit)h(celle)j(d'une)f(sup)q(erp)q(osition)g(de)g (c)o(hamps)e(de)i(donn)o(\023)-21 b(ees,)19 b(soit)h(celle)h(d'une)g (suite)f(d'op)o(\023)-21 b(e-)189 2276 y(rations.)15 b(p)o(ar)g(exemple,)i fk(plus)d fa(b)i fk(\(a)23 b(/&/)8 b(b\))15 b fq(ou)h(encore)g fk(\(carre)23 b fa(b)i fk(a)o(\))16 b fa(c)f fk(shift)g fq(son)o(t)g(des)189 2332 y(expressions)h(de)f(c)o (hamp)g(de)h(donn)o(\023)-21 b(ees.)143 2486 y({)23 b(les)15 b(op)o(\023)-21 b(erations)6 b(:)14 b(nous)h(a)o(v)o(ons)f(vu)h(que)g (les)g(op)o(\023)-21 b(erations)13 b(consistaien)o(t)j(en)f (l'application)h(d'une)189 2542 y(fonction)21 b(sur)f(un)h(c)o(hamp)g (de)g(donn)o(\023)-21 b(ees.)20 b(la)h(notation)f(des)h(fonctions)g(en) g fn(pei)f fq(est)g(d)o(\023)-21 b(eriv)o(\023)g(ee)p

eop %%page: 18 20 18 19 bop 75 269 a fq(18)587 b fl(chapitre)16 b(2.)45 b(d)1091 257 y(\023)1085 269 y(efinition)16 b(du)f(f)o(ormalisme)h fn(pei)189 397 y fq(du)k(lam)o(b)q(da-calcul.)i(une)e(fonction)g fk(f)g fq(de)g(domaine)g fm(dom)p fq(\()p fk(f)p fq(\))g(=)g ff(f)p fk(x)p ff(j)f fk(p\(x\))p ff(g)g fq(est)h(not)o(\023)-21 b(ee)19 b(par)189 453 y(l'expression)h fm(\025)p fk (\(x\)|p\(x\).f\(x\))m fq(.)f(p)o(ar)f(commo)q(dit)o(\023)-21 b(e,)17 b(nous)i(p)q(ouv)o(ons)f(\023)-21 b(ecrire)19 b fm(\025)p fk(\(x\).f\(x\))e fq(au)189 510 y(lieu)f(de)g fm(\025)p fk(\(x\)|true.f\(x\))n fq(,)f(et)g fm(\025)p fk(\(x\)|p\(x\))e fq(au)i(lieu)i(de)f fm(\025)p fk(\(x\)|p\(x\).\(x\))n fq(.)189 590 y(une)i(fonction)g fk(f)g fq(d)o(\023)-21 b(e\014nie)18 b(sur)g(des)g(sous-domaines)h(disjoin)o(ts)f(est)g(not)o (\023)-21 b(ee)16 b fk(f1)10 b fq(#)g fk(f2)p fq(,)17 b(et)h(le)h(do-)189 646 y(maine)c(d'une)h(fonction)g(comp)q(os)o(\023) -21 b(ee)14 b fk(f)h ff(\016)g fk(g)g fq(est)g ff(f)p fk(x)d ff(2)h fm(dom)p fq(\()p fk(g)p fq(\))f ff(j)i fk(g\(x\))h ff(2)e fm(dom)p fq(\()p fk(f)p fq(\))p ff(g)p fq(.)189 726 y(nous)19 b(utilisons)i(\023)-21 b(egalemen)o(t)19 b(la)h(fonction)g(iden)o(ti\014)o(\023)-21 b(ee)21 b(par)e(le)i(mot-cl) o(\023)-21 b(e)19 b fk(id)p fq(,)g(qui)i(\014ltre)f(les)g(s)o(\023)-21 b(e-)189 783 y(quences)16 b(\022)-23 b(a)14 b(un)h(\023)-21 b(el)o(\023)g(emen)o(t.)13 b(cette)i(fonction)g(appliqu)o(\023)-21 b(ee)16 b(\022)-23 b(a)15 b(un)g(\023)-21 b(el)o(\023)g(emen)o(t)13 b(rend)j(cet)e(\023)-21 b(el)o(\023)g(emen)o(t.)13 b(elle)189 839 y(est)20 b(fr)o(\023)-21 b(equemmen)o(t)19 b(utilis)o(\023)-21 b(ee)22 b(p)q(our)f(d)o(\023)-21 b(e\014nir)20 b(des)h(fonctions)g(r)o (\023)-21 b(ecursiv)o(es)20 b(dans)h(les)g(op)o(\023)-21 b(erations)189 896 y(fonctionnelles.)20 b(ainsi,)f(la)g(somme)f(r)o (\023)-21 b(ecursiv)o(e)19 b(d'une)g(s)o(\023)-21 b(equence)18 b(de)h(v)m(aleurs)h(s')o(\023)-21 b(ecrit)6 b(:)18 b fk(sum)g fq(=)189 952 y fk(id)30 b fq(#)10 b fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+sum\(b\)\))n fq(.)20 b(dans)h(cette)f(d)o (\023)-21 b(e\014nition,)21 b(la)g(fonction)g fk(sum)p fq(,)f(sur)h(le)g(domaine)189 1009 y(des)16 b(s)o(\023)-21 b(equences)17 b(\022)-23 b(a)15 b(un)h(\023)-21 b(el)o(\023)g(emen)o (t,)15 b(rend)h(cet)g(\023)-21 b(el)o(\023)g(emen)o(t.)14 b(sur)j(le)g(domaine)g(des)f(s)o(\023)-21 b(equences)17 b(\022)-23 b(a)16 b(plus)189 1065 y(d'un)f(\023)-21 b(el)o(\023)g(emen) o(t,)15 b(elle)j(rend)f(la)g(somme)f(du)g(premier)g(\023)-21 b(el)o(\023)g(emen)o(t)16 b(plus)h(la)g(somme)f(du)g(reste)h(de)f(la) 189 1122 y(s)o(\023)-21 b(equence.)189 1258 y(une)18 b(fonction)g(p)q(eut)h(appara)-5 b(^)-18 b(\020tre)18 b(directemen)o(t)g(dans)g(l'expression)h(de)g(c)o(hamp)f(de)g(donn)o (\023)-21 b(ees,)189 1314 y(comme)13 b(dans)g(l'expression)i fk(a)p fq(=)p fk(b)g fa(c)g fm(\025)p fk(\(i\).\(i+1\))o fq(,)d(ou)i(bien)f(^)-21 b(etre)12 b(r)o(\023)-21 b(ef)o(\023)g(erenc)o (\023)g(ee)11 b(par)i(un)h(nom)f(de)189 1371 y(fonction,)e(la)h(d)o (\023)-21 b(e\014nition)12 b(de)g(la)f(fonction)g(\023)-21 b(etan)o(t)10 b(rep)q(ort)o(\023)-21 b(ee)10 b(en)i(dehors)g(du)g(syst) o(\022)-21 b(eme)10 b(d')o(\023)-21 b(equations.)189 1427 y(c'est)10 b(la)i(notation)f(que)h(nous)g(utilisons)h(le)f(plus)h (fr)o(\023)-21 b(equemmen)o(t)10 b(p)q(our)i(des)g(raisons)f(de)h (lisibil)q(it)o(\023)-21 b(e.)75 1584 y ft(2.3)66 b(s)o(\023)-32 b(eman)n(tique)146 1691 y fq(nous)17 b(allons)i(main)o(tenan)o(t)e(exp) q(oser)h(la)h(s)o(\023)-21 b(eman)o(tique)17 b(des)h(d)o(\023)-21 b(e\014nitions)18 b fn(pei)p fq(,)g(toutes)f(bas)o(\023)-21 b(ees)17 b(sur)75 1747 y(celle)23 b(du)e(c)o(hamp)f(de)h(donn)o(\023)

-21 b(ees.)21 b(rapp)q(elons)h(qu'un)f(c)o(hamp)g(de)g(donn)o(\023)-21 b(ees)20 b(est)h(un)g(placemen)o(t)g(des)75 1803 y(v)m(aleurs)e(d'un)g (m)o(ulti-ensem)o(ble)h(de)f(v)m(aleurs)g fm(v)28 b fq(dans)18 b(un)h(r)o(\023)-21 b(ef)o(\023)g(eren)o(tiel,)17 b(tel)i(que)f(l'on)h (puisse)g(distin-)75 1860 y(guer)d(c)o(hacune)h(des)f(v)m(aleurs)h(par) f(des)h(co)q(ordonn)o(\023)-21 b(ees)15 b(en)o(ti)o(\022)-21 b(eres.)15 b(ce)i(placemen)o(t)f(est)g(une)h(application)75 1916 y fm(v)d fq(:)f fe(z)170 1900 y fw(n)202 1916 y ff(7!)g fm(v)d fq(.)k(nous)g(a)o(v)o(ons)g(vu)g(en)h(in)o(tro)q (duction)h(que)f(les)g(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees)14 b fk(v)p fq(,)g(don)o(t)g(les)i(v)m(aleurs)75 1973 y ff(f)p fq(1)p fm(;)8 b fq(2)p fm(;)g ff(\000)p fq(3)p fm(;)g fq(1)p ff(g)15 b fq(son)o(t)i(plac)o(\023)-21 b(ees)19 b(dans)f(cet)g(ordre)g(sur)h(l'in)o(terv)m(alle)h([0)8 b fm(:)g(:)g(:)d fq(3])17 b(de)i fe(z)-13 b fq(,)15 b(et)j fk(m)p fq(,)g(qui)h(place)g(ces)75 2029 y(m)o(^)-21 b(emes)17 b(v)m(aleurs)j(sur)e(les)h(p)q(oin)o(ts)g(de)g(co)q(ordonn)o(\023)-21 b(ees)18 b(0)g ff(\024)g fm(i;)8 b(j)19 b(<)g fq(2)f(de)h fe(z)1357 2013 y fg(2)1373 2029 y fq(,)f(ne)h(di\013)o(\022)-21 b(eren)o(t)17 b(que)i(d'une)75 2086 y(bijection)e fm(\033)r fq(\()p fm(i)p fq(\))11 b(=)i(\()p fm(i)h(mod)h fq(2)p fm(;)8 b(i)14 b(div)i fq(2\).)146 2201 y(plus)f(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t,)13 b(p)q(our)h(un)h(m)o(^)-21 b(eme)14 b(m)o(ulti-ensem)o(ble)j(de)e(v)m(aleurs)h(donn)o(\023)-21 b(e,)13 b(si)j fm(v)e fq(:)e fe(z)1649 2185 y fw(n)1682 2201 y ff(7!)h fm(v)24 b fq(est)75 2258 y(le)14 b(placemen)o(t)g(d)o (\023)-21 b(e\014nissan)o(t)14 b(un)g(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)13 b fk(v)g fq(et)h fm(v)1107 2241 y fx(0)1131 2258 y fq(:)e fe(z)1189 2241 y fw(p)1218 2258 y ff(7!)h fm(v)23 b fq(celui)15 b(d)o(\023)-21 b(e\014nissan)o(t)14 b(un)g(c)o(hamp)75 2314 y(de)j(donn)o(\023)-21 b(ees)17 b fk(v')p fq(,)g(on)g(a)f fm(v)i fq(=)e fm(v)608 2298 y fx(0)630 2314 y ff(\016)c fm(\033)r fq(,)k(o)q(\022)-24 b(u)17 b fm(\033)h fq(est)f(une)h(bijection)g(de)g fm(dom)p fq(\()p fm(v)r fq(\))d(sur)i fm(dom)p fq(\()p fm(v)1609 2298 y fx(0)1620 2314 y fq(\).)f(la)h(\014gure)75 2370 y(2.8)d(mon)o(tre)h(cette)g(relation)h(sur)f(l'exemple)h(de)g(la)g (section)f(2.1.1)146 2486 y(ceci)g(v)o(eut)g(dire)h(que)g(tout)e(c)o (hoix)h(de)h(placemen)o(t)g(d)o(\023)-21 b(e\014nissan)o(t)14 b(un)i(c)o(hamp)f(de)g(donn)o(\023)-21 b(ees)15 b(est)g(arbi-)75 2542 y(traire)g(et)f(di\013)o(\022)-21 b(ere)15 b(d'une)g(bijection)h (par)f(rapp)q(ort)f(\022)-23 b(a)15 b(ce)g(qu'on)g(p)q(ourrait)g (consid)o(\023)-21 b(erer)14 b(comme)h(un)g fj(pla-)p eop %%page: 19 21 19 20 bop 75 269 a fl(2.3.)29 b(s)207 257 y(\023)201 269 y(emantique)1310 b fq(19)547 401 y 13052758 9267452 3420651 40587345 16379658 49796792 starttexfig 547 401 a %%begindocument: figures/sigma.eps /tgifdict 3 dict def tgifdict begin /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def

/dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def end tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 0 setgray 72 0 mul 72 11.00 mul translate 72 128 div 100 mul 100 div dup neg scale gsave % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 47.250 -74.250 translate 1.406 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0

0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 50 df<5a1207123f12c71207b3a5eafff80d1c7c9b15>49 d e end texdict begin 1 0 bop 62 195 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat

countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 33.891 -101.250 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2

chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 51 df<ea07c0ea1830ea201cea400c130eeaf00f12f81307a21270ea00 0f130ea2131ca213381370136013c0ea0180ea0300ea0601120c1218ea1002ea3ffe127f 12ff101c7e9b15>50 d e end texdict begin 1 0 bop 62 195 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 20.250 -83.250 translate 1.406 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def

/texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 50 df<5a1207123f12c71207b3a5eafff80d1c7c9b15>49

d e end texdict begin 1 0 bop 62 195 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % polygon/closed-spline 0 setgray newpath 240 80 moveto 224 128 lineto 240 144 lineto 288 160 lineto 304 144 lineto 336 128 lineto 336 96 lineto 336 64 lineto 304 64 lineto 288 80 lineto 256 64 lineto closepath gsave grestore gsave stroke grestore % box 0 setgray gsave 1.415 setmiterlimit newpath 96 224 moveto 224 224 lineto 224 256 lineto 96 256 lineto closepath stroke 1 setmiterlimit grestore % box 0 setgray gsave 1.415 setmiterlimit newpath 352 208 moveto 416 208 lineto 416 272 lineto 352 272 lineto closepath stroke 1 setmiterlimit grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin

/showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -60.750 -164.250 translate 1.406 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0

0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 50 df<5a1207123f12c71207b3a5eafff80d1c7c9b15>49 d e end texdict begin 1 0 bop 62 195 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 68.203 -92.250 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x

array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 52 df<ea07e0ea1830ea201ca2ea781e130e131e1238ea001ca2131813 301360ea07c0ea0030131ca2130e130fa2127012f8a3eaf00eea401c1220ea1830ea07c0 101d7e9b15>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 62 195 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat

countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -29.109 -164.250 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2

chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 51 df<ea07c0ea1830ea201cea400c130eeaf00f12f81307a21270ea00 0f130ea2131ca213381370136013c0ea0180ea0300ea0601120c1218ea1002ea3ffe127f 12ff101c7e9b15>50 d e end texdict begin 1 0 bop 62 195 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -3.797 -164.250 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def

/texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 52 df<ea07e0ea1830ea201ca2ea781e130e131e1238ea001ca2131813

301360ea07c0ea0030131ca2130e130fa2127012f8a3eaf00eea401c1220ea1830ea07c0 101d7e9b15>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 62 195 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -6.750 -164.250 translate 1.406 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{

rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 50 df<5a1207123f12c71207b3a5eafff80d1c7c9b15>49 d e end texdict begin 1 0 bop 62 195 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % poly/open-spline 0 setgray gsave newpath 128 224 moveto 128 256 lineto stroke grestore % poly/open-spline 0 setgray

gsave newpath 160 224 moveto 160 256 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 192 224 moveto 192 256 lineto stroke grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 114.891 -155.250 translate 1.266 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id

ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 51 df<ea07c0ea1830ea201cea400c130eeaf00f12f81307a21270ea00 0f130ea2131ca213381370136013c0ea0180ea0300ea0601120c1218ea1002ea3ffe127f 12ff101c7e9b15>50 d e end texdict begin 1 0 bop 62 195 a fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def

0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 83.250 -155.250 translate 1.406 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false}

ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 50 df<5a1207123f12c71207b3a5eafff80d1c7c9b15>49 d e end texdict begin 1 0 bop 62 195 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 104.203 -173.250 translate 1.078 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{

pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 52 df<ea07e0ea1830ea201ca2ea781e130e131e1238ea001ca2131813 301360ea07c0ea0030131ca2130e130fa2127012f8a3eaf00eea401c1220ea1830ea07c0 101d7e9b15>51 d e /fb 1 1 df0 d e end texdict begin 1 0 bop 62 195 a fb(\000)p fa(3)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin

% eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 101.250 -173.250 translate 1.406 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{

cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 50 df<5a1207123f12c71207b3a5eafff80d1c7c9b15>49 d e end texdict begin 1 0 bop 62 195 a fa(1)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % poly/open-spline 0 setgray gsave newpath 352 240 moveto 416 240 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 384 208 moveto 384 272 lineto stroke grestore % poly/open-spline 0 setgray gsave newpath 240 240 moveto 0 96 atan dup cos 8 mul 336 exch sub exch sin 8 mul 240 exch sub lineto

stroke grestore gsave newpath 336 240 8 3 96 0 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 160 208 moveto -64 64 atan dup cos 8 mul 224 exch sub exch sin 8 mul 144 exch sub lineto stroke grestore gsave newpath 224 144 8 3 64 -64 tgifarrowtip closepath fill grestore % poly/open-spline 0 setgray gsave newpath 368 192 moveto -48 -48 atan dup cos 8 mul 320 exch sub exch sin 8 mul 144 exch sub lineto stroke grestore gsave newpath 320 144 8 3 -48 -48 tgifarrowtip closepath fill grestore % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -25.279 -183.375 translate 1.015 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n

/x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 91 df<003fb512803921c08100ea260038580102386002041240495a12 00495a495aa2495a495aa20181c7fca2ea0102ea0204a248481340485a15c03910200140

ec028038204004384080181460b6fc1a1c7f9b2c>90 d e /fb 1 19 df<90387fff8048b5fcd80780c7fc000ec8fc12185aa25aa25aa71260a27ea27e120e 6c7e0001b512806c7e90c8fca7007fb51280a219227d9920>18 d e /fc 2 42 df<1380ea0100120212065aa25aa25aa35aa412e0ac1260a47ea37ea27ea2 7e12027eea0080092a7c9e10>40 d<7e12407e12307ea27ea27ea37ea41380ac1300a412 06a35aa25aa25a12205a5a092a7e9e10>i e /fd 4 119 df<eb07e01300a2eb01c0a4eb 0380a43801e700ea0717ea0c0f1218ea380e12301270a2485aa4eb3880a3ea6078383199 00ea1e0e131d7e9c16>100 d<39381f81f0394e20c618394640e81ceb80f0ea8f00008e 13e0120ea2391c01c038a315703938038071a215e115e23970070064d83003133820127e 9124>109 d<13f8ea030cea0e06487e1218123000701380a238e00700a3130ea25bea60 185bea30e0ea0f8011127e9114>111 d<ea1c03ea270712471303ea8701a2120ea2ea1c 02a3130412181308a2ea1c10ea0c20ea07c010127e9114>118 d e end texdict begin 1 0 bop 62 195 a fd(dom)p fc(\()p fd(v)q fc(\))13 b fb(\022)f fa(z)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -11.109 -219.375 translate 1.266 1.266 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx

fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 119 df<ea1c03ea270712471303ea8701a2120ea2ea1c02a31304121813 08a2ea1c10ea0c20ea07c010127e9114>118 d e end texdict begin 1 0 bop 62 195 a fa(v)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin

% eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 100.406 -156.068 translate 1.018 1.074 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin

0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 51 df<121fea60c01360eaf07013301260ea0070a2136013c012011380 ea02005aea08101210ea2020ea7fe012ff0c137e9211>50 d e /fb 1 91 df<003fb512803921c08100ea260038580102386002041240495a1200495a495aa2 495a495aa20181c7fca2ea0102ea0204a248481340485a15c03910200140ec0280382040 04384080181460b6fc1a1c7f9b2c>90 d e /fc 1 19 df<90387fff8048b5fcd80780c7 fc000ec8fc12185aa25aa25aa71260a27ea27e120e6c7e0001b512806c7e90c8fca7007f b51280a219227d9920>18 d e /fd 1 49 df<1204120ea2121ca31238a212301270a212 60a212c0a2070f7f8f0a>48 d e /fe 2 42 df<1380ea0100120212065aa25aa25aa35a a412e0ac1260a47ea37ea27ea27e12027eea0080092a7c9e10>40 d<7e12407e12307ea27ea27ea37ea41380ac1300a41206a35aa25aa25a12205a5a092a7e 9e10>i e /ff 4 119 df<eb07e01300a2eb01c0a4eb0380a43801e700ea0717ea0c0f12 18ea380e12301270a2485aa4eb3880a3ea607838319900ea1e0e131d7e9c16>100 d<39381f81f0394e20c618394640e81ceb80f0ea8f00008e13e0120ea2391c01c038a315 703938038071a215e115e23970070064d83003133820127e9124>109 d<13f8ea030cea0e06487e1218123000701380a238e00700a3130ea25bea60185bea30e0 ea0f8011127e9114>111 d<ea1c03ea270712471303ea8701a2120ea2ea1c02a3130412 181308a2ea1c10ea0c20ea07c010127e9114>118 d e end texdict begin 1 0 bop 62 195 a ff(dom)p fe(\()p ff(v)177 180 y fd(0)190 195 y fe(\))12 b fc(\022)f fb(z)292 180 y fa(2)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def

0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 100.125 -111.375 translate 1.125 1.125 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false}

ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 49 df<1204120ea2121ca31238a212301270a21260a212c0a2070f7f8f 0a>48 d e /fb 1 119 df<ea1c03ea270712471303ea8701a2120ea2ea1c02a3130412 181308a2ea1c10ea0c20ea07c010127e9114>118 d e end texdict begin 1 0 bop 62 195 a fb(v)83 180 y fa(0)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 51.863 -254.109 translate 1.237 1.266 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx

fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 28 df<eb7ffeea01ff1207380f0780ea1e03381c01c012381270a338e0 0380a3eb0700a2130eea600c6c5aea1860ea0f8017147e931a>27 d e end texdict begin 1 0 bop 71 199 a fa(\033)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin

% eps end /tgiflevel0 save def /tgifdictcount0 countdictstack def /tgifopcount0 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate -47.109 -218.137 translate 1.266 1.237 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{

cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa002km.dvi) @start /fa 1 87 df<38fe0fe0a338380380ea3c07001c1300a3ea1e0fea0e0ea46c5a a4ea031813b8a3ea01b013f0a26c5a13197f9816>86 d e end texdict begin 1 0 bop 62 195 a fa(v)p eop end userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount0 sub {pop} repeat countdictstack tgifdictcount0 sub {end} repeat tgiflevel0 restore tgifdict begin % eps end /tgiflevel1 save def /tgifdictcount1 countdictstack def /tgifopcount1 count 1 sub def userdict begin /showpage {} def /letter {} def /legal {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath 1 72 128 div 100 mul 100 div div dup neg scale 72 0 mul neg 72 11.00 mul neg translate 132.891 -236.137 translate 1.266 1.237 scale 0 rotate save countdictstack mark newpath /showpage {} def /texdict 250 dict def texdict begin /n{def}def /b{bind def}n /s{exch}n /x{s n}b /tr{translate}n /isls false n /vsize 11 72 mul n /hsize 8.5 72 mul n /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 resolution div 72 vresolution div neg scale

isls{landplus90{vresolution 72 div vsize mul 0 exch}{resolution -72 div hsize mul 0}ifelse tr}if resolution vresolution vsize -72 div 1 add mul tr matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}n /@landscape{/isls true n}b /@manualfeed{ statusdict /manualfeed true put}b /@copies{/#copies x}b /fmat[1 0 0 -1 0 0]n /fbb[0 0 0 0]n /nn 0 n /ie 0 n /ctr 0 n /df-tail{/nn 8 dict n nn begin /fonttype 3 n /fontmatrix fntrx n /fontbbox fbb n string /base x array /bitmaps x /buildchar{charbuilder}n /encoding ie n end dup{/foo setfont}2 array copy cvx n load 0 nn put /ctr 0 n[}b /df{/sf 1 n /fntrx fmat n df-tail}b /dfs{div /sf x /fntrx[sf 0 0 sf neg 0 0]n df-tail}b /e{ pop nn dup definefont setfont}b /ch-width{ch-data dup length 5 sub get} b /ch-height{ch-data dup length 4 sub get}b /ch-xoff{128 ch-data dup length 3 sub get sub}b /ch-yoff{ch-data dup length 2 sub get 127 sub}b /ch-dx{ch-data dup length 1 sub get}b /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add n}if}b /id 0 n /rw 0 n /rc 0 n /gp 0 n /cp 0 n /g 0 n /sf 0 n /charbuilder{save 3 1 roll s dup /base get 2 index get s /bitmaps get s get /ch-data x pop /ctr 0 n ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image n /rw ch-width 7 add 8 idiv string n /rc 0 n /gp 0 n /cp 0 n{ rc 0 ne{rc 1 sub /rc x rw}{g}ifelse}imagemask restore}b /g{{id gp get /gp gp 1 add n dup 18 mod s 18 idiv pl s get exec}loop}b /adv{cp add /cp x}b /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp x adv}b /nd{/cp 0 n rw exit}b /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and s 1 and or}ifelse}ifelse put 1 adv} b /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv s 128 and or}ifelse}ifelse put 1 adv}b /clr{rw cp 2 index string putinterval adv}b /set{rw cp fillstr 0 4 index getinterval putinterval adv}b /fillstr 18 string 0 1 17{2 copy 255 put pop}for n /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc x nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall n /d{/cc x dup type /stringtype ne{] }if nn /base get cc ctr put nn /bitmaps get s ctr s sf 1 ne{dup dup length 1 sub dup 2 index s get sf div put}if put /ctr ctr 1 add n}b /i{ cc 1 add d}b /bop{userdict /bop-hook known{bop-hook}if /si save n @rigin 0 0 moveto /v matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/qv}{/rv}ifelse load def pop pop}n /eop{si restore showpage userdict /eop-hook known{eop-hook}if}n /@start{userdict /start-hook known{start-hook}if pop /vresolution x /resolution x 1000 div /dvimag x /ie 256 array n 0 1 255{ie s 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize x 65781.76 div /hsize x}n /p{show}n /rmat[1 0 0 -1 0 0]n /bdot 260 string n /rulex 0 n /ruley 0 n /v{/ruley x /rulex x v}b /v {}b /rv statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(display)eq exch 0 4 getinterval(next)eq or}{pop false} ifelse}{false}ifelse end{{gsave tr -.1 -.1 tr 1 1 scale rulex ruley false rmat{bdot}imagemask grestore}}{{gsave tr -.1 -.1 tr rulex ruley scale 1 1 false rmat{bdot}imagemask grestore}}ifelse b /qv{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}b /a{moveto}b /delta 0 n /tail{dup /delta x 0 rmoveto}b /m{s p delta add tail}b /b{s p tail} b /c{-4 m}b /d{-3 m}b /e{-2 m}b /f{-1 m}b /g{0 m}b /h{1 m}b /i{2 m}b /j{ 3 m}b /k{4 m}b /w{0 rmoveto}b /l{p -4 w}b /m{p -3 w}b /n{p -2 w}b /o{p -1 w}b /q{p 1 w}b /r{p 2 w}b /s{p 3 w}b /t{p 4 w}b /x{0 s rmoveto}b /y{ 3 2 roll p a}b /bos{/ss save n}b /eos{ss restore}b end texdict begin 39158280 55380996 1000 300 300 (/tmp/tgifa0040t.dvi) @start /fa 1 78 df<38fc07e0eafe0fa2383a0b80ea3b1ba513bbea39b3a413f3ea38 e3a21303a538fe0fe0a313197f9816>77 d e end texdict begin 1 0 bop 62 195 a fa(m)p eop end

userdict /end-hook known{end-hook}if cleartomark countdictstack exch sub { end } repeat restore count tgifopcount1 sub {pop} repeat countdictstack tgifdictcount1 sub {end} repeat tgiflevel1 restore tgifdict begin grestore tgifsavedpage restore end %matchingcreationdate: mon oct 14 14:12:46 1996 %%enddocument endtexfig 425 1132 a fn(fig.)15 b fq(2.8)g(-)30 b fj(r)n(elation)16 b(entr)n(e)f(deux)i(plac)n(ements)d(\023)-22 b(e)n(quivalents)75 1283 y(c)n(ement)15 b(de)h(r)o(\023)-22 b(ef)o(\023)g(er)n(enc)n(e)p fq(.)12 b(en)j(in)o(t)o(\023)-21 b(egran)o(t)13 b(cette)h(bijection)i (\022)-23 b(a)14 b(la)h(s)o(\023)-21 b(eman)o(tique)14 b(des)h(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees,)75 1339 y(on)13 b(p)q(eut)h(ainsi)h(facilemen)o(t)f(mon)o(trer)e(la)i(relation) g(en)o(tre)f(deux)h(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees)13 b(repr)o(\023)-21 b(esen)o(tan)o(t)12 b(le)75 1396 y(m)o(^)-21 b(eme)16 b(m)o(ulti-ensem)o(ble)10 b(:)16 b(ceci)i(deviendra)g(explicite)i(dans)d(l'in)o(terpr)o(\023)-21 b(etation)16 b(fonctionnelle)j(d)o(\023)-21 b(e\014nie)75 1452 y(en)17 b(section)g(2.3.5.)e(nous)i(caract)o(\023)-21 b(erisons)15 b(donc)i(un)g(c)o(hamp)g(de)g(donn)o(\023)-21 b(ees)17 b fk(x)f fq(par)h(une)g(paire)g(de)g(fonc-)75 1509 y(tions)d(\()p fm(v)g fq(:)e fm(\033)r fq(\).)g(de)h(plus,)h(nous) g(imp)q(osons)g(la)g(con)o(train)o(te)f(suiv)m(an)o(te)8 b(:)13 b fm(dom)p fq(\()p fm(v)r fq(\),)f(que)i(nous)f(app)q(elons)i (le)75 1565 y fj(domaine)h(des)f(valeurs)f fq(doit)g(^)-21 b(etre)13 b(inclus)k(dans)d(le)i(domaine)f(de)g(la)f(bijection)i fm(\033)r fq(,)e(que)h(nous)f(app)q(elons)75 1622 y fj(domaine)i(de)h (r)o(\023)-22 b(ef)o(\023)g(er)n(enc)n(e)13 b fq(des)j(v)m(aleurs,)f (a\014n)h(qu'un)f(c)o(hangemen)o(t)f(\023)-21 b(ev)o(en)o(tuel)15 b(de)g(placemen)o(t)h(concerne)75 1678 y(toutes)f(les)h(v)m(aleurs)g (du)f(c)o(hamp)h(de)f(donn)o(\023)-21 b(ees.)75 1798 y fo(d)o(\023)c(e\014nition)19 b(2)31 b fq(\(c)o(hamp)14 b(de)i(donn)o(\023)-21 b(ees\))75 1854 y(soit)18 b fm(v)26 b fq(un)18 b(m)o(ulti-ensem)o(ble)i(de)e(v)m(aleurs.)g(soit)f fm(v)i fq(une)f(fonction)g(de)f fe(z)1337 1838 y fw(n)1373 1854 y ff(7!)f fm(v)27 b fq(et)17 b fm(\033)i fq(une)f(bijection)75 1911 y fe(z)108 1894 y fw(n)141 1911 y ff(7!)13 b fe(z)232 1894 y fw(p)248 1911 y fq(,)i(alors)g(la)g(paire)h(\()p fm(v)e fq(:)e fm(\033)r fq(\))j(d)o(\023)-21 b(e\014nit)15 b(un)h(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)15 b(si)g fm(dom)p fq(\()p fm(v)r fq(\))d ff(\022)h fm(dom)p fq(\()p fm(\033)r fq(\).)75 2072 y fw(2.3.1)55 b(s)o(\023)-27 b(eman)n(tique)17 b(des)h(op)o(\023)-27 b(erations)146 2171 y fq(ceci)15 b(nous)f(p)q(ermet)g(de)h(d)o(\023)-21 b(e\014nir)15 b(la)f(s)o(\023)-21 b(eman)o(tique)14 b(de)g (l'application)j(de)d(c)o(hacune)h(des)g(trois)f(op)o(\023)-21 b(era-)75 2227 y(tions)15 b(sur)h(un)f(c)o(hamp)g fk(x)p fq(=)e(\()p fm(v)h fq(:)e fm(\033)r fq(\))7 b(:)75 2387 y fo(op)o(\023)-25 b(eration)19 b(fonctionnelle)75 2486 y fq(soit)g fk(f)h fq(une)f(fonction)h(de)g fm(v)29 b

fq(dans)19 b(un)h(ensem)o(ble)g(de)g(v)m(aleurs)g fm(w)6 b fq(,)19 b(telle)i(que)e fm(img)r fq(\()p fm(v)r fq(\))e ff(\022)j fm(dom)p fq(\()p fk(f)p fq(\),)75 2542 y(l'op)o(\023)-21 b(eration)14 b(fonctionnelle)k fk(y)d fq(=)g fk(f)g fa(b)h fk(x)f fq(d)o(\023)-21 b(e\014nit)15 b fk(y)g fq(comme)g(\()p fk(f)10 b ff(\016)g fm(v)k fq(:)e fm(\033)r fq(\).)p eop %%page: 20 22 20 21 bop 75 269 a fq(20)587 b fl(chapitre)16 b(2.)45 b(d)1091 257 y(\023)1085 269 y(efinition)16 b(du)f(f)o(ormalisme)h fn(pei)75 397 y fo(op)o(\023)-25 b(eration)19 b(de)e(routage)h(\(ou)g (op)o(\023)-25 b(eration)18 b(g)o(\023)-25 b(eom)o(\023)g(etrique\))75 483 y fq(soit)17 b fk(g)f fq(une)h(fonction)f(de)h fe(z)568 466 y fw(n)602 483 y ff(7!)e fm(dom)p fq(\()p fm(v)r fq(\),)g(telle)i(que)g fm(dom)p fq(\()p fk(g)p fq(\))c ff(\022)i fm(dom)p fq(\()p fm(\033)r fq(\),)f(l'op)o(\023)-21 b(eration)16 b(de)h(routage)75 539 y fk(y)24 b(=)f(x)15 b fa(c)h fk(g)f fq(d)o(\023)-21 b(e\014nit)15 b fk(y)g fq(comme)f(\023)-21 b(etan)o(t)14 b(\()p fm(v)d ff(\016)f fk(g)j fq(:)f fm(\033)r fq(\).)75 652 y(la)19 b(condition)h(sur)f(les)h (domaines)f(de)h fk(g)e fq(et)h fm(\033)i fq(assure)d(que)i(le)f(r)o (\023)-21 b(esultat)18 b fk(y)h fq(est)g(bien)h(un)f(c)o(hamp)g(de)75 709 y(donn)o(\023)-21 b(ees,)15 b(c'est-\022)-23 b(a-dire)15 b(que)h(sa)f(paire)g(\()p fm(v)786 716 y fw(y)828 709 y fq(:)d fm(\033)879 716 y fw(y)908 709 y fq(\))i(resp)q(ecte)i fm(dom)p fq(\()p fm(v)1242 716 y fw(y)1270 709 y fq(\))d ff(\022)g fm(dom)p fq(\()p fm(\033)1479 716 y fw(y)1507 709 y fq(\).)75 822 y(r)o(\023)-21 b(ecipro)q(quemen)o(t,)13 b(soit)g fk(g)g fq(une)h(fonction)f(partielle)h fm(dom)p fq(\()p fm(v)r fq(\))e ff(7!)h fm(dom)p fq(\()p fm(\033)r fq(\),)e(la)i fj(r)o(\023)-22 b(e)n(duction)14 b(g)o(\023)-22 b(e)n(om)o(\023)g(etrique)75 878 y fq(d)o(\023)h(e\014nit)18 b fk(g)d fq(;)8 b fa(b)15 b fk(x)j fq(comme)e(\023)-21 b(etan)o(t)16 b(un)j(c)o(hamp)f(de)g(donn)o(\023)-21 b(ees)18 b(\()p fm(w)f fq(:)g fm(\033)r fq(\))g(tel)h(que)h fm(dom)p fq(\()p fm(w)q fq(\))c(=)j fk(g)p fq(\()p fm(dom)p fq(\()p fm(v)r fq(\)\))75 934 y(et)d fm(w)q fq(\()p fm(z)r fq(\))f(est)h(une)h(s)o(\023)-21 b(equence)15 b(des)g(v)m(aleurs)h fm(v)r fq(\()p fm(y)r fq(\),)e(telle)i(que)g fk(g)p fq(\()p fm(y)r fq(\))11 b(=)i fm(z)r fq(.)i(p)o(ar)f(cons)o(\023)-21 b(equen)o(t)15 b(la)g(propri)o(\023)-21 b(et)o(\023)g(e)75 991 y(suiv)m(an)o(te)16 b(lie)h(la)e(r)o(\023)-21 b(eduction)15 b(g)o(\023)-21 b(eom)o(\023)g(etrique)14 b(et)h(l'op)o(\023)-21 b(eration)14 b(g)o(\023)-21 b(eom)o(\023)g(etrique.)75 1104 y fo(prop)q(osition)19 b(1)47 b fq(le)16 b(c)o(hamp)g(de)g(donn)o (\023)-21 b(ees)16 b fk(g)f fq(;)8 b fa(b)14 b fk(x)i fq(est)e(\023)-21 b(egal)15 b(\022)-23 b(a)16 b fk(x)f fa(c)g fk(g)1341 1087 y fq(-1)1394 1104 y fq(si)h(et)g(seulemen)o(t)g (si)g fk(g)g fq(est)75 1160 y(bijectiv)o(e.)75 1278 y fo(op)o(\023)-25 b(eration)19 b(de)e(c)o(hangemen)o(t)g(de)h(base)75 1364 y fq(soit)d fk(h)g fq(une)g(bijection)h(d'une)f(partie)g(de)g fe(z)825 1347 y fw(n)859 1364 y fq(dans)g fe(z)998 1347 y fw(p)1014 1364 y fq(,)g(le)g(c)o(hangemen)o(t)g(de)g(base)g fk(y)23 b(=)h(h)8 b(::)f(x)14 b fq(d)o(\023)-21 b(e\014nit)75 1420 y fk(y)15 b fq(comme)f(\023)-21 b(etan)o(t)13 b(\()p fm(v)k ff(\016)e fk(h)504 1404 y fq(-1)541 1420 y fq(:)d fm(\033)17 b ff(\016)e fk(h)671 1404 y fq(-1)708 1420 y fq(\),)g(si)g fm(dom)p fq(\()p fm(v)r fq(\))d ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\).)75 1596 y fw(2.3.2)55 b(s)o(\023)-27 b(eman)n(tique)17 b(de)h(la)h(sup)r(erp)r(osition)146

1682 y fq(soien)o(t)f(deux)g(fonctions)g fm(f)k fq(et)c fm(g)r fq(,)e(nous)i(notons)f fm(f)5 b ff(j)1045 1689 y fw(dom)p fg(\()p fw(g)q fg(\))1170 1682 y fq(la)18 b(partie)g(de)g fm(f)23 b fq(d)o(\023)-21 b(e\014nie)18 b(sur)g(dom\()p fm(g)r fq(\).)75 1738 y(ainsi)13 b fm(f)5 b ff(j)229 1745 y fw(dom)p fg(\()p fw(g)q fg(\))349 1738 y fq(=)13 b fm(g)g fq(signi\014e)g(que)f fm(dom)p fq(\()p fk(g)p fq(\))g ff(\022)h fm(dom)p fq(\()p fk(f)p fq(\))e(et)g(que)i fm(f)k fq(et)11 b fm(g)i fq(son)o(t)e(\023)-21 b(egales)11 b(sur)h fm(dom)p fq(\()p fm(g)r fq(\).)d(deux)75 1795 y(c)o(hamps)i(de)g(donn)o(\023)-21 b(ees)11 b fk(x1)p fq(=)i(\()p fm(v)594 1802 y fg(1)625 1795 y fq(:)f fm(\033)676 1802 y fg(1)695 1795 y fq(\))e(et)h fk(x2)p fq(=)i(\()p fm(v)908 1802 y fg(2)939 1795 y fq(:)f fm(\033)990 1802 y fg(2)1009 1795 y fq(\))f(p)q(euv)o(en)o(t)f(^)-21 b(etre)10 b(sup)q(erp)q(os)o(\023)-21 b(es)11 b(si)h fm(\033)1584 1802 y fg(1)1602 1795 y ff(j)1615 1802 y fw(dom)p fg(\()p fw(\033)1709 1806 y fp(2)1726 1802 y fg(\))1754 1795 y fq(=)h fm(\033)1828 1802 y fg(2)75 1851 y fq(ou)k fm(\033)166 1858 y fg(2)185 1851 y ff(j)198 1858 y fw(dom)p fg(\()p fw(\033)292 1862 y fp(1)309 1858 y fg(\))340 1851 y fq(=)f fm(\033)417 1858 y fg(1)435 1851 y fq(.)h(la)g(s)o(\023)-21 b(eman)o(tique)17 b(de)g(l'op)o(\023)-21 b(erateur)16 b(de)h(sup)q(erp)q(osition)i(est)e(alors)g(d)o(\023)-21 b(e\014nie,)17 b(p)q(our)75 1907 y fk(\(x1/&/)7 b(x2\))p fq(,)15 b(par)7 b(:)146 1968 y fx(8)146 2005 y(>)146 2018 y(<)146 2093 y(>)146 2105 y(:)207 2010 y fq(\()p fm(v)247 2017 y fg(1)278 2010 y fq(:)13 b fm(\033)r fq(\))49 b(sur)g fm(dom)p fq(\()p fm(v)635 2017 y fg(1)653 2010 y fq(\))p ff(n)p fm(dom)p fq(\()p fm(v)820 2017 y fg(2)838 2010 y fq(\))207 2066 y(\()p fm(w)13 b fq(:)g fm(\033)r fq(\))56 b(sur)49 b fm(dom)p fq(\()p fm(v)635 2073 y fg(1)653 2066 y fq(\))10 b ff(\\)h fm(dom)p fq(\()p fm(v)848 2073 y fg(2)866 2066 y fq(\))207 2123 y(\()p fm(v)247 2130 y fg(2)278 2123 y fq(:)i fm(\033)r fq(\))49 b(sur)g fm(dom)p fq(\()p fm(v)635 2130 y fg(2)653 2123 y fq(\))p ff(n)p fm(dom)p fq(\()p fm(v)820 2130 y fg(1)838 2123 y fq(\))146 2224 y(o)q(\022)-24 b(u)15 b fm(w)q fq(\()p fm(z)r fq(\))d(=)h fm(v)384 2231 y fg(1)402 2224 y fq(\()p fm(z)r fq(\);)o fm(v)495 2231 y fg(2)514 2224 y fq(\()p fm(z)r fq(\))i(et)g fm(\033)f fq(=)729 2152 y fx(\()787 2196 y fm(\033)813 2203 y fg(1)847 2196 y fq(si)i fm(dom)p fq(\()p fm(\033)1023 2203 y fg(2)1041 2196 y fq(\))c ff(\022)h fm(dom)p fq(\()p fm(\033)1249 2203 y fg(1)1267 2196 y fq(\))787 2252 y fm(\033)813 2259 y fg(2)847 2252 y fq(si)j fm(dom)p fq(\()p fm(\033)1023 2259 y fg(1)1041 2252 y fq(\))c ff(\022)h fm(dom)p fq(\()p fm(\033)1249 2259 y fg(2)1267 2252 y fq(\).)75 2400 y fw(2.3.3)55 b(s)o(\023)-27 b(eman)n(tique)17 b(des)g(\023)-27 b(equations)146 2486 y fq(nous)14 b(a)o(v)o(ons)g(vu)h(qu'une)g(\023)-21 b(equation)14 b(d)o(\023)-21 b(e\014nit)15 b(l')o(\023)-21 b(egalit)o(\023)g(e)14 b(des)h(expressions)h(de)f(c)o(hamps)g(de)g (donn)o(\023)-21 b(ees)75 2542 y(de)16 b(part)f(et)g(d'autre)g(du)h (sym)o(b)q(ole)f(\023)-21 b(egale.)14 b(nous)i(pr)o(\023)-21 b(ecisons)15 b(main)o(tenan)o(t)g(la)h(s)o(\023)-21 b(eman)o(tique)15 b(asso)q(ci)o(\023)-21 b(ee)15 b(\022)-23 b(a)p eop %%page: 21 23 21 22 bop 75 269 a fl(2.3.)29 b(s)207 257 y(\023)201 269 y(emantique)1310 b fq(21)75 397 y(une)13 b(\023)-21 b(equation,)13 b(qui)h(s'exprime)h(elle)g(aussi)f(\022)-23

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2563 y fr(x)1109 2556 y fq(\)\))12 b(=)h fm(img)r fq(\()p fk(g)p fq(\))p eop %%page: 23 25 23 24 bop 75 269 a fl(2.3.)29 b(s)207 257 y(\023)201 269 y(emantique)1310 b fq(23)143 397 y fj({)23 b(ou)17 b(bien)e fk(h)8 b(::)f(\(x)24 b fa(c)g fk(g\))16 b fj(=)g fk(h)707 380 y fx(0)726 397 y fk(::)8 b(y)48 b fj(et)h fm(dom)p fq(\()p fm(v)1065 404 y fr(x)1082 397 y fq(\))12 b(=)h fm(img)r fq(\()p fk(g)p fq(\))p fj(.)75 481 y fq(p)o(our)h(d)o (\023)-21 b(emon)o(trer)13 b(cette)h(caract)o(\023)-21 b(erisation)14 b(nous)g(mon)o(trons)g(d'ab)q(ord)g(la)h(propri)o(\023) -21 b(et)o(\023)g(e)13 b(suiv)m(an)o(te,)i(qui)g(lie)75 537 y(un)h(c)o(hamp)f(de)g(donn)o(\023)-21 b(ees)15 b fk(x)g fq(et)g(un)h(c)o(hamp)f fk(y)g fq(d)o(\023)-21 b(e\014ni)16 b(par)f(une)h(expression)g(fonction)f(de)h fk(x)p fq(.)75 650 y fo(propri)o(\023)-25 b(et)o(\023)g(e)19 b(1)50 b fq(soien)o(t)17 b fk(x)g fq(un)g(c)o(hamp)f(de)i(donn)o(\023) -21 b(ees)16 b(et)h fk(y)f fq(une)h(expression)h(de)f(c)o(hamp)g(de)g (donn)o(\023)-21 b(ees)75 707 y(fonction)16 b(de)f fk(x)p fq(.)g(il)h(existe)g(alors)f(une)h(paire)f(de)h(bijections)g(\()p fk(h)p fq(,)p fk(h)1186 690 y fx(0)1197 707 y fq(\))f(telles)h(que)680 756 y fx(8)680 794 y(>)680 806 y(<)680 881 y(>)680 893 y(:)738 800 y fm(\033)764 807 y fr(x)791 800 y ff(\016)10 b fk(h)848 783 y fq(-1)898 800 y fq(=)j fm(\033)972 807 y fr(y)999 800 y ff(\016)d fk(h)1056 783 y fx(0)1068 777 y fq(-1)1157 800 y fq(\()p fm(a)p fq(\))738 856 y fm(dom)p fq(\()p fm(v)864 863 y fr(x)880 856 y fq(\))j ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\))56 b(\()p fm(b)p fq(\))738 913 y fm(dom)p fq(\()p fm(v)864 920 y fr(y)880 913 y fq(\))13 b ff(\022)g fm(dom)p fq(\()p fk(h)1087 896 y fx(0)1097 913 y fq(\))46 b(\()p fm(c)p fq(\))75 1025 y fj(pr)n(euve)p fq(.)15 b fj(cf.)g fq(annexe)h(a)75 1138 y fj(pr)n(euve.)f fq(\(caract)o(\023)-21 b(erisation\))75 1278 y fo(\()p ff(\))p fo(\))24 b fq(supp)q(osons)e fk(x)e fq(et)h fk(y)g fq(faiblemen)o(t)g(\023)-21 b(equiv)m(alen)o(ts.)21 b(il)h(existe)f(alors)g(une)h(bijection)g fm(\013)f fq(telle)h(que)189 1335 y fm(v)211 1342 y fr(y)241 1335 y fq(=)13 b fm(v)311 1342 y fr(x)338 1335 y ff(\016)d fm(\013)p fq(,)15 b fm(img)r fq(\()p fm(\013)p fq(\))c(=)i fm(dom)p fq(\()p fm(v)758 1342 y fr(x)775 1335 y fq(\))i(et)g fm(dom)p fq(\()p fm(\013)p fq(\))d(=)h fm(dom)p fq(\()p fm(v)1198 1342 y fr(y)1215 1335 y fq(\).)189 1408 y(d'apr)o(\022)-21 b(es)17 b(la)j(propri)o(\023)-21 b(et)o(\023)g(e)18 b(1,)h(il)h(existe) g(donc)g(deux)g(bijections)h fk(h)f fq(et)f fk(h)1404 1392 y fx(0)1435 1408 y fq(telles)h(que)g fm(\033)1670 1415 y fr(x)1698 1408 y ff(\016)10 b fk(h)1755 1392 y fq(-1)1811 1408 y fq(=)189 1465 y fm(\033)215 1472 y fr(y)242 1465 y ff(\016)g fk(h)299 1448 y fx(0)311 1442 y fq(-1)363 1465 y fq(a)o(v)o(ec)15 b fm(dom)p fq(\()p fm(v)589 1472 y fr(x)606 1465 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))h(et)h fm(dom)p fq(\()p fm(v)1023 1472 y fr(y)1040 1465 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(h)1246 1448 y fx(0)1257 1465 y fq(\).)189 1538 y(soit)j fk(g)g fq(la)g(bijection)i fk(h)10 b ff(\016)g fm(\013)g ff(\016)g fk(h)724 1521 y fx(0)735 1515 y fq(-1)773 1538 y fq(,)15 b(et)h(consid)o(\023)-21 b(erons)16 b(le)h(c)o(hamp)f fk(\(h)8 b(::)f(x\))15 b fa(c)h fk(g)p fq(.)f(mon)o(trons)g(que)189 1594 y(ce)g(c)o(hamp)g(est)g(correctemen)o(t)g(d)o(\023)-21 b(e\014ni)8 b(:)243 1684 y({)23 b fm(dom)p fq(\()p fm(v)415 1691 y fr(x)431 1684 y fq(\))13 b ff(\022)g fm(dom)p

fq(\()p fk(h)p fq(\).)243 1753 y({)23 b fm(dom)p fq(\()p fk(g)p fq(\))14 b ff(\022)h fm(dom)p fq(\()p fm(\033)629 1760 y fr(x)656 1753 y ff(\016)10 b fk(h)713 1737 y fq(-1)750 1753 y fq(\).)16 b(en)h(e\013et,)f(par)h(d)o(\023)-21 b(e\014nition)18 b fm(dom)p fq(\()p fk(g)p fq(\))c(=)h fk(h)1505 1737 y fx(0)1517 1753 y fq(\()p fm(dom)p fq(\()p fk(h)9 b ff(\016)h fm(\013)p fq(\)\))d(;)16 b(or)289 1810 y fm(dom)p fq(\()p fk(h)9 b ff(\016)h fm(\013)p fq(\))i ff(\022)h fm(dom)p fq(\()p fm(\013)p fq(\))f(=)h fm(dom)p fq(\()p fm(v)903 1817 y fr(y)920 1810 y fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\033)1128 1817 y fr(y)1145 1810 y fq(\).)c(on)i(a)f(donc)g fm(dom)p fq(\()p fk(g)p fq(\))i ff(\022)h fk(h)1622 1793 y fx(0)1633 1810 y fq(\()p fm(dom)p fq(\()p fm(\033)1781 1817 y fr(y)1798 1810 y fq(\)\))f(=)289 1866 y fm(dom)p fq(\()p fm(\033)419 1873 y fr(y)446 1866 y ff(\016)d fk(h)502 1850 y fx(0)514 1844 y fq(-1)551 1866 y fq(\))k(=)f fm(dom)p fq(\()p fm(\033)759 1873 y fr(x)786 1866 y ff(\016)e fk(h)843 1850 y fq(-1)880 1866 y fq(\).)243 1935 y({)23 b fm(img)r fq(\()p fk(g)p fq(\))10 b ff(\022)j fm(dom)p fq(\()p fm(v)613 1942 y fr(x)640 1935 y ff(\016)d fk(h)697 1919 y fq(-1)734 1935 y fq(\).)k(on)g(a)g(m)o(^)-21 b(eme)14 b(l')o(\023)-21 b(egalit)o(\023)g(e)6 b(:)14 b(en)g(e\013et,)g(par)g(d) o(\023)-21 b(e\014nition)15 b fm(img)r fq(\()p fk(g)p fq(\))10 b(=)289 1992 y fk(h)p fq(\()p fm(img)r fq(\()p fm(\013)e ff(\016)i fk(h)523 1975 y fx(0)534 1969 y fq(-1)572 1992 y fq(\)\).)f(on)i(a)e(donc)i fm(dom)p fq(\()p fm(\013)p fq(\))h(=)h fm(dom)p fq(\()p fm(v)1174 1999 y fr(y)1191 1992 y fq(\))f ff(\022)h fm(dom)p fq(\()p fk(h)1397 1975 y fx(0)1408 1992 y fq(\),)c(donc)i fm(img)r fq(\()p fm(\013)d ff(\016)i fk(h)1744 1975 y fx(0)1756 1969 y fq(-1)1793 1992 y fq(\))i(=)289 2048 y fm(img)r fq(\()p fm(\013)p fq(\),)h(c'est-\022)-23 b(a-dire)16 b fm(img)r fq(\()p fk(g)p fq(\))10 b(=)j fk(h)p fq(\()p fm(img)r fq(\()p fm(\013)p fq(\)\))e(=)i fk(h)p fq(\()p fm(dom)p fq(\()p fm(v)1323 2055 y fr(x)1339 2048 y fq(\)\))f(=)h fm(dom)p fq(\()p fm(v)1561 2055 y fr(x)1588 2048 y ff(\016)d fk(h)1645 2032 y fq(-1)1682 2048 y fq(\).)189 2138 y(le)15 b(c)o(hamp)h(est)e (\023)-21 b(egal)14 b(\022)-23 b(a)7 b(:)227 2191 y fk(\(h)h(::)g(x)o (\))15 b fa(c)h fk(g)209 2247 y fq(=)42 b(\()p fm(v)326 2254 y fr(x)354 2247 y ff(\016)9 b fk(h)410 2231 y fq(-1)458 2247 y ff(\016)h fk(g)i fq(:)g fm(\033)578 2254 y fr(x)606 2247 y ff(\016)e fk(h)663 2231 y fq(-1)700 2247 y fq(\))209 2307 y(=)42 b(\()p fm(v)326 2314 y fr(x)354 2307 y ff(\016)9 b fk(h)410 2290 y fq(-1)458 2307 y ff(\016)h fk(h)g ff(\016)f fm(\013)i ff(\016)f fk(h)654 2290 y fx(0)665 2284 y fq(-1)715 2307 y fq(:)i fm(\033)766 2314 y fr(x)794 2307 y ff(\016)e fk(h)851 2290 y fq(-1)888 2307 y fq(\))256 b(\(d)o(\023)-21 b(e\014nition)16 b(de)f fk(g)p fq(\))209 2366 y(=)42 b(\()p fm(v)326 2373 y fr(x)354 2366 y ff(\016)9 b fm(\013)i ff(\016)f fk(h)483 2350 y fx(0)494 2344 y fq(-1)544 2366 y fq(:)i fm(\033)595 2373 y fr(x)623 2366 y ff(\016)e fk(h)680 2350 y fq(-1)717 2366 y fq(\))272 b(\(car)14 b fm(dom)p fq(\()p fm(v)1226 2373 y fr(x)1243 2366 y fq(\))e ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\)\))209 2426 y(=)42 b(\()p fm(v)326 2433 y fr(y)354 2426 y ff(\016)9 b fk(h)410 2409 y fx(0)422 2403 y fq(-1)472 2426 y fq(:)j fm(\033)523 2433 y fr(x)551 2426 y ff(\016)e fk(h)608 2409 y fq(-1)645 2426 y fq(\))501 b(\(d)o(\023)-21 b(e\014nition)16 b(d')f fm(\013)p fq(\))209 2485 y(=)42 b(\()p fm(v)326

2492 y fr(y)354 2485 y ff(\016)9 b fk(h)410 2469 y fx(0)422 2463 y fq(-1)472 2485 y fq(:)j fm(\033)523 2492 y fr(y)551 2485 y ff(\016)e fk(h)608 2469 y fx(0)619 2463 y fq(-1)656 2485 y fq(\))273 b(\(de)15 b(la)h(d)o(\023)-21 b(e\014nition)16 b(de)g fk(h)f fq(et)g fk(h)1455 2469 y fx(0)1466 2485 y fq(\))209 2542 y(=)42 b fk(h)310 2525 y fx(0)329 2542 y fk(::)8 b(y)586 b fq(\(car)15 b fm(dom)p fq(\()p fm(v)1215 2549 y fr(y)1231 2542 y fq(\))e ff(\022)g fm(dom)p fq(\()p fk(h)1438 2525 y fx(0)1448 2542 y fq(\)\))p eop %%page: 24 26 24 25 bop 75 269 a fq(24)587 b fl(chapitre)16 b(2.)45 b(d)1091 257 y(\023)1085 269 y(efinition)16 b(du)f(f)o(ormalisme)h fn(pei)189 397 y fq(de)f(mani)o(\022)-21 b(ere)15 b(analogue,)g(on)g(p) q(eut)h(d)o(\023)-21 b(e\014nir)15 b fk(g)g fq(comme)g fm(\013)10 b ff(\016)g fk(h)1231 380 y fx(0)1243 375 y fq(-1)1290 397 y ff(\016)g fk(h)15 b fq(et)g(en)h(d)o(\023)-21 b(eduire)8 b(:)189 453 y fk(h)f(::)g(\(x)24 b fa(c)g fk(g\))15 b fq(=)h fk(h)544 437 y fx(0)563 453 y fk(::)7 b(y)75 545 y fo(\()p ff(\()p fo(\))24 b fq(supp)q(osons)16 b(que)h fk(\(h)7 b(::)h(x\))15 b fa(c)g fk(g)h fq(=)g fk(h)833 529 y fx(0)852 545 y fk(::)7 b(y)16 b fq(a)o(v)o(ec)f fk(h)p fq(\()p fm(dom)p fq(\()p fm(v)1215 552 y fr(x)1232 545 y fq(\)\))e(=)h fm(img)r fq(\()p fk(g)p fq(\).)f(alors,)j(nous)g(a) o(v)o(ons)189 602 y fm(v)211 609 y fr(x)238 602 y ff(\016)10 b fk(h)295 585 y fq(-1)342 602 y ff(\016)g fk(g)16 b fq(=)h fm(v)489 609 y fr(y)517 602 y ff(\016)9 b fk(h)573 585 y fx(0)585 580 y fq(-1)639 602 y fq(c'est-\022)-23 b(a-dire)16 b fm(v)894 609 y fr(y)926 602 y fq(=)f fm(v)998 609 y fr(x)1026 602 y ff(\016)10 b fk(h)1083 585 y fq(-1)1130 602 y ff(\016)g fk(g)g ff(\016)g fk(h)1254 585 y fx(0)1281 602 y fq(car)16 b fm(dom)p fq(\()p fm(v)1484 609 y fr(y)1501 602 y fq(\))e ff(\022)h fm(dom)p fq(\()p fk(h)1711 585 y fx(0)1722 602 y fq(\).)g(soit)189 658 y fm(\013)g fq(la)h(bijection)g fk(h)493 642 y fq(-1)540 658 y ff(\016)10 b fk(g)g ff(\016)g fk(h)664 642 y fx(0)676 658 y fq(.)k(il)j(vien)o(t)e fm(v)883 665 y fr(y)913 658 y fq(=)e fm(v)983 665 y fr(x)1011 658 y ff(\016)d fm(\013)p fq(.)189 733 y(p)o(ar)f(d)o(\023)-21 b(e\014nition)11 b(de)f(l'op)o(\023)-21 b(eration)10 b(g)o(\023)-21 b(eom)o(\023)g(etrique,)8 b fm(dom)p fq(\()p fk(g)p fq(\))j ff(\022)i fm(dom)p fq(\()p fm(\033)1343 740 y fr(x)1370 733 y ff(\016)d fk(h)1427 716 y fq(-1)1464 733 y fq(\))i(=)h fm(dom)p fq(\()p fm(\033)1672 740 y fr(y)1699 733 y ff(\016)d fk(h)1756 716 y fx(0)1768 710 y fq(-1)1805 733 y fq(\))i ff(\022)189 789 y fm(dom)p fq(\()p fk(h)317 773 y fx(0)327 767 y fq(-1)365 789 y fq(\))g(=)h fm(img)r fq(\()p fk(h)565 773 y fx(0)575 789 y fq(\).)f(on)g(p)q(eut)h(donc)g(conclure)g(que)g fm(img)r fq(\()p fm(\013)p fq(\))e(=)i fk(h)1384 773 y fq(-1)1421 789 y fq(\()p fm(img)r fq(\()p fk(g)8 b ff(\016)i fk(h)1626 773 y fx(0)1638 789 y fq(\)\))h(est)h(donc)188 846 y(\023)-21 b(egale)12 b(\022)-23 b(a)13 b fk(h)360 829 y fq(-1)397 846 y fq(\()p fm(img)r fq(\()p fk(g)p fq(\)\))d(=)j fk(h)655 829 y fq(-1)703 846 y ff(\016)d fk(h)p fq(\()p fm(dom)p fq(\()p fm(v)904 853 y fr(x)920 846 y fq(\)\))i(=)h fm(dom)p fq(\()p fm(v)1142 853 y fr(x)1159 846 y fq(\).)f fk(x)h fq(et)g fk(y)g fq(son)o(t)g(donc)g (faiblemen)o(t)g(\023)-21 b(equi-)189 902 y(v)m(alen)o(ts.)146 991 y fa(\003)75 1104 y fo(exemple)17 b(4)f fj(il)r(lustr)n(ation)f(de) h(la)h(c)n(ar)n(act)o(\023)-22 b(eristation)15 b(de)i(l')o(\023)-22 b(e)n(quivalenc)n(e)14 b(faible)146 1217 y fq(la)19 b(\014gure)h(2.10)e

(mon)o(tre)h(deux)i(c)o(hamps)e(faiblemen)o(t)h(\023)-21 b(equiv)m(alen)o(ts)20 b fk(x)f fq(et)h fk(y)p fq(.)f(les)h(domaines)g (de)75 1274 y(v)m(aleurs)g(de)f(ces)g(deux)g(c)o(hamps)f(son)o(t)g (supp)q(os)o(\023)-21 b(es)18 b(\023)-21 b(egaux)17 b(\(repr)o(\023)-21 b(esen)o(t)o(\023)g(es)17 b(par)h(la)h(partie)g(gris)o(\023)-21 b(ee\))17 b(mais)75 1330 y(leurs)j(domaines)g(de)g(r)o(\023)-21 b(ef)o(\023)g(erence)17 b(di\013)o(\022)-21 b(eren)o(t,)19 b(celui)i(de)e fk(x)f fq(\023)-21 b(etan)o(t)18 b(le)i(v)o(ecteur,)f (et)g(celui)i(de)f fk(y)e fq(\023)-21 b(etan)o(t)17 b(la)75 1387 y(premi)o(\022)-21 b(ere)13 b(face)g(du)g(cub)q(e.)h(p)o(our)e (mon)o(trer)g(que)i(ces)f(c)o(hamps)g(son)o(t)f(faiblemen)o(t)h(\023) -21 b(equiv)m(alen)o(ts)14 b(plusieurs)75 1443 y(p)q(ossiblit)o(\023) -21 b(es)14 b(s'o\013ren)o(t)d(p)q(our)i(le)h(c)o(hoix)g(des)f (fonctions)g fk(h)p fq(,)f fk(h)1072 1427 y fx(0)1097 1443 y fq(et)g fk(g)h fq(p)q(our)f(\023)-21 b(etablir)13 b(l')o(\023)-21 b(egalit)o(\023)g(e)12 b fk(\(h)c(::)f(x\))15 b fa(c)h fk(g)75 1500 y fq(=)g fk(h)150 1483 y fx(0)169 1500 y fk(::)7 b(y)15 b fq(de)h(la)f(caract)o(\023)-21 b(erisation.)143 1600 y({)23 b(on)13 b(p)q(eut)h(c)o(hoisir)g fk(h)f fq(=)h fk(g)f fq(=)g fm(\025)p fk(\(i\).\(i\))f fq(et)h fk(h)947 1583 y fx(0)972 1600 y fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(j)6 b(&)i(k)p fq(=)p fk(0\).\(i\))o fq(,)13 b(p)q(our)h(r)o(\023)-21 b(eduire)189 1656 y(le)14 b(domaine)g(de)g(r)o(\023)-21 b(ef)o(\023)g(erence)12 b(de)i(l'expression)g(\022)-23 b(a)13 b(droite)h(de)g(l')o(\023)-21 b(egalit)o(\023)g(e)12 b(\022)-23 b(a)14 b(celui)h(du)f(v)o(ecteur)f fk(x)g fq(dans)189 1713 y fe(z)-13 b fq(.)143 1805 y({)23 b(la)17 b(deuxi)o(\022)-21 b(eme)18 b(p)q(ossibilit)o(\023)-21 b(e)20 b(est)d(de)h(c)o(hoisir)g fk(h)f fq(=)h fm(\025)p fk(\(i\).\(i,i\))n fq(,)g fk(g)f fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p fq(=)p fk(j\).\(i,j\))m fq(,)189 1861 y(et)c fk(h)265 1845 y fx(0)291 1861 y fq(=)g fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(j)6 b(&)i(k)p fq(=)p fk(0\).\(i,i\))o fq(,)14 b(p)q(our)h(que)g(les)g(domaines)f(de)h (r)o(\023)-21 b(ef)o(\023)g(erence)13 b(des)i(deux)189 1918 y(expressions)h(soien)o(t)f(repr)o(\023)-21 b(esen)o(t)o(\023)g (es)13 b(dans)j fe(z)930 1901 y fg(2)946 1918 y fq(.)143 2010 y({)23 b(en\014n,)10 b(on)g(p)q(eut)h(c)o(hoisir)g fk(h)f fq(=)h fm(\025)p fk(\(i\).\(i,i,0\))d fq(et)i fk(g)g fq(=)g fk(h')p fq(=)g fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(j)c(&)i(k)p fq(=)p fk(0\).\(i,j,k\))189 2066 y fq(p)q(our)15 b(que)h(le)g(domaine)g(de)f(r)o(\023)-21 b(ef)o(\023)g(erence)14 b(de)i(l'expression)g(de)g(gauc)o(he)f(soit)f (\023)-21 b(egal)14 b(\022)-23 b(a)15 b(celui)i(de)f(l'ex-)189 2123 y(pression)g(de)f(droite)h(dans)f fe(z)695 2106 y fg(3)711 2123 y fq(.)75 2223 y fa(\006)75 2400 y fw(2.3.5)55 b(in)n(terpr)o(\023)-27 b(etation)18 b(fonctionnelle)146 2486 y fq(a)13 b(partir)f(des)i(d)o(\023)-21 b(e\014nitions)13 b(s)o(\023)-21 b(eman)o(tiques)13 b(des)g(op)o(\023)-21 b(erations)12 b(on)g(p)q(eut)i(pr)o(\023)-21 b(eciser)13 b(une)h fj(interpr)o(\023)-22 b(etation)75 2542 y(fonctionnel)r(le)15 b fq([)-8 b([)p fk(x)p fq(])g(])17 b(d'un)g(c)o(hamp)g(de)h(donn)o (\023)-21 b(ees)17 b fk(x)p fq(=\()p fm(v)g fq(:)e fm(\033)r fq(\),)h(c'est-\022)-23 b(a-dire)18 b(donner)g(une)f(fonction)h(qui)p eop %%page: 25 27 25 26 bop 75 269 a fl(2.3.)29 b(s)207 257 y(\023)201 269 y(emantique)1310 b fq(25)492 889 y @beginspecial @setspecial

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b(=)h([)-8 b([)p fk(x)p fq(])g(])10 b ff(\016)f fm(\033)j ff(\016)e fk(g)g ff(\016)g fm(\033)577 546 y fx(\000)p fg(1)75 703 y fo(exemple)17 b(5)f fj(interpr)o(\023)-22 b(etation)16 b(fonctionnel)r(le)e(d'un)j(champ)g(r)o(\023)-22 b(esultat)75 816 y fq(consid)o(\023)h(erons)15 b(le)h(programme)e(suiv) m(an)o(t)8 b(:)75 929 y fk(p)24 b(:)f(v)15 b ff(7!)h fk(b)75 941 y fx(8)75 979 y(>)75 991 y(<)75 1066 y(>)75 1078 y(:)137 983 y fk(v)f fq(=)g fk(align)7 b(::)h(m)137 1040 y(a)15 b fq(=)g fk(m)g fa(c)h fk(spread)137 1096 y(b)f fq(=)g fk(inc)g fa(b)h fk(a)100 1171 y(align)73 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p fq(=)p fk(0\).\(j\))100 1228 y(spread)f fq(=)h fm(\025)p fk(\(i,j\).\(0,j\)) 100 1284 y(inc)121 b fq(=)50 b fm(\025)p fk(\(a\).\(a+1\))146 1374 y fq(nous)15 b(c)o(herc)o(hons)g(l'in)o(terpr)o(\023)-21 b(etation)14 b(fonctionnelle)j(du)f(c)o(hamp)f(r)o(\023)-21 b(esultat)14 b fk(b)h fq(en)g(fonction)g(de)h(l'en-)75 1430 y(tr)o(\023)-21 b(ee)18 b fk(v)p fq(.)g(de)h(l')o(\023)-21 b(equation)18 b fk(v)h fq(=)g fk(align)7 b(::)g(m)19 b fq(et)f(de)i(la)f(s)o(\023)-21 b(eman)o(tique)18 b(du)h(c)o(hangemen) o(t)g(de)g(base,)f(nous)75 1486 y(tirons)i(les)g(\023)-21 b(egalit)o(\023)g(es)6 b(:)20 b fm(v)489 1493 y fw(v)539 1486 y fq(=)h fm(v)617 1493 y fw(m)664 1486 y ff(\016)10 b fk(align)816 1469 y fx(\000)p fg(1)881 1486 y fq(et)20 b fm(\033)965 1493 y fw(v)1015 1486 y fq(=)h fm(\033)1097 1493 y fw(m)1144 1486 y ff(\016)10 b fk(align)1296 1469 y fx(\000)p fg(1)1341 1486 y fq(.)19 b(comme)h(il)h(n'y)f(a)g(pas)g(de) 75 1543 y(c)o(hangemen)o(t)15 b(de)h(base)f(en)o(tre)g(les)h(c)o(hamps) f fk(a)p fq(,)g fk(b)g fq(et)g fk(m)p fq(,)f(nous)i(a)o(v)o(ons)e (aussi)h fm(\033)1395 1550 y fw(a)1435 1543 y fq(=)e fm(\033)1509 1550 y fw(b)1550 1543 y fq(=)g fm(\033)1624 1550 y fw(m)1661 1543 y fq(.)i(de)g(l\022)-23 b(a,)766 1621 y fm(\033)792 1628 y fw(b)862 1621 y fq(=)42 b fm(\033)965 1628 y fw(v)1004 1621 y ff(\016)9 b fk(align)75 1700 y fq(nous)15 b(a)o(v)o(ons)f(d'autre)h(part)7 b(:)342 1775 y([)-8 b([)p fk(b)p fq(])g(])41 b(=)h([)-8 b([)p fk(inc)14 b fa(b)i fk(a)p fq(])-8 b(])12 b(=)h fk(inc)d ff(\016)g fq([)-8 b([)p fk(a)p fq(])g(])443 1831 y(=)42 b fk(inc)10 b ff(\016)g fq([)-8 b([)p fk(m)14 b fa(c)i fk(spread)o fq(])-8 b(])12 b(=)h fk(inc)d ff(\016)g fq([)-8 b([)p fk(m)p fq(])g(])9 b ff(\016)h fm(\033)1206 1838 y fw(m)1253 1831 y ff(\016)g fk(spread)f ff(\016)h fm(\033)1498 1838 y fw(m)1535 1815 y fx(\000)p fg(1)443 1888 y fq(=)42 b fk(inc)10 b ff(\016)g fm(v)657 1895 y fw(m)704 1888 y ff(\016)f fm(\033)762 1895 y fw(m)799 1871 y fx(\000)p fg(1)854 1888 y ff(\016)h fm(\033)913 1895 y fw(m)960 1888 y ff(\016)f fk(spread)h ff(\016)g fm(\033)1205 1895 y fw(m)1241 1871 y fx(\000)p fg(1)443 1944 y fq(=)42 b fk(inc)10 b ff(\016)g fm(v)657 1951 y fw(m)704 1944 y ff(\016)f fk(spread)h ff(\016)g fm(\033)949 1951 y fw(m)985 1928 y fx(\000)p fg(1)443 2001 y fq(=)42 b fk(inc)10 b ff(\016)g fm(v)657 2008 y fw(v)696 2001 y ff(\016)f fk(align)h ff(\016)g fk(spread)f ff(\016)h fm(\033)1103 2008 y fw(m)1140 1984 y fx(\000)p fg(1)75 2075 y fq(c'est-\022)-23 b(a-dire)8 b(:)502 2132 y fm(v)524 2139 y fw(b)563 2132 y ff(\016)i fm(\033)622 2139 y fw(b)650 2113 y fx(\000)p fg(1)707 2132 y fq(=)j fk(inc)d ff(\016)g fm(v)892 2139 y fw(v)931 2132 y ff(\016)g fk(align)f ff(\016)h fk(spread)f ff(\016)h fm(\033)1338 2139 y fw(m)1375 2113 y fx(\000)p fg(1)75 2204 y fq(d'o)q(\022)-24 b(u)7 b(:)333 b fm(v)536 2211 y fw(b)578 2204 y fq(=)13 b fk(inc)c ff(\016)h fm(v)762

2211 y fw(v)801 2204 y ff(\016)g fk(align)f ff(\016)h fk(spread)75 2316 y fq(nous)15 b(p)q(ouv)o(ons)g(ainsi)g(\023)-21 b(ev)m(aluer)15 b(la)g(v)m(aleur)h(asso)q(ci)o(\023)-21 b(ee)14 b(\022)-23 b(a)15 b(tout)f(p)q(oin)o(t)h fm(v)1260 2323 y fw(b)1289 2316 y fq(\()p fm(i;)8 b(j)s fq(\))k(en)j(calculan)o (t)h(\()p fk(inc)24 b ff(\016)10 b fm(v)1818 2323 y fw(v)85 2373 y ff(\016)g fm(\025)p fk(\(i,j\).\(j\))n fq(\))21 b(\()p fm(i;)8 b(j)s fq(\).)18 b(l'in)o(terpr)o(\023)-21 b(etation)20 b(fonctionnelle)j(est)e(la)g(base)g(du)h(traducteur)e fn(pei)h fq(v)o(ers)75 2429 y fn(caml)p fq([60)o(])15 b(inclus)j(dans)e(l'en)o(vironnemen)o(t)h fn(pei)p fq(,)e(don)o(t)h (nous)g(parlerons)g(en)g(d)o(\023)-21 b(etail)16 b(dans)g(la)g(section) 75 2486 y(3.5.)29 b fa(\006)p eop %%page: 27 29 27 28 bop 1801 269 a fq(27)75 709 y fi(chapitre)32 b(3)75 918 y fp(ra\016nemen)m(t)37 b(et)i(abstraction)75 1159 y ft(3.1)66 b(d)o(\023)-32 b(e\014nition)23 b(g)o(\023)-32 b(en)o(\023)g(erale)20 b(du)j(ra\016nemen)n(t)146 1262 y fq(le)16 b(ra\016nemen)o(t)g(est)f(un)i(pro)q(c)o(\023)-21 b(ed)o(\023)g(e)15 b(classique)i(de)f(construction)h(ou)e(de)i (transformation)d(de)j(pro-)75 1319 y(grammes)d([2,)h(44)o(],)g(qui)h (consiste)g(\022)-23 b(a)15 b(obtenir)h(un)f(programme)g(satisfaisan)o (t)f(toute)h(sp)o(\023)-21 b(eci\014cation)16 b(que)75 1375 y(satisfaisait)h(un)h(programme)e(initial.)j(la)e(d)o(\023)-21 b(e\014nition)18 b(du)g(ra\016nemen)o(t)e(est)h(comm)o(un)o(\023)-21 b(emen)o(t)15 b(\023)-21 b(etablie)75 1432 y(de)16 b(la)f(mani)o(\022) -21 b(ere)15 b(suiv)m(an)o(te,)h(\022)-23 b(a)14 b(partir)i(d'une)f (propri)o(\023)-21 b(et)o(\023)g(e)14 b(de)i(satisfaction)f(c)o (hoisie,)h(not)o(\023)-21 b(ee)14 b fl(sat)i fq(:)75 1546 y fo(d)o(\023)-25 b(e\014nition)22 b(6)37 b fq(soien)o(t)18 b(les)h(programmes)e fm(p)24 b fq(et)18 b fm(p)999 1529 y fx(0)1011 1546 y fq(,)g(on)g(dit)g(que)h fm(p)24 b fq(est)18 b(ra\016n)o(\023)-21 b(e)17 b(par)h fm(p)1657 1529 y fx(0)1669 1546 y fq(,)g(et)f(nous)75 1602 y(notons)e fm(p)k ff(v)13 b fm(p)353 1586 y fx(0)365 1602 y fq(,)i(si)g(et)g (seulemen)o(t)i(si,)e(p)q(our)g(toute)g(sp)o(\023)-21 b(eci\014cation)16 b fm(s)s fq(,)768 1700 y fm(p)21 b fl(sat)16 b fm(s)f ff(\))e fm(p)1025 1684 y fx(0)1053 1700 y fl(sat)i fm(s)146 1798 y fq(si)21 b(la)h(propri)o(\023)-21 b(et)o(\023)g(e)19 b(c)o(hoisie)j(est)f(la)g(correction)g(totale)g(par) f(exemple,)i(on)f(utilisera)h(les)g fj(we)n(akest)75 1854 y(pr)n(e)n(c)n(onditions)c fq(de)i(dijkstra,)f(not)o(\023)-21 b(ees)18 b fm(w)q(p)p fq(\()p fm(p)q(;)8 b(post)p fq(\),)19 b(o)q(\022)-24 b(u)19 b fm(p)26 b fq(est)20 b(un)f(\023)-21 b(enonc)o(\023)g(e)18 b(et)h fm(post)i fq(un)f(pr)o(\023)-21 b(edicat)7 b(:)75 1911 y fm(w)q(p)p fq(\()p fm(p)q(;)h(post)p fq(\))16 b(est)h(le)g(plus)h(faible)g(pr)o(\023)-21 b(edicat)17 b(p)q(our)g(lequel)h(l'ex)o(\023)-21 b(ecution)17 b(de)h fm(p)23 b fq(termine)17 b(dans)g(un)f(\023)-21 b(etat)75 1967 y(satisfaisan)o(t)15 b fm(post)p fq(.)g(la)g(d)o(\023)-21 b(e\014nition)16 b(g)o(\023)-21 b(en)o(\023)g(erale)14 b(du)i(ra\016nemen)o(t)f(devien)o(t)h(alors)7 b(:)587 2065 y fm(p)19 b ff(v)13 b fm(p)718 2049 y fx(0)745 2065 y fq(ssi)j ff(8)p fm(r)g fq(\()p fm(w)q(p)p fq(\()p fm(p)q(;)8 b(r)q fq(\))j ff(\))i fm(w)q(p)p fq(\()p fm(p)1234 2049 y fx(0)1245 2065 y fm(;)8 b(r)q fq(\)\).)146 2163 y(la)j(notion)g(de)g (ra\016nemen)o(t)g(est)g(fr)o(\023)-21 b(equemmen)o(t)10 b(utilis)o(\023)-21 b(ee)12 b(dans)f(un)h(pro)q(c)o(\023)-21 b(ed)o(\023)g(e)10 b(con)o(tin)o(u)h(de)g(construc-)75 2219 y(tion)k(d')o(\023)-21 b(enonc)o(\023)g(es)14 b(successifs,)i

(depuis)g(une)g(sp)o(\023)-21 b(eci\014cation)16 b(jusqu'\022)-23 b(a)15 b(un)g(programme)f(ex)o(\023)-21 b(ecutable,)15 b(pre-)75 2276 y(nan)o(t)20 b(en)i(compte)e(tel)h(ou)g(tel)g(mo)q(d)o (\022)-21 b(ele)21 b(d'ex)o(\023)-21 b(ecution.)21 b(elle)h(d)o(\023) -21 b(e\014nit)21 b(une)g(relation)g(d'ordre)g(partiel)75 2332 y(en)o(tre)f(les)g(\023)-21 b(enonc)o(\023)g(es,)19 b(dite)i fj(r)n(elation)g(de)g(r)n(a\016nement)p fq(.)e(le)i (ra\016nemen)o(t)f(est)g(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t)18 b(\023)-21 b(etendu)75 2388 y(aux)18 b(sp)o(\023)-21 b(eci\014cations)20 b(elles-m)o(^)-21 b(emes,)18 b(a)o(v)o(ec,)g(p)q (our)h(des)g(sp)o(\023)-21 b(eci\014cations)19 b(de)g(t)o(yp)q(e)f([)p fm(pr)q(e)p fq(,)p fm(post)p fq(],)g(les)h(deux)75 2445 y(cas)c(particuliers)9 b(:)189 2542 y([)p fm(p;)f(q)r fq(])j ff(v)h fq([)p fm(p)375 2526 y fx(0)387 2542 y fm(;)c(q)r fq(])14 b(si)h fm(p)e ff(\))g fm(p)619 2526 y fx(0)645 2542 y fq(\(a\013aiblissemen)o(t)j(des)g(pr)o(\023)-21 b(econditions\))15 b(et)p eop %%page: 28 30 28 29 bop 75 269 a fq(28)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)189 397 y fq([)p fm(p;)8 b(q)r fq(])j ff(v)h fq([)p fm(p;)c(q)418 380 y fx(0)429 397 y fq(])15 b(si)g fm(q)524 380 y fx(0)548 397 y ff(\))f fm(q)j fq(\(renforcemen)o(t)d(des)i(p)q(ostconditions\).) 146 506 y(knapp)d([31)o(])f(pr)o(\023)-21 b(esen)o(te)11 b(la)i(d)o(\023)-21 b(e\014nition)13 b(du)g(ra\016nemen)o(t)f(de)h(sp)o (\023)-21 b(eci\014cations)12 b(sous)h(une)g(autre)f(forme,)75 562 y(p)q(our)19 b(l'appliquer)h(aux)f(programmes)e fn(unity)p fq([10)n(].)h(bien)i(qu')o(\023)-21 b(equiv)m(alen)o(te,)19 b(la)g(form)o(ulation)f(en)h(est)75 619 y(di\013)o(\023)-21 b(eren)o(te)11 b(du)i(fait)e(de)i(la)f(forme)f(des)h(sp)o(\023)-21 b(eci\014cations)13 b(utilis)o(\023)-21 b(ees)13 b(\(en)f(logique)h(mo) q(dale\))8 b(:)j(la)h(s)o(\023)-21 b(eman)o(tique)75 675 y(d')p fn(unity)13 b fq(est)h(d)o(\023)-21 b(e\014nie)15 b(sous)f(la)h(forme)e(d'un)i(syst)o(\022)-21 b(eme)13 b(de)h(transitions)h(d')o(\023)-21 b(etats)12 b(et)i(la)g(logique)i (utilis)o(\023)-21 b(ee)75 732 y(p)q(ermet)16 b(d'exprimer)h(des)g (propri)o(\023)-21 b(et)o(\023)g(es)15 b(sur)h(ces)f(\023)-21 b(etats.)14 b(soien)o(t)j fm(s)g fq(et)f fm(s)1283 715 y fx(0)1311 732 y fq(deux)g(\023)-21 b(etats,)14 b(on)i(note)g fm(s)p fq(\()p fm(p)p fq(\))g(et)75 788 y fm(s)96 772 y fx(0)108 788 y fq(\()p fm(q)r fq(\))d(le)h(fait)f(que)h(les)f(\023) -21 b(etats)11 b(satisfon)o(t)i(les)h(propri)o(\023)-21 b(et)o(\023)g(es)11 b fm(p)j fq(et)f fm(q)i fq(resp)q(ectiv)o(emen)o (t.)f(in)o(tuitiv)o(emen)o(t,)f(dans)75 844 y(ce)j(t)o(yp)q(e)f (d'appro)q(c)o(he,)g(une)h(sp)o(\023)-21 b(eci\014cation)16 b fm(s)h fq(a)e(la)h(forme)e(suiv)m(an)o(te)9 b(:)663 969 y fm(s)15 b fq(:)d ff(8)p fm(s)p fq(\()p fm(p)p fq(\))j ff(9)p fm(s)897 950 y fx(0)910 969 y fq(\()p fm(q)r fq(\))p fm(:)p fq(\()p fm(s)c fa( )i fm(s)1110 950 y fx(0)1122 969 y fq(\))60 b(\(1\))75 1065 y(o)q(\022)-24 b(u)20 b(la)h(notation)f fm(s)h fa( )g fm(s)515 1049 y fx(0)547 1065 y fq(signi\014e)h(que)f fm(s)815 1049 y fx(0)847 1065 y fq(est)f(un)g(\023)-21 b(etat)18 b(\\atteignable")j(\022)-23 b(a)19 b(partir)i(de)f(l')o(\023)-21 b(etat)19 b fm(s)p fq(.)h(le)75 1122 y(ra\016nemen)o(t)15 b(est)g(alors)g(d)o(\023)-21 b(e\014ni)15 b(de)h(la)g(mani)o(\022)-21 b(ere)14 b(suiv)m(an)o(te)9 b(:)75 1238 y fo(d)o(\023)-25 b(e\014nition)19 b(7)c fq(soien)o(t)h(deux)g(sp)o(\023)-21 b(eci\014cations)16 b fm(s)h fq(et)e fm(s)1013 1222 y fx(0)1040 1238 y fq(de)g(la)h(forme)e (\(1\),)g fm(s)h ff(v)e fm(s)1486 1222 y fx(0)1513 1238

y fq(ssi)i fm(s)1607 1222 y fx(0)1631 1238 y ff(\))e fm(s)75 1403 y ft(3.2)66 b(ra\016nemen)n(t)23 b(des)e(\023)-32 b(enonc)o(\023)g(es)19 b fs(pei)75 1513 y fw(3.2.1)55 b(d)o(\023)-27 b(e\014nitions)146 1606 y fq(la)14 b(d)o(\023)-21 b(e\014nition)15 b(du)g(ra\016nemen)o(t)g(des)e(\023)-21 b(enonc)o(\023)g(es)14 b fn(pei)g fq(s'inspire)i(de)f(la)f(d)o(\023)-21 b(e\014nition)16 b(de)f(knapp,)f(mais)75 1663 y(en)i(restan)o(t)e(dans) i(la)f(logique)i(du)e(premier)h(ordre.)f(consid)o(\023)-21 b(erons)15 b(un)g(\023)-21 b(enonc)o(\023)g(e)14 b fn(pei)p fq(,)h(don)o(t)g(les)h(en)o(tr)o(\023)-21 b(ees)75 1719 y(et)15 b(les)h(sorties)f(son)o(t)f fk(d)h fq(et)h fk(r)f fq(resp)q(ectiv)o(emen)o(t)8 b(:)75 1780 y fk(p)24 b(:)f(d)15 b ff(7!)h fk(r)75 1785 y fx(\()164 1829 y fm(s)192 1836 y fg(1)164 1885 y fm(s)192 1892 y fg(2)146 1974 y fq(nous)k(p)q(ouv)o (ons)h(consid)o(\023)-21 b(erer,)20 b(parmi)h(les)f(\023)-21 b(equations)20 b(du)h(syst)o(\022)-21 b(eme,)18 b(deux)k(t)o(yp)q(es)e (d')o(\023)-21 b(equations)7 b(:)75 2030 y(celles)14 b(qui)g(p)q(orten)o(t)e(uniquemen)o(t)i(sur)e(des)h(en)o(tr)o(\023)-21 b(ees,)12 b(et)g(celles)i(qui)g(comp)q(orten)o(t)e(des)h(c)o(hamps)f (de)h(don-)75 2087 y(n)o(\023)-21 b(ees)13 b(in)o(term)o(\023)-21 b(ediaires)14 b(ou)f(des)h(sorties.)f(nous)h(app)q(elons)g(les)g(premi) o(\022)-21 b(eres)14 b fj(pr)o(\023)-22 b(e-)o(\023)g(e)n(quations)p fq(,)13 b(les)h(secondes)75 2143 y fj(p)n(ost-)o(\023)-22 b(e)n(quations)p fq(.)18 b(dans)h(la)h(structure)f(sc)o(h)o(\023)-21 b(ematique)19 b(de)g(l')o(\023)-21 b(enonc)o(\023)g(e)19 b fk(p)g fq(ci-dessus,)h(les)g(pr)o(\023)-21 b(e-)o(\023)g(equations)75 2200 y(appartiennen)o(t)14 b(\022)-23 b(a)13 b(l'ensem)o(ble)i(d')o (\023)-21 b(equations)12 b fm(s)885 2207 y fg(1)904 2200 y fq(,)h(les)h(p)q(ost-)o(\023)-21 b(equations)12 b(\022)-23 b(a)13 b(l'ensem)o(ble)i(d')o(\023)-21 b(equations)13 b fm(s)1816 2207 y fg(2)1834 2200 y fq(.)75 2316 y fo(d)o(\023)-25 b(e\014nition)19 b(8)31 b fq(\(pr)o(\023)-21 b(e-)o(\023)g(equation)13 b(et)i(p)q(ost-)o(\023)-21 b(equation\))75 2373 y(une)14 b fj(pr)o(\023)-22 b(e-)o(\023)g(e)n(quation)13 b fq(d'un)f(\023)-21 b(enonc)o(\023)g(e)12 b fk(p)h fq(est)g(une)g(\023)-21 b(equation)12 b(lian)o(t)i(uniquemen)o(t)g(des)f(c)o(hamps)g(de)h(donn) o(\023)-21 b(ees)75 2429 y(en)20 b(en)o(tr)o(\023)-21 b(ee)19 b(de)h fk(p)p fq(.)f(une)h(\023)-21 b(equation)19 b(d'un)g(\023)-21 b(enonc)o(\023)g(e)19 b fk(p)g fq(est)h(une)g fj(p)n(ost-)o(\023)-22 b(e)n(quation)20 b fq(si)g(elle)i(n'est)d(pas)h (une)75 2486 y(pr)o(\023)-21 b(e-)o(\023)g(equation.)p eop %%page: 29 31 29 30 bop 75 269 a fl(3.2.)29 b(raffinement)15 b(des)651 257 y(\023)645 269 y(enonc)818 257 y(\023)812 269 y(es)h fn(pei)831 b fq(29)146 397 y(nous)19 b(p)q(ouv)o(ons)h(main)o(tenan)o (t)f(pr)o(\023)-21 b(eciser)20 b(ce)g(que)g(sp)o(\023)-21 b(eci\014e)21 b(un)e(\023)-21 b(enonc)o(\023)g(e)18 b fn(pei)p fq(.)h(nous)h(asso)q(cions)g(\022)-23 b(a)75 453 y(l')o(\023)i(enonc)o(\023)g(e)16 b fk(p)p fq(,)h(une)g(propri)o (\023)-21 b(et)o(\023)g(e)16 b fm(sat)p fq(\()p fk(p)p fq(\),)h(don)o(t)f(nous)i(dirons)f(qu'elle)i(est)e(satisfaite)g(par)g fk(p)p fq(.)f(cette)h(pro-)75 510 y(pri)o(\023)-21 b(et)o(\023)g(e)14 b(pr)o(\023)-21 b(ecise)16 b(que)f(p)q(our)h(tout)f(c)o(hamp)g(de)h (donn)o(\023)-21 b(ees)15 b fk(d)g fq(v)o(\023)-21 b(eri\014an)o(t)15 b fm(s)1270 517 y fg(1)1288 510 y fq(,)g(la)h(solution)g(est)f(form)o (\023)-21 b(ee)14 b(d'un)75 566 y(c)o(hamp)h(de)h(donn)o(\023)-21 b(ees)15 b fk(r)g fq(v)o(\023)-21 b(eri\014an)o(t)14 b fm(s)699 573 y fg(2)718 566 y fq(.)g(ce)i(que)f(nous)h(noterons)e (par)8 b(:)800 735 y fm(sat)p fq(\()p fk(p)p fq(\))k(:)g

fm(s)986 742 y fg(1)1018 735 y ff(\))h fm(s)1104 742 y fg(2)75 873 y fq(p)o(ar)h(exemple,)i(\022)-23 b(a)15 b(l')o(\023)-21 b(enonc)o(\023)g(e)15 b(suiv)m(an)o(t)8 b(:)75 1001 y fk(p1)24 b(:)f(a)15 b ff(7!)h fk(b)75 1015 y fx(\()133 1059 y fk(a)f fq(=)h fk(matrix)7 b(::)g(a)133 1116 y(b)15 b fq(=)h fk(a)f fa(c)g fk(transp)96 1182 y(matrix)40 b fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))96 1238 y(transp)e fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(j,i\))75 1331 y fq(nous)15 b(asso)q(cions)h (la)f(propri)o(\023)-21 b(et)o(\023)g(e)6 b(:)313 1474 y(\()p fm(v)353 1481 y fr(a)385 1474 y fq(=)16 b fm(v)458 1481 y fr(a)485 1474 y ff(\016)10 b fk(matrix)661 1456 y fq(-1)714 1474 y ff(^)15 b fm(\033)785 1481 y fr(a)818 1474 y fq(=)h fm(\033)895 1481 y fr(a)922 1474 y ff(\016)10 b fk(matrix)1098 1456 y fq(-1)1151 1474 y ff(^)15 b fm(dom)p fq(\()p fm(v)1322 1481 y fr(a)1339 1474 y fq(\))g ff(\022)g fm(dom)p fq(\()p fk(matrix)p fq(\)\))218 1554 y ff(\))50 b fq(\()p fm(v)353 1561 y fr(b)385 1554 y fq(=)16 b fm(v)458 1561 y fr(a)485 1554 y ff(\016)10 b fk(transp)15 b ff(^)g fm(\033)748 1561 y fr(b)765 1554 y fq(=)p fm(\033)826 1561 y fr(a)844 1554 y fq(\))g ff(^)313 1610 y fm(dom)p fq(\()p fk(transp)p fq(\))e ff(\022)j fm(dom)p fq(\()p fm(\033)773 1617 y fr(a)789 1610 y fq(\))f ff(^)h fm(img)r fq(\()p fk(transp)p fq(\))c ff(\022)k fm(dom)p fq(\()p fm(v)1317 1617 y fr(a)1334 1610 y fq(\)\))75 1754 y(ce)g(que)f(nous)g (noterons)g(sous)g(la)h(forme)7 b(:)582 1922 y(\()p fk(a)15 b fq(=)g fk(matrix)7 b(::)g(a)p fq(\))15 b ff(\))g fq(\()p fk(b)p fq(=)p fk(a)g fa(c)h fk(transp)o fq(\))75 2076 y fo(d)o(\023)-25 b(e\014nition)19 b(9)31 b fq(\(ra\016nemen)o(t)14 b(d')o(\023)-21 b(enonc)o(\023)g(es\))75 2132 y(nous)19 b(disons)h(que)g fk(p)g fq(est)f fj(r)n(a\016n)o(\023)-22 b(e)19 b fq(par)g fk(p)798 2116 y fx(0)809 2132 y fq(,)g(ou)h(que)g fk(p)f fj(abstr)n(ait)g fk(p)1231 2116 y fx(0)1243 2132 y fq(,)g(et)g(nous)g(notons)g fk(p)h ff(v)f fk(p)1715 2116 y fx(0)1727 2132 y fq(,)g(si)h(et)75 2189 y(seulemen)o(t)c(si)g fm(sat)p fq(\()p fk(p)435 2172 y fx(0)447 2189 y fq(\))489 2164 y fx(\021)480 2189 y ff(\))p fm(sat)p fq(\()p fk(p)p fq(\).)75 2316 y(la)g(notation)335 2291 y fx(\021)325 2316 y ff(\))h fq(est)f(une)h(relation)g(d'implication)h(p)q(ermettan)o (t)e(de)g(prendre)h(en)g(compte)f(l')o(\023)-21 b(equiv)m(a-)75 2373 y(lence)14 b(des)g(c)o(hamps)f(de)g(donn)o(\023)-21 b(ees,)13 b(don)o(t)f(la)i(d)o(\023)-21 b(e\014nition)14 b(est)f(la)g(suiv)m(an)o(te.)g(dans)g(cette)g(d)o(\023)-21 b(e\014nition,)13 b(nous)75 2429 y(notons)h fm(s)s fq([)p fk(x)p ff(n)p fk(x)336 2413 y fx(0)346 2429 y fq(])g(le)h(syst)o(\022) -21 b(eme)13 b(d')o(\023)-21 b(equations)14 b(don)o(t)g(une)h(o)q (ccurrence)g(de)g fk(x)f fq(a)f(\023)-21 b(et)o(\023)g(e)12 b(substitu)o(\023)-21 b(ee)14 b(par)g fk(x)1770 2413 y fx(0)1782 2429 y fq(,)g(et)75 2486 y fm(s)s fq([)p fk(x)p ff(n)p fk(x)190 2469 y fx(0)200 2486 y fq(])213 2469 y fw(?)247 2486 y fq(quand)i(toutes)e(les)i(o)q(ccurrences)g(de)g fk(x)f fq(on)o(t)f(\023)-21 b(et)o(\023)g(e)13 b(remplac)o(\023)-21 b(ees)15 b(par)g fk(x)1419 2469 y fx(0)1446 2486 y fq(dans)g fm(s)s fq(.)p eop %%page: 30 32 30 31 bop 75 269 a fq(30)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)445 1095 y @beginspecial @setspecial %%begindocument: raffinement.pstex

/$f2psdict 200 dict def $f2psdict begin $f2psdict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def /drawsplinesection { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def /xa x1 x2 x1 sub 0.666667 mul add def /ya y1 y2 y1 sub 0.666667 mul add def /xb x3 x2 x3 sub 0.666667 mul add def /yb y3 y2 y3 sub 0.666667 mul add def x1 y1 lineto xa ya xb yb x3 y3 curveto } def end /$f2psbegin {$f2psdict begin /$f2psenteredstate save def} def /$f2psend {$f2psenteredstate restore end} def $f2psbegin 0 setlinecap 0 setlinejoin -18.0 198.0 translate 0.900 -0.900 scale 0.500 setlinewidth % polyline n 177 57 m 230 84 l 230 164 l 177 137 l 177 57 l gs col-1 s gr % polyline n 187 54 m 240 81 l 240 161 l 187 134 l 187 54 l gs col-1 s gr % polyline n 230 84 m 240 81 l gs col-1 s gr % polyline n 177 57 m 187 54 l gs col-1 s gr

% polyline n 230 164 m 240 161 l gs col-1 s gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 240 81 m 294 64 l gs col-1 s gr [] 0 setdash 0 setlinecap 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 187 54 m 250 34 l gs col-1 s gr [] 0 setdash 0 setlinecap 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 240 161 m 294 141 l gs col-1 s gr [] 0 setdash 0 setlinecap % polyline n 19 61 m 72 88 l 72 168 l 19 141 l 19 61 l gs col-1 s gr % polyline n 63 59 m 116 86 l 116 166 l 63 139 l 63 59 l gs col-1 s gr n 77.000 201.000 m 69.000 199.000 l 77.000 197.000 l gs 2 setlinejoin col-1 s gr % polyline n 69 199 m 119 199 l gs col-1 s gr n 111.000 197.000 m 119.000 199.000 l 111.000 201.000 l gs 2 setlinejoin col-1 s gr n 127.000 201.000 m 119.000 199.000 l 127.000 197.000 l gs 2 setlinejoin col-1 s gr % polyline n 119 199 m 224 199 l gs col-1 s gr n 216.000 197.000 m 224.000 199.000 l 216.000 201.000 l gs 2 setlinejoin col-1 s gr % open spline n 194.000 94.000 m 169.000 74.000 l 169.000 74.000 144.000 54.000 124.000 66.500 drawsplinesection 104.000 79.000 l gs col-1 s gr n 111.844 76.456 m 104.000 79.000 l 109.724 73.064 l gs 2 setlinejoin col-1 s gr % open spline n 119.000 144.000 m 136.500 151.500 l 136.500 151.500 154.000 159.000 179.000 139.000 drawsplinesection 204.000 119.000 l gs col-1 s gr n 196.504 122.436 m 204.000 119.000 l 199.002 125.559 l gs 2 setlinejoin col-1 s gr $f2psend %%enddocument @endspecial 1326 945 a fq(x)558 963 y(x)839 476 y(h)857 464 y(-1)914 814 y(h)764 963 y(x)782 952 y fk(0)652 1095 y fs(ra\016nemen)o(t)123 b(substitution)340 1238 y fn(fig.)15 b fq(3.1)g(-)30 b fj(r)n(a\016nement)16 b(p)n(ar)g(substitution)g(de)h (champs)e(\023)-22 b(e)n(quivalents)146 1417 y fo(d)o(\023)d (e\014nition)19 b(10)30 b fq(\(implication)17 b(mo)q(dulo)f(l')o(\023) -21 b(equiv)m(alence\))146 1473 y(soit)18 b fk(p1)g fq(et)g fk(p2)f fq(deux)h(\023)-21 b(enonc)o(\023)g(es)17 b(et)h fm(s)i fq(le)f(syst)o(\022)-21 b(eme)17 b(d')o(\023)-21 b(equations)17 b(de)i fk(p1)p fq(.)e(nous)h(disons)h(que)f fm(sat)p fq(\()p fk(p2)p fq(\))146 1530 y fj(implique)j fm(sat)p fq(\()p fk(p1)p fq(\))g fj(mo)n(dulo)h(l')o(\023)-22 b(e)n(quivalenc)n(e)19 b fq(\(forte)i(ou)g(faible\))h(de)g(c)o(hamps)f (de)g(donn)o(\023)-21 b(ees,)21 b(et)g(nous)146 1586 y(notons)14 b fm(sat)p fq(\()p fk(p2)p fq(\))462 1561 y fx(\021)452 1586 y ff(\))q fm(sat)p fq(\()p fk(p1)p fq(\))h(si)g(et)g(seulemen)o(t)i(si)e fm(sat)p fq(\()p fk(p2)p fq(\))g ff(\))h fm(sat)p fq(\()p fk(p1')p fq(\))f(est)g(v)o

(\023)-21 b(eri\014)o(\023)g(ee,)14 b(a)o(v)o(ec)7 b(:)201 1680 y(1.)22 b(ou)12 b(bien)i fk(p1')d fq(=)i fk(p1)p ff(f)p fm(s)s fq([)p fk(x)p ff(n)p fm(e)p fq(\()p fk(x)p fq(\)])802 1663 y fw(?)818 1680 y ff(g)f fq(o)q(\022)-24 b(u)12 b fm(e)p fq(\()p fk(x)p fq(\))f(est)h(une)h(expression)g(d)o (\023)-21 b(ep)q(endan)o(t)12 b(de)g fk(x)p fq(,)g(satisfaisan)o(t)259 1736 y(la)20 b(propri)o(\023)-21 b(et)o(\023)g(e)18 b(suiv)m(an)o(te)9 b(:)19 b(p)q(our)g(tout)g(c)o(hamp)h(de)g(donn)o(\023)-21 b(ees)19 b(d)o(\023)-21 b(enot)o(\023)g(e)18 b(par)h(cet)h fk(x)p fq(,)f(il)i(existe)f(un)259 1793 y(c)o(hamp)f(de)f(donn)o(\023) -21 b(ees,)18 b(disons)h fk(x)823 1776 y fx(0)835 1793 y fq(,)f(qui)h(lui)h(est)e(fortemen)o(t)f(\023)-21 b(equiv)m(alen)o(t,) 18 b(tel)h(que)g(l'expression)259 1849 y fm(e)p fq(\()p fk(x)322 1833 y fx(0)334 1849 y fq(\))14 b(d)o(\023)-21 b(enote)15 b(le)h(m)o(^)-21 b(eme)14 b(c)o(hamp)h(de)h(donn)o(\023)-21 b(ees)15 b(que)h fk(x)p fq(,)201 1943 y(2.)22 b(ou)14 b(bien)h fk(p1')e fq(=)g fk(p1)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fm(e)p fq(\()p fk(e)p fq(\)])p ff(g)e fq(o)q(\022)-24 b(u)13 b fm(e)p fq(\()p fk(e)p fq(\))g(est)h(une)g(expression)g(d)o (\023)-21 b(ep)q(endan)o(t)14 b(de)g fk(e)p fq(,)f(satisfaisan)o(t)259 1999 y(la)k(propri)o(\023)-21 b(et)o(\023)g(e)15 b(suiv)m(an)o(te)8 b(:)16 b(p)q(our)h(tout)e(c)o(hamp)i(de)g(donn)o(\023)-21 b(ee)16 b(d)o(\023)-21 b(enot)o(\023)g(e)14 b(par)j(cette)f(expression) h fk(e)p fq(,)f(il)259 2056 y(existe)i(un)f(c)o(hamp)f(de)i(donn)o (\023)-21 b(ees,)16 b(disons)h fk(e)1013 2039 y fx(0)1025 2056 y fq(,)f(qui)i(lui)g(est)e(faiblemen)o(t)h(\023)-21 b(equiv)m(alen)o(t,)17 b(et)g(tel)g(que)259 2112 y(l'expression)g fm(e)p fq(\()p fk(e)567 2096 y fx(0)578 2112 y fq(\))e(d)o(\023)-21 b(enote)14 b(le)i(m)o(^)-21 b(eme)14 b(c)o(hamp)i(de)f(donn)o(\023)-21 b(ees)15 b(que)h fk(e)p fq(.)146 2200 y(d'apr)o(\022)-21 b(es)18 b(la)j(d)o(\023)-21 b(e\014nition)22 b(10.1,)d(on)h(p)q(eut)h (\023)-21 b(evidemmen)o(t)20 b(remplacer)h(les)h(o)q(ccurrences)f(de)g fk(x)g fq(par)75 2257 y(l'expression)16 b fm(e)p fq(\()p fk(x)382 2240 y fx(0)394 2257 y fq(\))f(qui)h(lui)g(est)e(\023)-21 b(egale.)15 b(in)o(tuitiv)o(emen)o(t,)g(le)h(ra\016nemen)o(t)f(consite) h(alors)f(\022)-23 b(a)15 b(substituer)75 2313 y fm(e)p fq(\()p fk(x)p fq(\))10 b(\022)-23 b(a)10 b fm(e)p fq(\()p fk(x)262 2297 y fx(0)273 2313 y fq(\),)g(sac)o(han)o(t)f(que)i fk(x')f fq(est)g(fortemen)o(t)f(\023)-21 b(equiv)m(alen)o(t)11 b(\022)-23 b(a)10 b fk(x)p fq(.)f(l'explication)k(in)o(tuitiv)o(e)e (est)g(similaire)75 2370 y(p)q(our)k(le)h(cas)f(10.2)f(de)i(l')o(\023) -21 b(equiv)m(alence)17 b(faible.)75 2429 y(la)11 b(\014gure)g(3.1)g (illustre)h(l'explication)i(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)6 b(:)11 b(le)h(c)o(hamp)f fk(x)g fq(exprim)o(\023)-21 b(e)11 b(dans)g fe(z)1495 2413 y fg(3)1522 2429 y fq(est)f(\023)-21 b(egal)10 b(\022)-23 b(a)11 b fk(h)c(::)h(x)1823 2413 y fx(0)1834 2429 y fq(,)75 2486 y(o)q(\022)-24 b(u)15 b fk(x)162 2469 y fx(0)189 2486 y fq(est)g(exprim)o(\023)-21 b(e)15 b(dans)g fe(z)572 2469 y fg(2)588 2486 y fq(.)g(le)h (ra\016nemen)o(t)e(consiste)i(\022)-23 b(a)15 b(substituer)h fk(h)7 b(::)g(x)15 b fq(\022)-23 b(a)15 b fk(h)8 b(::)f(x)1613 2469 y fx(0)1625 2486 y fq(.)p eop %%page: 31 33 31 32 bop 75 269 a fl(3.2.)29 b(raffinement)15 b(des)651 257 y(\023)645 269 y(enonc)818 257 y(\023)812 269 y(es)h fn(pei)831 b fq(31)75 397 y(l')o(\023)-21 b(equiv)m(alence)17 b(de)f(deux)e(\023)-21 b(enonc)o(\023)g(es)14 b(est)h(d)o(\023)-21 b(e\014nie)16 b(\022)-23 b(a)15 b(partir)g(du)h(ra\016nemen)o(t)7 b(:)75 520 y fo(d)o(\023)-25 b(e\014nition)19 b(11)31 b fq(\()o(\023)-21 b(equiv)m(alence)16 b(d')o(\023)-21

b(enonc)o(\023)g(es\))75 576 y(nous)15 b(dirons)h(que)g(deux)e(\023)-21 b(enonc)o(\023)g(es)14 b fk(p)h fq(et)g fk(p')g fq(son)o(t)f fj(\023)-22 b(e)n(quivalents)p fq(,)13 b(et)i(on)g(note)g fk(p)h ff(\021)f fk(p')g fq(si)h(et)f(seulemen)o(t)75 633 y(si)h(ils)g(se)f(ra\016nen)o(t)g(m)o(utuellemen)o(t.)75 756 y fo(exemple)i(6)f fj(r)n(a\016nement)f(p)n(ar)i(substitution)f (d'un)g(champ)h(fortement)f(\023)-22 b(e)n(quivalent)75 879 y fq(consid)o(\023)h(erons)14 b(les)h(deux)g(programmes)e(suiv)m (an)o(ts,)h fk(p2)g fq(et)g fk(p3)p fq(.)g(nous)g(c)o(herc)o(hons)g (\022)-23 b(a)14 b(mon)o(trer)g(que)g fk(p2)g ff(v)75 935 y fk(p3)h fq(en)g(utilisan)o(t)i(l'implication)g(mo)q(dulo)f(l')o (\023)-21 b(equiv)m(alence)17 b(de)f(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees.)125 1073 y fk(p2)23 b(:)h(x)15 b ff(7!)h fk(y)125 1092 y fx(\()183 1136 y fk(x)f fq(=)h fk(carre)7 b(::)g(x)183 1192 y(y)15 b fq(=)h fk(x)146 1254 y(carre)c fq(=)g fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))951 1045 y(p3)23 b(:)h(x)15 b ff(7!)h fk(y)951 1064 y fx(\()1009 1108 y fk(x)f fq(=)g fk(carre)7 b(::)h(x)1009 1164 y(transp)f(::)g(y)15 b fq(=)h fk(x)971 1226 y(carre)c fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))971 1283 y(transp)f fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(j,i\))75 1394 y fq(consid)o(\023)-21 b(erons)15 b fk(p2')f fq(obten)o(u)h(\022)-23 b(a)15 b(partir)g(de)g fk(p2)g fq(en)h(rempla\030)-20 b(can)o(t)15 b fk(y)g fq(par)f(\()p fk(transp)7 b(::)g(y)p fq(\),)14 b(conform)o(\023)-21 b(emen)o(t)75 1450 y(\022)e(a)19 b(la)g(partie)h(1)f(de)g(la)h(d)o(\023)-21 b(e\014nition)20 b(10.)e(nous)h(v)o(\023)-21 b(eri\014ons)19 b(imm)o(\023)-21 b(ediatemen)o(t)19 b fm(sat)p fq(\()p fk(p3)p fq(\))g ff(\))h fm(sat)p fq(\()p fk(p2)1737 1434 y fx(0)1749 1450 y fq(\))f(car)75 1507 y fk(p3)h fq(et)h fk(p2')f fq(son)o(t)g(les)h(m)o(^)-21 b(emes)19 b(\023)-21 b(enonc)o(\023)g(es.) 19 b(d'autre)g(part,)h(soit)g fk(y)1211 1490 y fx(0)1243 1507 y fq(le)i(c)o(hamp)e(\()p fk(transp)1608 1490 y fq(-1)1653 1507 y fk(::)8 b(y)o fq(\))g(:)20 b(sa-)75 1563 y(c)o(han)o(t)d(de)h(l')o(\023)-21 b(enonc)o(\023)g(e)17 b fk(p2)g fq(que)h fm(dom)p fq(\()p fm(\033)723 1570 y fr(y)740 1563 y fq(\))e ff(\022)h fm(dom)p fq(\()p fk(transp)1073 1547 y fq(-1)1110 1563 y fq(\),)g fk(y)1182 1547 y fx(0)1211 1563 y fq(est)h(alors)f(fortemen)o(t)f(\023)-21 b(equiv)m(alen)o(t)18 b(\022)-23 b(a)75 1620 y fk(y)18 b fq(et)g fk(transp)7 b(::)g(y)403 1603 y fx(0)433 1620 y fq(=)19 b fk(transp)7 b(::)g(\(transp)860 1603 y fq(-1)905 1620 y fk(::)h(y\))18 b fq(=)h fk(y)p fq(.)e(nous)i(a)o(v)o(ons)e(par)h (cons)o(\023)-21 b(equen)o(t)18 b fm(sat)p fq(\()p fk(p3)p fq(\))85 1651 y fx(\021)75 1676 y ff(\))p fm(sat)p fq(\()p fk(p2)p fq(\))d(et)g(par)g(cons)o(\023)-21 b(equen)o(t)15 b fk(p2)g ff(v)g fk(p3)p fq(.)75 1733 y fa(\006)146 1856 y fq(cet)f(exemple)h(illustre)i(les)e(propri)o(\023)-21 b(et)o(\023)g(es)13 b(suiv)m(an)o(tes,)i(directemen)o(t)g(issues)g(des) g(d)o(\023)-21 b(e\014nitions)15 b(du)g(raf-)75 1912 y(\014nemen)o(t)e(\(9)f(et)g(10\))g(en)g(utilisan)o(t)i(l'implication)h (mo)q(dulo)e(l')o(\023)-21 b(equiv)m(alence.)14 b(elles)g(p)q(ermetten) o(t)e(en)h(e\013et)75 1968 y(d'obtenir)18 b(des)f(programmes)g(ra\016n) o(\023)-21 b(es)16 b(en)i(substituan)o(t)f(un)h(c)o(hamp)f(par)g(un)h (c)o(hamp)f(fortemen)o(t)g(ou)75 2025 y(faiblemen)o(t)e(\023)-21 b(equiv)m(alen)o(t.)75 2148 y fo(propri)o(\023)c(et)o(\023)g(e)20 b(2)54 b fq(soit)18 b fk(p)g fq(un)f(\023)-21 b(enonc)o(\023)g(e)17 b(et)h fk(x)f fq(un)i(iden)o(ti\014cateur)g(de)f(c)o(hamp.)g(si)h

fm(dom)p fq(\()p fm(\033)1605 2155 y fr(x)1621 2148 y fq(\))e ff(\022)h fm(img)r fq(\()p fk(h)p fq(\))75 2204 y(alors)d fk(p)d ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(x)p ff(n)p fk(h)6 b(::)h(x)p fq(])528 2188 y fw(?)547 2204 y ff(g)75 2306 y fj(pr)n(euve)p fq(.)15 b fj(cf.)g fq(annexe)h(a.)75 2429 y fj(r)n(emar)n(que)p fq(.)g(la)g(preuv)o(e)g(met)f(en)h(\023)-21 b(evidence)16 b(un)h(c)o(hamp)f(de)g(donn)o(\023)-21 b(ees)15 b fk(x)1319 2413 y fx(0)1347 2429 y fq(=)h fk(h)1422 2413 y fq(-1)1467 2429 y fk(::)7 b(x)16 b fq(qui)g(est)g(tel)g(que)75 2486 y fk(x)f fq(=)h fk(h)7 b(::)g(x)275 2469 y fx(0)287 2486 y fq(.)15 b(le)g(ra\016nemen)o(t)g(a)g(donc)h(consist)o(\023)-21 b(e)14 b(\022)-23 b(a)15 b(substituer)h fk(h)7 b(::)h(x)15 b fq(\022)-23 b(a)15 b fk(h)7 b(::)g(x)1458 2469 y fx(0)1470 2486 y fq(.)p eop %%page: 32 34 32 33 bop 75 269 a fq(32)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)75 397 y fo(propri)o(\023)-25 b(et)o(\023)g(e)26 b(3)67 b fq(soit)23 b fk(p)f fq(un)g(\023)-21 b(enonc)o(\023)g(e)21 b(o)q(\022)-24 b(u)23 b fk(x)f fq(est)g(un)h(iden)o(ti\014cateur)h(de)f(c)o(hamp)f(n'apparaissan)o(t) 75 453 y(que)g(sous)e(la)i(forme)e fk(h)8 b(::)f(x)p fq(.)21 b(si)h fm(dom)p fq(\()p fm(\033)794 460 y fr(x)811 453 y fq(\))g ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))n ff(^)f fm(img)r fq(\()p fk(h)p fq(\))e ff(\022)j fm(dom)p fq(\()p fk(h)p fq(\))d(nous)h(a)o(v)o(ons)g(alors)7 b(:)75 510 y fk(p)12 b ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(h)7 b(::)g(x)p ff(n)p fk(x)p fq(])420 493 y fw(?)438 510 y ff(g)75 604 y fj(pr)n(euve)p fq(.)15 b fj(cf.)g fq(annexe)h(a.)75 719 y fo(propri)o(\023)-25 b(et)o(\023)g(e)17 b(4)46 b fq(soit)15 b fk(p)g fq(un)g(\023)-21 b(enonc)o(\023)g(e.)13 b(nous)j(a)o(v)o(ons)6 b(:)370 809 y(\(1\))49 b(si)15 b fk(h)g fq(est)g(une)h(in)o(v)o(erse)g(gauc)o(he)f(de)h fk(g)p fq(,)478 866 y fm(dom)p fq(\()p fm(v)604 873 y fr(e)620 866 y fq(\))d(=)g fm(dom)p fq(\()p fk(g)p fq(\))h(et)h fk(h)p fq(\()p fm(dom)p fq(\()p fk(g)p fq(\)\))10 b ff(\022)j fm(dom)p fq(\()p fm(\033)1306 873 y fr(e)1323 866 y fq(\))f ff(\022)h fm(img)r fq(\()p fk(h)p fq(\).)478 922 y fk(p)f ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fk(h)6 b(::)h(\(e)24 b fa(c)g fk(g)940 906 y fq(-1)977 922 y fk(\))p fq(])p ff(g)370 1035 y fq(\(2\))49 b(si)15 b fk(h)g fq(est)g(une)h(in)o(v)o(erse)g(droite)f(de)h fk(g)p fq(,)478 1092 y fm(dom)p fq(\()p fm(v)604 1099 y fr(e)620 1092 y fq(\))d(=)g fm(img)r fq(\()p fk(g)p fq(\))g(et)i fm(dom)p fq(\()p fk(g)p fq(\))c ff(\022)i fm(dom)p fq(\()p fm(\033)1240 1099 y fr(e)1257 1092 y fq(\))f ff(\022)h fm(img)r fq(\()p fk(h)p fq(\).)478 1148 y fk(p)f ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fk(\(h)6 b(::)i(e\))15 b fa(c)g fk(g)947 1132 y fq(-1)984 1148 y fq(])p ff(g)75 1294 y fj(pr)n(euve)p fq(.)g fj(cf.)g fq(annexe)h(a.)75 1394 y(notons)h(\023)-21 b(egalemen)o(t)17 b(un)i(cas)g(particulier)h (de)f(cette)f(derni)o(\022)-21 b(ere)19 b(prop)q(osition,)g(p)q(our)f (lequel)j fk(h)d fq(=)h fk(g)1797 1378 y fq(-1)1834 1394 y fq(.)75 1451 y(dans)c(ce)g(cas)g(la)h(propri)o(\023)-21 b(et)o(\023)g(e)14 b(4)g(devien)o(t)9 b(:)75 1566 y fo(propri)o(\023) -25 b(et)o(\023)g(e)14 b(5)37 b fq(soit)13 b fk(p)f fq(un)g(\023)-21 b(enonc)o(\023)g(e,)11 b fk(e)h fq(une)i(expression)f(de)g(c)o(hamp)f (de)h(donn)o(\023)-21 b(ees,)12 b(et)g fk(h)g fq(une)i(bijection.)75 1622 y(si)i fm(dom)p fq(\()p fm(v)254 1629 y fr(e)271 1622 y fq(\))c(=)h fm(dom)p fq(\()p fm(\033)479 1629 y fr(e)496 1622 y fq(\))f(=)h fm(dom)p fq(\()p fk(h)p

fq(\))f(=)h fm(img)r fq(\()p fk(h)p fq(\))g(nous)i(a)o(v)o(ons)7 b(:)621 1730 y(\(1\))41 b fk(p)g ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fk(h)6 b(::)h(\(e)24 b fa(c)g fk(h\))p fq(])p ff(g)621 1786 y fq(\(2\))41 b fk(p)g ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fk(\(h)6 b(::)i(e\))15 b fa(c)g fk(h)p fq(])p ff(g)146 1975 y fq(cette)21 b(derni)o(\022)-21 b(ere)23 b(propri)o(\023)-21 b(et)o(\023)g(e)21 b(est)h(celle)i(que)e(nous)h(a)o(vions)f(utilis)o (\023)-21 b(ee)23 b(dans)g(le)g(c)o(hapitre)f(2)g(p)q(our)75 2031 y(in)o(tro)q(duire)16 b(l')o(\023)-21 b(equiv)m(alence)18 b(faible)e(et)f(est)g(illustr)o(\023)-21 b(ee)16 b(par)f(la)h(\014gure) f(2.9.)75 2165 y fw(3.2.2)55 b(cas)20 b(particuliers)146 2256 y fq(nous)d(donnons)h(ici)i(un)e(ensem)o(ble)h(d'exemples)g (classiques)g(qui)g(n'utilisen)o(t)g(que)f(l'implication)75 2312 y(simple.)75 2427 y fo(exemple)f(7)f fj(r)n(a\016nement)f(p)n(ar)i (intr)n(o)n(duction)f(d'un)g(champ)h(interm)o(\023)-22 b(e)n(diair)n(e)75 2542 y fq(consid)o(\023)h(erons)15 b(les)h(deux)f(programmes)g(suiv)m(an)o(ts,)g fk(p2)g fq(et)g fk(p4)p fq(.)p eop %%page: 33 35 33 34 bop 75 269 a fl(3.2.)29 b(raffinement)15 b(des)651 257 y(\023)645 269 y(enonc)818 257 y(\023)812 269 y(es)h fn(pei)831 b fq(33)125 416 y fk(p2)23 b(:)h(x)15 b ff(7!)h fk(y)125 434 y fx(\()183 478 y fk(x)f fq(=)h fk(carre)7 b(::)g(x)183 534 y(y)15 b fq(=)h fk(x)146 597 y(carre)c fq(=)g fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))951 391 y(p4)23 b(:)h(x)15 b ff(7!)h fk(y)951 408 y fx(8)951 445 y(>)951 458 y(<)951 532 y(>)951 545 y(:)1012 450 y fk(x)f fq(=)h fk(carre)7 b(::)g(x)1012 506 y(z)15 b fq(=)h fk(x)1012 563 y(y)f fq(=)h fk(z)971 622 y(carre)c fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))75 745 y fq(nous)g(sommes)f(ici)j(dans)e(le)h(cas)e(particulier)j(o)q (\022)-24 b(u)13 b fk(p2')g fq(est)f(le)i(m)o(^)-21 b(eme)11 b(\023)-21 b(enonc)o(\023)g(e)12 b(que)h fk(p2)p fq(.)g fk(p2)f fq(ne)i(di\013)o(\022)-21 b(ere)12 b(de)75 802 y fk(p4)f fq(que)h(par)g(la)g(pr)o(\023)-21 b(esence)11 b(du)h(c)o(hamp)g(de)g(donn)o(\023)-21 b(ees)11 b(in)o(term)o(\023)-21 b(ediaire)13 b fk(z)p fq(,)e(et)g(leurs)i(propri)o(\023)-21 b(et)o(\023)g(es)10 b(resp)q(ectiv)o(es)75 858 y(son)o(t)d(:)506 969 y fm(sat)p fq(\()p fk(p2)p fq(\))g(:)15 b(\()p fk(x)g fq(=)g fk(carre)7 b(::)h(x)o fq(\))15 b ff(\))h fq(\()p fk(y)f fq(=)g fk(x)p fq(\))506 1025 y fm(sat)p fq(\()p fk(p4)p fq(\))7 b(:)15 b(\()p fk(x)g fq(=)g fk(carre)7 b(::)h(x)o fq(\))15 b ff(\))h fq(\()p fk(z)f fq(=)g fk(x)g ff(^)h fk(y)f fq(=)g fk(z)p fq(\))146 1207 y(nous)24 b(v)o(\023)-21 b(eri\014ons)24 b(imm)o(\023)-21 b(ediatemen)o(t)24 b fm(sat)p fq(\()p fk(p4)p fq(\))g ff(\))h fm(sat)p fq(\()p fk(p2)p fq(\))f(car)g(la)h(relation)g(d')o(\023)-21 b(egalit)o(\023)g (e)23 b(sur)h(les)75 1264 y(c)o(hamps)15 b(de)h(donn)o(\023)-21 b(ees)15 b(est)g(naturellemen)o(t)h(transitiv)o(e,)f(et)g fk(p2)g ff(v)g fk(p4)p fq(.)146 1343 y(de)g(plus,)h(de)g(la)g(d)o(\023) -21 b(e\014nition)16 b(9,)f(puisqu'il)j(existe)e(un)g(c)o(hamp)f(de)h (donn)o(\023)-21 b(ees)15 b fk(z)h fq(tel)g(que)f(\()p fk(y)g fq(=)h fk(x)p fq(\))f ff(\))75 1399 y fq(\()p fk(z)e fq(=)h fk(x)g ff(^)g fk(y)g fq(=)g fk(z)p fq(\))f(est)g(vrai,)h (nous)g(a)o(v)o(ons)f(aussi)h(la)g(relation)g fk(p4)h ff(v)h fk(p2)o fq(,)e(ce)g(qui)g(signi\014e)i(que)e(ces)g(deux)74 1456 y(\023)-21 b(enonc)o(\023)g(es)14 b(son)o(t)f(\023)-21 b(equiv)m(alen)o(ts.)146 1534 y(p)o(ar)13 b(cons)o(\023)-21

b(equen)o(t,)14 b(l'a)s(jout)f(ou)i(la)g(suppression)g(d'un)g(c)o(hamp) g(in)o(term)o(\023)-21 b(ediare)14 b(donne)h(toujours)f(un)75 1591 y(nouv)o(el)h(\023)-21 b(enonc)o(\023)g(e)13 b(\023)-21 b(equiv)m(alen)o(t.)75 1647 y fa(\006)146 1782 y fq(des)20 b(pro)q(c)o(\023)-21 b(ed)o(\023)g(es)19 b(classiques)i(de)g (ra\016nemen)o(t)f(consisten)o(t)g(\022)-23 b(a)20 b fj(\()-9 b(\()q fq(a\013aiblir)p fj(\))g(\))21 b fq(une)g(pr)o(\023)-21 b(e-)o(\023)g(equation,)19 b(ou)75 1839 y fj(\()-9 b(\()q fq(renforcer)p fj(\))g(\))14 b fq(une)h(p)q(ost-)o(\023)-21 b(equation.)14 b(en)h fn(pei)p fq(,)f(ces)g(pro)q(c)o(\023)-21 b(ed)o(\023)g(es)14 b(son)o(t)f(\023)-21 b(egalemen)o(t)13 b(applicables,)k(\022)-23 b(a)14 b(tra)o(v)o(ers)75 1895 y(la)i(propri)o(\023)-21 b(et)o(\023)g(e)13 b(suiv)m(an)o(te)c(:)75 2031 y fo(propri)o(\023)-25 b(et)o(\023)g(e)14 b(6)37 b fq(soien)o(t)13 b fk(p)p ff(f)p fm(s)579 2038 y fg(1)597 2031 y fm(;)8 b(s)646 2038 y fg(2)664 2031 y ff(g)k fq(et)g fk(p')p ff(f)p fm(s)851 2014 y fx(0)848 2042 y fg(1)866 2031 y fm(;)c(s)918 2014 y fx(0)915 2042 y fg(2)933 2031 y ff(g)k fq(deux)g(\023)-21 b(enonc)o(\023)g(es,)11 b(o)q(\022)-24 b(u)12 b fm(s)1340 2038 y fg(1)1371 2031 y fq(et)g fm(s)1452 2014 y fx(0)1449 2042 y fg(1)1480 2031 y fq(son)o(t)g(des)g(ensem)o (bles)75 2087 y(de)k(pr)o(\023)-21 b(e-equations,)14 b(et)h fm(s)511 2094 y fg(2)530 2087 y fq(,)f fm(s)588 2070 y fx(0)585 2098 y fg(2)619 2087 y fq(des)i(ensem)o(bles)g(de)g(p)q (ost-equations.)f(nous)g(a)o(v)o(ons)7 b(:)170 2209 y fk(p)31 b ff(v)f fk(p')50 b fq(si)113 b fm(s)559 2216 y fg(2)590 2209 y fq(=)13 b fm(s)669 2193 y fx(0)666 2221 y fg(2)715 2209 y fq(et)30 b(\()p fm(s)829 2216 y fg(1)860 2209 y ff(\))13 b fm(s)949 2193 y fx(0)946 2221 y fg(1)965 2209 y fq(\))i(est)g(v)o(\023)-21 b(eri\014)o(\023)g (ee)388 2266 y(ou)15 b(si)50 b fm(s)559 2273 y fg(1)590 2266 y fq(=)13 b fm(s)669 2249 y fx(0)666 2277 y fg(1)715 2266 y fq(et)30 b(\()p fm(s)832 2249 y fx(0)829 2277 y fg(2)860 2266 y ff(\))13 b fm(s)946 2273 y fg(2)965 2266 y fq(\))i(est)g(v)o(\023)-21 b(eri\014)o(\023)g(ee)75 2373 y fj(pr)n(euve.)12 b fq(supp)q(osons)i fm(s)486 2380 y fg(1)517 2373 y ff(\))f fm(s)606 2356 y fx(0)603 2384 y fg(1)622 2373 y fq(,)f(alors)g(\()p fm(s)802 2356 y fx(0)799 2384 y fg(1)830 2373 y ff(\))h fm(s)916 2380 y fg(2)935 2373 y fq(\))f ff(\))h fq(\()p fm(s)1069 2380 y fg(1)1100 2373 y ff(\))g fm(s)1186 2380 y fg(2)1204 2373 y fq(\).)f(de)h(m)o(^)-21 b(eme,)11 b(supp)q(osons)i fm(s)1701 2356 y fx(0)1698 2384 y fg(2)1729 2373 y ff(\))h fm(s)1816 2380 y fg(2)1834 2373 y fq(,)75 2429 y(alors)h(\()p fm(s)230 2436 y fg(1)261 2429 y ff(\))e fm(s)350 2413 y fx(0)347 2441 y fg(2)365 2429 y fq(\))g ff(\))g fq(\()p fm(s)500 2436 y fg(1)531 2429 y ff(\))g fm(s)617 2436 y fg(2)635 2429 y fq(\).)75 2486 y fa(\003)p eop %%page: 34 36 34 35 bop 75 269 a fq(34)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)75 397 y fo(exemple)j(8)f fj(a\013aiblissement)e(des)i(pr)o(\023)-22 b(e-)o(\023)g(e)n(quations) 75 510 y fk(p4)24 b(:)f(x)15 b ff(7!)h fk(y)75 516 y fx(\()133 559 y fk(x)f fq(=)h fk(vecteur)6 b(::)i(x)133 616 y(y)15 b fq(=)h fk(x)75 677 y fq(sp)o(\023)-21 b(eci\014)o(\023)g (e)15 b(par)g fm(sat)p fq(\()p fk(p4)p fq(\))7 b(:)75 733 y(\()p fk(x)15 b fq(=)g fk(vecteur)7 b(::)g(x)p fq(\))15 b ff(\))g fq(\()p fk(y)p fq(=)p fk(x)p fq(\))962 510 y fk(p5)24 b(:)g(x)15 b ff(7!)g fk(y)962 519 y fx(n)1015 566 y fk(y)g fq(=)g fk(x)962 631 y fq(sp)o(\023)-21 b(eci\014)o(\023)g

(e)15 b(par)g fm(sat)p fq(\()p fk(p5)p fq(\))8 b(:)962 687 y(vrai)16 b ff(\))f fq(\()p fk(y)p fq(=)p fk(x)p fq(\))75 850 y(la)e(propri)o(\023)-21 b(et)o(\023)g(e)11 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)12 b(a\016rme)h(que)g fk(p4)f ff(v)i fk(p5)p fq(,)e(ce)h(qui)h(p)q(eut)e(\023)-21 b(egalemen)o(t)11 b(^)-21 b(etre)11 b(\023)-21 b(etabli)13 b(par)g fm(sat)p fq(\()p fk(p5)p fq(\))75 907 y ff(\))21 b fm(sat)p fq(\()p fk(p4)p fq(\))8 b(:)20 b(en)h(e\013et,)g fm(sat)p fq(\()p fk(p5)p fq(\))f(se)h(simpli\014e)i(en)f(\()p fk(y)e fq(=)h fk(x)p fq(\))g(et)f(la)i(propri)o(\023)-21 b(et)o(\023)g(e)19 b(\()p fk(y)h fq(=)i fk(x)p fq(\))e ff(\))h fq(\(\()p fk(x)f fq(=)75 963 y fk(vecteur)7 b(::)g(x)p fq(\))15 b ff(\))g fq(\()p fk(y)g fq(=)g fk(x)p fq(\)\))g(est)g(v)o (\023)-21 b(eri\014)o(\023)g(ee.)75 1020 y fa(\006)75 1112 y fo(exemple)17 b(9)f fj(r)n(enfor)n(c)n(ement)e(des)i(p)n(ost-)o (\023)-22 b(e)n(quations)146 1225 y fk(p6)23 b(:)h(x)15 b ff(7!)h fk(y)146 1231 y fx(\()204 1274 y fk(vecteur)7 b(::)g(z)15 b fq(=)g fk(x)204 1331 y(y)g fq(=)g fk(vecteur)7 b(::)g(z)146 1415 y fq(sp)o(\023)-21 b(eci\014)o(\023)g(e)14 b(par)h fm(sat)p fq(\()p fk(p6)p fq(\))8 b(:)15 b(vrai)g ff(\))g fq(\()p fk(vecteur)7 b(::)g(z)15 b fq(=)h fk(x)f ff(^)g fk(y)g fq(=)h fk(vecteur)6 b(::)h(z)p fq(\))75 1528 y(nous)14 b(a)o(v)o(ons)g fk(p5)f ff(v)i fk(p6)f fq(car)g fm(sat)p fq(\()p fk(p6)p fq(\))e ff(\))h fm(sat)p fq(\()p fk(p5)q fq(\))g(s')o(\023)-21 b(ecrit)14 b(\()p fk(vecteur)6 b(::)h(z)14 b fq(=)h fk(x)f ff(^)h fk(y)f fq(=)g fk(vecteur)7 b(::)g(z)p fq(\))75 1585 y ff(\))16 b fq(\()p fk(y)e fq(=)i fk(x)p fq(\),)e(qui)i(est)f(v)o(\023)-21 b(eri\014)o(\023)g(ee.)75 1641 y fa(\006)75 1841 y ft(3.3)66 b(calcul)23 b(de)g(ra\016nemen)n(t)146 1942 y fq(de)15 b(mani)o(\022)-21 b(ere)16 b(plus)h(op)o(\023)-21 b(erationnelle,)16 b(nous)h(d)o(\023)-21 b(e\014nissons)16 b(main)o(tenan)o(t)f(un)i fj(c)n(alcul)f(de)h(r)n(a\016nement)75 1998 y fq(qui)e(p)q(ermet)f(de)g (construire)g(un)g(programme)f(ra\016nan)o(t)g(un)h(programme)f(donn)o (\023)-21 b(e,)13 b(sans)g(utiliser)j(l'im-)75 2055 y(plication)k(mo)q (dulo)e(l')o(\023)-21 b(equiv)m(alence.)20 b(ce)e(calcul)i(stipule)f (que)f(le)h(ra\016nemen)o(t)f(d'un)f(\023)-21 b(enonc)o(\023)g(e)17 b(p)q(eut)i(se)75 2111 y(faire)d(par)g(la)h(substitution)g(d'une)f (\023)-21 b(equation)16 b(de)g(cet)g(\023)-21 b(enonc)o(\023)g(e)15 b(par)h(une)h(autre.)e(nous)h(notons)g(par)g(la)75 2168 y(suite)g fk(p)p ff(f)p fm(s)s fq([)p fm(e)s ff(n)p fm(e)372 2151 y fx(0)381 2168 y fq(])p ff(g)f fq(l')o(\023)-21 b(enonc)o(\023)g(e)15 b fk(p)g fq(dans)h(lequel)i(l')o(\023)-21 b(equation)15 b fm(e)j fq(a)d(\023)-21 b(et)o(\023)g(e)14 b(substitu)o(\023)-21 b(ee)15 b(par)h(l')o(\023)-21 b(equations)15 b fm(e)1823 2151 y fx(0)1834 2168 y fq(.)75 2224 y(nous)h(pr)o(\023)-21 b(ecisons)16 b(main)o(tenan)o(t)g(les)h(mo)o(y)o(ens)e(d'obtenir,)i (\022)-23 b(a)15 b(partir)h(d'une)g(\023)-21 b(equation)16 b fm(e)s fq(,)f(un)g(\023)-21 b(equation)75 2281 y fm(e)112 2264 y fx(0)139 2281 y fq(telle)17 b(que)f fk(p)p ff(f)p fm(s)s fq([)p fm(e)s fq(])p ff(g)d(v)j fk(p)p ff(f)p fm(s)s fq([)p fm(e)s ff(n)p fm(e)739 2264 y fx(0)747 2281 y fq(])p ff(g)p fq(.)f(nous)h(in)o(tro)q(duisons)h(la)f(notation)f fm(e)h ff(\000)-7 b(!)14 b fm(e)1593 2264 y fx(0)1604 2281 y fq(,)h(qui)i(signi\014e)75 2337 y(que)f(l'on)f(substitue)h fm(e)h fq(par)e fm(e)612 2321 y fx(0)638 2337 y fq(en)h(utilisan)o(t)g (un)g(des)f(deux)h(pro)q(c)o(\023)-21 b(ed)o(\023)g(es)14 b(suiv)m(an)o(t)8 b(:)131 2429 y(1.)22 b(l'application)17 b(d'un)g(c)o(hangemen)o(t)e(de)i(base)f(aux)g(deux)g(mem)o(bres)g (d'une)g(\023)-21 b(equation,)15 b(conduit)189 2486 y(aux)g(deux)h(r)o

(\022)-21 b(egles)14 b(de)i(substitution)g(suiv)m(an)o(tes)8 b(:)p eop %%page: 35 37 35 36 bop 75 269 a fl(3.3.)29 b(calcul)16 b(de)f(raffinement)990 b fq(35)p 189 353 1623 2 v 189 732 2 380 v 228 405 a(\(i\))125 b fk(y)15 b fq(=)h fk(x)60 b ff(\000)-7 b(!)80 b fk(h)8 b(::)f(y)15 b fq(=)h fk(h)7 b(::)g(x)732 461 y fq(et)31 b fm(dom)p fq(\()p fm(\033)931 468 y fr(x)947 461 y fq(\))13 b ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\))8 b ff(^)j fm(dom)p fq(\()p fm(\033)1351 468 y fr(y)1368 461 y fq(\))h ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))203 546 y(et)i(r)o(\023)-21 b(ecipro)q(quemen)o(t,)228 646 y(\(i)q(i\))126 b fk(h)7 b(::)g(y)15 b fq(=)h fk(h)7 b(::)h(x)75 b ff(\000)-7 b(!)81 b fk(y)15 b fq(=)g fk(x)939 703 y fq(et)30 b fm(dom)p fq(\()p fm(v)1133 710 y fr(x)1150 703 y fq(\))12 b ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))p 1810 732 v 189 734 1623 2 v 131 799 a(2.)22 b(l'application)16 b(d'une)g(des)f(r)o(\022)-21 b(egles)13 b(\023)-21 b(el)o(\023)g(emen)o(taires)14 b(don)o(t)g(nous)h(donnons)g(la)h(liste)f(dans)g(les)h(trois)189 856 y(tables)k(qui)h(suiv)o(en)o(t.)f(quand)h(des)f(conditions)i (apparaissen)o(t)e(dans)g(la)g(partie)g(droite)h(de)f(la)189 912 y(r)o(\022)-21 b(egle,)16 b(on)i(doit)g(s'assurer,)e(p)q(our)i(p)q (ouv)o(oir)g(appliquer)h(la)f(r)o(\022)-21 b(egle,)17 b(que)h(le)g(programme)f(obten)o(u)189 969 y(apr)o(\022)-21 b(es)15 b(substitution,)h(requiert)g(\023)-21 b(egalemen)o(t)15 b(ces)i(conditions)g(\(elles)g(doiv)o(en)o(t)g(donc)f(p)q(ouv)o(oir)h (se)189 1025 y(d)o(\023)-21 b(eduire)11 b(des)h(autres)e(\023)-21 b(equations)10 b(du)i(programme)e(obten)o(u\).)g(notons)h(que)g(si)h (on)f(p)q(eut)h(appliquer)189 1082 y(la)j(r)o(\022)-21 b(egle,)14 b(le)i(programme)e(obten)o(u)h(est)e(\023)-21 b(equiv)m(alen)o(t)16 b(puisque)g(la)g(r)o(\022)-21 b(egle)14 b(r)o(\023)-21 b(ecipro)q(que)15 b(ne)h(supp)q(ose)189 1138 y(aucune)g(condition.)146 1214 y(a)d(titre)h(d'exemple,)h (examinons)f(les)h(r)o(\022)-21 b(egles)13 b(\(5a\))f(et)i(\(5b\).)f (dans)g(la)h(r)o(\022)-21 b(egle)14 b(\(5a\))e(la)i(partie)g(droite)75 1270 y(de)i(la)g(r)o(\022)-21 b(egle)14 b(n'est)i(une)g(expression)g (correcte)f(que)h(si)g(\()p fk(x)f fa(c)g fk(f1)p fq(\))g(est)g(un)h(c) o(hamp)f(de)h(donn)o(\023)-21 b(ees,)15 b(ce)h(qui)75 1326 y(n'est)i(pas)h(requis)g(par)f(la)h(partie)g(gauc)o(he)f(de)h(la)g (r)o(\022)-21 b(egle.)18 b(consid)o(\023)-21 b(erons)18 b(par)g(exemple)i(les)f(fonctions)75 1383 y fk(f1)p fq(=)p fm(\025)p fk(\(i\))p ff(j)p fk(\(2)p ff(\024)p fk(i)p ff(\024)p fk(n\).\(i-1\))10 b fq(et)i fk(f2)p fq(=)p fm(\025)p fk(\(i\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n-1\).\(i+1\))n fq(.)f(de)h(la)h(d)o(\023)-21 b(e\014nition)13 b(de)f(l'op)o(\023)-21 b(eration)75 1439 y(g)o(\023)g(eom)o(\023)g(etrique,)14 b(la)i(partie)g(gauc)o(he)g (de)g(la)g(r)o(\022)-21 b(egle)16 b(est)f(correcte)h(si)g fm(dom)p fq(\()p fk(f1)10 b ff(\016)f fk(f2)p fq(\))16 b ff(\022)g fm(dom)p fq(\()p fm(\033)r fq(\).)e(dans)h(la)75 1496 y(partie)g(droite,)f(l'expression)i(doit)f(v)o(\023)-21 b(eri\014er)14 b(\022)-23 b(a)15 b(la)g(fois)f fm(dom)p fq(\()p fk(f1)p fq(\))g ff(\022)h fm(dom)p fq(\()p fm(\033)r fq(\))e(et)h fm(dom)p fq(\()p fk(f2)o fq(\))e ff(\022)h fm(dom)p fq(\()p fm(\033)r fq(\),)75 1552 y(qui)g(se)f(simplife)h(en)g fm(dom)p fq(\()p fk(f1)o fq(\))s ff([)s fm(dom)p fq(\()p fk(f2)o fq(\))g ff(\022)g fm(dom)p fq(\()p fm(\033)r fq(\).)d(en)i(l'o)q(ccurrence,)g(ceci)h(signi\014e)h(que)e(le)h(c)o (hamp)75 1609 y(de)k(la)g(partie)g(droite)g(doit)g(v)o(\023)-21 b(eri\014er)17 b([1)8 b fm(:)g(:)g(:)t(n)p fq(])15 b

ff(\022)h fm(dom)p fq(\()p fm(\033)r fq(\))f(alors)h(que)i(la)f(partie) g(gauc)o(he)f(ne)h(n)o(\023)-21 b(ecessite)75 1665 y(que)18 b([1)8 b fm(:)g(:)g(:)t(n)p ff(\000)q fq(1])16 b ff(\022)h fm(dom)p fq(\()p fm(\033)r fq(\).)e(le)k(fait)e(que)h(le)g(c)o(hamp)g (soit)f(correctemen)o(t)g(d)o(\023)-21 b(e\014ni)18 b(par)f (l'expression)75 1722 y(en)f(partie)f(droite,)g(implique)j(qu'il)e(est) f(bien)h(d)o(\023)-21 b(e\014ni)16 b(par)f(l'expression)h(en)g(partie)f (gauc)o(he.)75 1778 y(si)f(en)h(plus,)f(on)f(a)s(joute)g(une)h(con)o (train)o(te)g(imp)q(osan)o(t)f fm(dom)p fq(\()p fm(\033)r fq(\))e(=)i([1)8 b fm(:)g(:)g(:)d(n)p fq(],)13 b(par)g(exemple)i(en)f (a)s(joutan)o(t)75 1835 y(l')o(\023)-21 b(equation)11 b fk(x)h fq(=)g fm(\025)p fk(\(i\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n\))6 b(::)i(x)p fq(,)j(alors)g(on)h(d)o (\023)-21 b(eduit)12 b(de)g(cette)g(information)g(con)o(textuelle)h (qu'on)75 1891 y(p)q(eut)i(\023)-21 b(egalemen)o(t)14 b(appliquer)j(la)f(r)o(\022)-21 b(egle)14 b(dans)i(le)g(sens)f(opp)q (os)o(\023)-21 b(e,)14 b(comme)h(l'indique)j(la)d(r)o(\022)-21 b(egle)15 b(\(5b\).)146 2004 y(le)20 b(lien)h(en)o(tre)e(la)h(r)o(\023) -21 b(e)o(\023)g(ecriture)18 b(d'une)h(\023)-21 b(equation)19 b(d'un)f(\023)-21 b(enonc)o(\023)g(e)19 b(et)g(le)h(ra\016nemen)o(t)f (est)f(\023)-21 b(etabli)20 b(\022)-23 b(a)75 2060 y(l'aide)16 b(de)g(la)f(propri)o(\023)-21 b(et)o(\023)g(e)14 b(suiv)m(an)o(te)8 b(:)75 2173 y fo(propri)o(\023)-25 b(et)o(\023)g(e)17 b(7)31 b fq(soien)o(t)15 b(deux)g(\023)-21 b(equations)15 b fm(e)i fq(et)e fm(e)958 2157 y fx(0)984 2173 y fq(et)g(un)g(\023)-21 b(enonc)o(\023)g(e)14 b fk(p)p ff(f)p fm(s)s ff(g)6 b fq(:)189 2249 y(si)16 b fm(e)h fq(est)e(une)h(pr)o(\023)-21 b(e-)o(\023)g(equation)14 b(et)h fm(e)30 b ff(\000)-7 b(!)28 b fm(e)969 2232 y fx(0)995 2249 y fq(alors)15 b fk(p)p ff(f)p fm(s)s fq([)p fm(e)s ff(n)p fm(e)1292 2232 y fx(0)1300 2249 y fq(])p ff(g)g(v)g fk(p)189 2335 y fq(si)h fm(e)h fq(est)e(une)h(p)q(ost-)o(\023)-21 b(equation)14 b(et)h fm(e)30 b ff(\000)-7 b(!)28 b fm(e)990 2319 y fx(0)1016 2335 y fq(alors)15 b fk(p)g ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fm(e)s ff(n)p fm(e)1403 2319 y fx(0)1411 2335 y fq(])p ff(g)75 2410 y fj(pr)n(euve.)d fq(examinons)h(les)g(deux) h(pro)q(c)o(\023)-21 b(ed)o(\023)g(es)12 b(p)q(ermettan)o(t)h (d'e\013ectuer)g(une)i(substitution)f(d')o(\023)-21 b(equation)7 b(:)131 2486 y(1.)122 b fo(r)o(\022)-25 b(egle)16 b(\(i\))e fq(il)h(faut)e(d'ab)q(ord)f(\023)-21 b(etablir)14 b(la)g(preuv)o(e)g(p) q(our)g(le)g(premier)h(cas)e(de)h(substitution.)289 2542 y(soien)o(t)19 b fm(p)g fq(les)h(conditions)h(induites)g(par)e(la)h (partie)f(gauc)o(he)g(de)h(la)g(r)o(\022)-21 b(egle)18 b(\(i\),)h(et)g fm(p)1765 2526 y fx(0)1796 2542 y fq(les)p eop %%page: 36 38 36 37 bop 75 269 a fq(36)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)p 75 353 1800 2 v 75 1620 2 1267 v 247 405 a fj(\(fonctionnel)r(le-b)n(ase\))114 461 y fq(\(1a\))51 b fk(y)15 b fq(=)h fk(f1)f fa(b)g fk(\(f2)8 b(::)f(x\))207 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f2)7 b(::)g(\(f1)24 b fa(b)g fk(x\))114 518 y fq(\(1b\))49 b fk(y)15 b fq(=)h fk(f2)7 b(::)g(\(f1)24 b fa(b)g fk(x\))190 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f1)e fa(b)i fk(\(f2)8 b(::)f(x\))247 631 y fj(\(g)o(\023)-22 b(e)n(om)o(\023)g(etrique-fonctionnel)r(le\))114 687 y fq(\(2a\))51 b fk(y)15 b fq(=)h fk(\(f1)23 b fa(b)h fk(x\))15 b fa(c)h fk(f2)187 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f1)e fa(b)i fk(\(x)24 b fa(c)g fk(f2\))114 744 y fq(\(2b\))49 b fk(y)15 b fq(=)h fk(f1)f fa(b)g

fk(\(x)24 b fa(c)g fk(f2\))187 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(f1)23 b fa(b)h fk(x\))15 b fa(c)h fk(f2)247 857 y fj(\(b)n(ase-g)o(\023)-22 b(e)n(om)o(\023)g(etrique\)) 114 913 y fq(\(3a\))51 b fk(y)15 b fq(=)h fk(\(f2)7 b(::)h(x\))15 b fa(c)g fk(f1)207 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f2)7 b(::)g(\(x)24 b fa(c)g fk(f2)1342 897 y fx(\000)p fg(1)1396 913 y ff(\016)10 b fk(f1)g ff(\016)g fk(f2\))963 969 y fq(et)36 b fm(dom)p fq(\()p fk(f1)o fq(\))10 b ff([)g fm(img)r fq(\()p fk(f1)n fq(\))j ff(\022)g fm(img)r fq(\()p fk(f2)n fq(\))i ff(^)1037 1026 y fm(dom)p fq(\()p fm(v)1163 1033 y fr(x)1180 1026 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(f2)o fq(\))114 1139 y(\(3b\))49 b fk(y)15 b fq(=)h fk(f2)7 b(::)g(\(x)24 b fa(c)g fk(f1\))190 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(f2)7 b(::)h(x\))15 b fa(c)g fk(f2)10 b ff(\016)g fk(f1)g ff(\016)g fk(f2)1531 1122 y fx(\000)p fg(1)963 1195 y fq(et)15 b fm(dom)p fq(\()p fk(f1)o fq(\))10 b ff([)g fm(img)r fq(\()p fk(f1)o fq(\))i ff(\022)h fm(dom)p fq(\()p fk(f2)o fq(\))247 1252 y fj(\(r)o(\023)-22 b(e)n(duction-b)n(ase\))114 1308 y fq(\(3c\))54 b fk(y)15 b fq(=)h fk(f1)f fq(;)8 b fa(b)14 b fk(\(f2)8 b(::)f(x\))187 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f2)7 b(::)g(\(f2)1235 1292 y fx(\000)p fg(1)1289 1308 y ff(\016)j fk(f1)g ff(\016)g fk(f2)24 b fq(;)8 b fa(b)23 b fk(x\))963 1365 y fq(et)36 b fm(dom)p fq(\()p fk(f1)o fq(\))10 b ff([)g fm(img)r fq(\()p fk(f1)n fq(\))j ff(\022)g fm(img)r fq(\()p fk(f2)n fq(\))i ff(^)1037 1421 y fm(dom)p fq(\()p fm(v)1163 1428 y fr(x)1180 1421 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(f2)o fq(\))114 1534 y(\(3d\))49 b fk(y)15 b fq(=)h fk(f2)7 b(::)g(\(f1)24 b fq(;)8 b fa(b)23 b fk(x\))170 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f2)9 b ff(\016)h fk(f1)g ff(\016)g fk(f2)1282 1518 y fx(\000)p fg(1)1341 1534 y fq(;)e fa(b)15 b fk(\(f2)8 b(::)f(x\))963 1590 y fq(et)15 b fm(dom)p fq(\()p fk(f1)o fq(\))10 b ff([)g fm(img)r fq(\()p fk(f1)o fq(\))i ff(\022)h fm(dom)p fq(\()p fk(f2)o fq(\))p 1873 1620 v 75 1622 1800 2 v 351 1699 a fn(t)l(ab.)i fq(3.1)f(-)31 b fj(r)o(\022)-22 b(e)n(gles)15 b(impliquant)h(des)g(op)o(\023)-22 b(er)n(ations)16 b(de)h(typ)n(e)f(di\013)o(\023)-22 b(er)n(ent)289 1841 y fq(conditions)16 b(induites)h(par)e(la)g(partie)h(droite)f(de)h (cette)f(m)o(^)-21 b(eme)14 b(r)o(\022)-21 b(egle)8 b(:)554 1947 y fm(p)61 b fq(=)50 b(\()p fm(v)763 1954 y fr(y)796 1947 y fq(=)15 b fm(v)868 1954 y fr(x)901 1947 y ff(^)g fm(\033)972 1954 y fr(y)1005 1947 y fq(=)g fm(\033)1081 1954 y fr(x)1099 1947 y fq(\))554 2004 y fm(p)577 1987 y fx(0)638 2004 y fq(=)50 b(\()p fm(v)763 2011 y fr(y)791 2004 y ff(\016)9 b fk(h)847 1987 y fq(-1)900 2004 y fq(=)15 b fm(v)972 2011 y fr(x)1000 2004 y ff(\016)10 b fk(h)1057 1987 y fq(-1)1109 2004 y ff(^)16 b fm(\033)1181 2011 y fr(y)1208 2004 y ff(\016)10 b fk(h)1265 1987 y fq(-1)1317 2004 y fq(=)16 b fm(\033)1394 2011 y fr(x)1421 2004 y ff(\016)10 b fk(h)1478 1987 y fq(-1)723 2060 y ff(^)16 b fm(dom)p fq(\()p fm(v)895 2067 y fr(x)911 2060 y fq(\))f ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))e ff(^)h fm(dom)p fq(\()p fm(v)1326 2067 y fr(y)1343 2060 y fq(\))g ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\))723 2117 y ff(^)h fm(dom)p fq(\()p fm(\033)899 2124 y fr(x)915 2117 y fq(\))f ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))e ff(^)h fm(dom)p fq(\()p fm(\033)1334 2124 y fr(y)1351 2117 y fq(\))p ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\)\))289 2234 y(consid)o(\023)-21 b(erons)15 b(d'ab)q(ord)h(que)g fk(x)g fq(=)g fk(y)g

fq(est)g(une)g(pr)o(\023)-21 b(e-)o(\023)g(equation)15 b(dans)h(un)g(\023)-21 b(enonc)o(\023)g(e)14 b fk(p)p fq(.)i(alors,)289 2290 y(la)f(propri)o(\023)-21 b(et)o(\023)g(e)13 b(de)i fk(p)p fq(,)f(et)g(celle)j(de)e(l')o(\023)-21 b(enonc)o(\023)g(e)13 b fk(p')h fq(obten)o(ue)h(par)g(application)h(de) f(la)g(r)o(\022)-21 b(egle)14 b(\(i\))289 2347 y(on)o(t)g(la)i(forme)e (suiv)m(an)o(te)9 b(:)314 2458 y fm(sat)p fq(\()p fk(p)p fq(\))61 b(:)49 b(\()p fm(p)599 2465 y fg(1)628 2458 y ff(^)10 b fm(p)p fq(\))15 b ff(\))g fm(p)807 2465 y fg(2)314 2514 y fm(sat)p fq(\()p fk(p)417 2497 y fx(0)428 2514 y fq(\))50 b(:)f(\()p fm(p)599 2521 y fg(1)628 2514 y ff(^)10 b fm(p)691 2497 y fx(0)703 2514 y fq(\))k ff(\))i fm(p)819 2521 y fg(2)p eop %%page: 37 39 37 38 bop 75 269 a fl(3.3.)29 b(calcul)16 b(de)f(raffinement)990 b fq(37)p 75 353 1800 2 v 75 1225 2 872 v 247 405 a fj(\(c)n(omp)n (osition)15 b(fonctionnel)r(le\))114 461 y fq(\(4a\))51 b fk(y)15 b fq(=)h fk(\(f1)9 b ff(\016)h fk(f2\))15 b fa(b)g fk(x)226 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f1)f fa(b)g fk(\(f2)24 b fa(b)g fk(x\))114 518 y fq(\(4b\))49 b fk(y)15 b fq(=)h fk(f1)f fa(b)g fk(\(f2)24 b fa(b)g fk(x\))185 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(f1)10 b ff(\016)f fk(f2\))15 b fa(b)h fk(x)247 631 y fj(\(c)n(omp)n(osition)f (g)o(\023)-22 b(e)n(om)o(\023)g(etrique\))114 687 y fq(\(5a\))51 b fk(y)15 b fq(=)h fk(x)f fa(c)g fk(\(f1)10 b ff(\016)g fk(f2\))225 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(x)23 b fa(c)i fk(f1)o(\))15 b fa(c)h fk(f2)114 744 y fq(\(5b\))49 b fk(y)15 b fq(=)h fk(\(x)23 b fa(c)h fk(f1\))15 b fa(c)h fk(f2)185 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(x)f fa(c)g fk(\(f1)10 b ff(\016)g fk(f2\))961 800 y fq(et)15 b fm(img)r fq(\()p fk(f2)o fq(\))d ff(\022)h fm(dom)p fq(\()p fk(f1)o fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\033)1596 807 y fr(x)1613 800 y fq(\))247 857 y fj(\(c)n(omp)n(osition)i(r)o (\023)-22 b(e)n(duction\))114 913 y fq(\(6\))74 b fk(y)15 b fq(=)h fk(\(f1)9 b ff(\016)h fk(f2\))15 b fq(;)8 b fa(b)15 b fk(x)205 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f1)f fq(;)8 b fa(b)14 b fk(\(f2)24 b fq(;)8 b fa(b)23 b fk(x\))247 1026 y fj(\(c)n(omp)n(osition)15 b(b)n(ase\))114 1082 y fq(\(7a\))51 b fk(y)15 b fq(=)h fk(\(f1)9 b ff(\016)h fk(f2\))d(::)h(x)228 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f1)7 b(::)g(\(f2)h(::)g(x\))114 1139 y fq(\(7b\))49 b fk(y)15 b fq(=)h fk(f1)7 b(::)g(\(f2)h(::)g(x)o(\))208 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(f1)10 b ff(\016)f fk(f2\))f(::)f(x)961 1195 y fq(et)15 b fk(f1)g fq(et)g fk(f2)g fq(bijectiv)o(es)p 1873 1225 v 75 1227 1800 2 v 378 1304 a fn(t)l(ab.)f fq(3.2)h(-)30 b fj(r)o(\022)-22 b(e)n(gles)15 b(c)n(onc)n(ernant)g(la)h(c)n(omp)n(osition)g(des)g (fonctions)289 1449 y fq(o)q(\022)-24 b(u)17 b fm(p)377 1456 y fg(1)412 1449 y fq(et)g fm(p)490 1456 y fg(2)525 1449 y fq(traduisen)o(t)g(resp)q(ectiv)o(emen)o(t)h(les)g(pr)o(\023)-21 b(e-)16 b(et)h(p)q(ost-)o(\023)-21 b(equations)17 b(de)g fk(p)p fq(.)f(d'apr)o(\022)-21 b(es)289 1506 y(la)15 b(propri)o(\023)-21 b(et)o(\023)g(e)14 b(6,)g(nous)h(dev)o(ons)h(mon)o (trer)e(que)h(\()p fm(p)1138 1513 y fg(1)1167 1506 y ff(^)10 b fm(p)1230 1489 y fx(0)1241 1506 y fq(\))15 b ff(\))h fq(\()p fm(p)1376 1513 y fg(1)1404 1506 y ff(^)10 b fm(p)p fq(\))15 b(est)g(toujours)f(vraie)289 1562 y(p)q(our)h (conclure)i(que)e fk(p')g ff(v)g fk(p)p fq(.)g(il)h(su\016t)f(donc)h (de)f(mon)o(trer)g(que)g fm(p)1433 1546 y fx(0)1457 1562 y ff(\))e fm(p)p fq(.)289 1618 y(en)g(e\013et,)g(de)h

fm(v)550 1625 y fr(y)577 1618 y ff(\016)c fk(h)634 1602 y fq(-1)685 1618 y fq(=)j fm(v)755 1625 y fr(x)783 1618 y ff(\016)d fk(h)840 1602 y fq(-1)890 1618 y fq(et)k fm(\033)968 1625 y fr(y)995 1618 y ff(\016)c fk(h)1052 1602 y fq(-1)1103 1618 y fq(=)j fm(\033)1177 1625 y fr(x)1205 1618 y ff(\016)d fk(h)1262 1602 y fq(-1)1299 1618 y fq(,)j(nous)h(d)o (\023)-21 b(eduisons)13 b(resp)q(ectiv)o(e-)289 1675 y(men)o(t)g fm(v)424 1682 y fr(x)442 1675 y ff(j)455 1682 y fw(dom)p fg(\()p fw(h)p fg(\))579 1675 y fq(=)i fm(v)651 1682 y fr(y)668 1675 y ff(j)681 1682 y fw(dom)p fg(\()p fw(h)p fg(\))806 1675 y fq(et)f fm(\033)884 1682 y fr(x)901 1675 y ff(j)914 1682 y fw(dom)p fg(\()p fw(h)p fg(\))1038 1675 y fq(=)h fm(\033)1114 1682 y fr(y)1131 1675 y ff(j)1144 1682 y fw(dom)p fg(\()p fw(h)p fg(\))1255 1675 y fq(,)e(et)h(comme)g fm(dom)p fq(\()p fm(v)1612 1682 y fr(x)1629 1675 y fq(\),)g fm(dom)p fq(\()p fm(v)1800 1682 y fr(y)1816 1675 y fq(\),)289 1731 y fm(dom)p fq(\()p fm(\033)419 1738 y fr(x)435 1731 y fq(\))h(et)f fm(dom)p fq(\()p fm(\033)650 1738 y fr(y)667 1731 y fq(\))g(son)o(t)g(inclus)i (dans)f fm(dom)p fq(\()p fk(h)p fq(\),)e(il)j(vien)o(t)f fm(v)1377 1738 y fr(x)1409 1731 y fq(=)g fm(v)1481 1738 y fr(y)1513 1731 y fq(et)f fm(\033)1591 1738 y fr(x)1623 1731 y fq(=)h fm(\033)1699 1738 y fr(y)1717 1731 y fq(.)f(donc)289 1788 y fm(sat)p fq(\()p fk(p)392 1771 y fx(0)404 1788 y fq(\))g ff(\))i fm(sat)p fq(\()p fk(p)p fq(\))f(et)g fk(p')g ff(v)h fk(p)p fq(.)289 1913 y(dans)i(le)h(cas)f(o)q(\022)-24 b(u)19 b(les)f(\023)-21 b(equations)18 b(de)h(la)g(r)o(\022)-21 b(egle)18 b(\(i\))g(son)o(t)g(des)h(p)q(ost-)o(\023)-21 b(equations,)18 b(alors)g(les)289 1969 y(propri)o(\023)-21 b(et)o(\023)g(es)13 b(de)j fk(p)f fq(et)g fk(p')g fq(on)o(t)f(la)i (forme)e(suiv)m(an)o(te)9 b(:)314 2082 y fm(sat)p fq(\()p fk(p)p fq(\))61 b(:)49 b fm(p)581 2089 y fg(1)615 2082 y ff(\))16 b fq(\()p fm(p)717 2089 y fg(2)745 2082 y ff(^)10 b fm(p)p fq(\))314 2138 y fm(sat)p fq(\()p fk(p)417 2122 y fx(0)428 2138 y fq(\))50 b(:)f fm(p)581 2145 y fg(1)615 2138 y ff(\))16 b fq(\()p fm(p)717 2145 y fg(2)745 2138 y ff(^)10 b fm(p)808 2122 y fx(0)835 2138 y fq(\))289 2235 y(nous)i(a)o(v)o(ons)g(mon)o(tr)o(\023)-21 b(e)11 b(que)i fm(p)782 2219 y fx(0)806 2235 y ff(\))g fm(p)g fq(et)g(donc)g(ici,)g fm(sat)p fq(\()p fk(p)1230 2219 y fx(0)1242 2235 y fq(\))g ff(\))g fm(sat)p fq(\()p fk(p)p fq(\))g(d'apr)o(\022)-21 b(es)11 b(la)i(propri)o(\023)-21 b(et)o(\023)g(e)289 2292 y(6.)14 b(p)o(ar)g(cons)o(\023)-21 b(equen)o(t,)15 b fk(p)g ff(v)g fk(p')p fq(.)289 2429 y fo(r)o(\022)-25 b(egle)15 b(\(i)q(i\))e fq(dans)f(ce)h(qui)g(suit,)g (nous)g(ne)g(n')o(\023)-21 b(etablissons)12 b(la)h(preuv)o(e)g(que)g(p) q(our)f(le)i(cas,)e(o)q(\022)-24 b(u)289 2486 y(les)13 b(\023)-21 b(equations)13 b(son)o(t)g(des)h(p)q(ost-)o(\023)-21 b(equations.)13 b(la)h(d)o(\023)-21 b(emonstration)12 b(p)q(our)i(une)h(pr)o(\023)-21 b(e-)o(\023)g(equation)289 2542 y(est)15 b(la)g(m)o(^)-21 b(eme)14 b(que)i(pr)o(\023)-21 b(ec)o(\023)g(edemmen)o(t.)p eop %%page: 38 40 38 39 bop 75 269 a fq(38)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)p 75 353 1800 2 v 75 1225 2 872 v 270 405 a fj(\(distributivit)o(\023)-22 b(e)16 b(g)o(\023)-22 b(e)n(om)o(\023)g(etrique\))114 461 y fq(\(8a\))74 b fk(y)15 b fq(=)g fk(\(x1)24 b(/&/)7 b(x2\))15 b fa(c)h fk(f)221 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(x1)23 b fa(c)i fk(f\))e(/&/)8 b(\(x2)23

b fa(c)h fk(f\))114 518 y fq(\(8b\))72 b fk(y)15 b fq(=)g fk(\(x1)24 b fa(c)g fk(f\))g(/&/)7 b(\(x2)24 b fa(c)g fk(f\))49 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(x1)23 b(/&/)8 b(x2\))15 b fa(c)g fk(f)1040 574 y fq(et)36 b fm(dom)p fq(\()p fk(f)p fq(\))12 b ff(\022)h fm(dom)p fq(\()p fm(\033)1450 581 y fr(x1)1482 574 y fq(\))d ff(\\)g fm(dom)p fq(\()p fm(\033)1680 581 y fr(x2)1712 574 y fq(\))15 b ff(^)1114 631 y fm(img)r fq(\()p fk(f)p fq(\))c ff(\022)i fm(dom)p fq(\()p fm(v)1439 638 y fr(x1)1471 631 y fq(\))d ff(\\)g fm(dom)p fq(\()p fm(v)1665 638 y fr(x2)1698 631 y fq(\))270 687 y fj(\(p)n(artition)16 b(g)o(\023)-22 b(e)n(om)o(\023)g(etrique\))114 744 y fq(\(9a\))74 b fk(y)15 b fq(=)g fk(x)g fa(c)h fk(\(f1)10 b fq(#)g fk(f2\))266 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(x)24 b fa(c)g fk(f1\))f(/&/)8 b(\(x)23 b fa(c)i fk(f2)o(\))114 800 y fq(\(9b\))72 b fk(y)15 b fq(=)g fk(\(x)24 b fa(c)g fk(f1\))g(/&/)7 b(\(x)24 b fa(c)g fk(f2\))49 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(x)f fa(c)h fk(\(f1)9 b fq(#)h fk(f2\))1040 857 y fq(et)16 b fm(dom)p fq(\()p fk(f1)o fq(\))9 b ff(\\)i fm(dom)p fq(\()p fk(f2)o fq(\))h(=)h ff(fg)270 913 y fj(\(p)n(artition)j(r)o (\023)-22 b(e)n(duction\))114 969 y fq(\(10\))74 b fk(y)15 b fq(=)g fk(\(f1)10 b fq(#)g fk(f2\))15 b fq(;)8 b fa(b)15 b fk(x)246 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(f1)23 b fq(;)8 b fa(b)24 b fk(x\))f(/&/)8 b(\(f2)23 b fq(;)8 b fa(b)24 b fk(x\))270 1082 y fj(\(distributivit)o(\023)-22 b(e)16 b(b)n(ase\))114 1139 y fq(\(11a\))51 b fk(y)15 b fq(=)g fk(\(f)8 b(::)g(x1\))23 b(/&/)8 b(\(f)f(::)h(x2\))88 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(f)7 b(::)h(\(x1)23 b(/&/)8 b(x2\))114 1195 y fq(\(11b\))49 b fk(y)15 b fq(=)g fk(f)8 b(::)f(\(x1)24 b(/&/)7 b(x2\))224 b ff(\000)-7 b(!)50 b fk(y)15 b fq(=)h fk(\(f)8 b(::)f(x1\))24 b(/&/)7 b(\(f)h(::)g(x2)o(\))p 1873 1225 v 75 1227 1800 2 v 504 1304 a fn(t)l(ab.)14 b fq(3.3)g(-)31 b fj(r)o(\022)-22 b(e)n(gles)15 b(c)n(onc)n(ernant)g(l'op)o(\023)-22 b(er)n(ateur)17 b fk(/&/)289 1433 y fq(soien)o(t)f(les)g(\023)-21 b(enonc)o(\023)g(es,) 15 b fk(p)h fq(a)o(y)o(an)o(t)e fk(h)8 b(::)f(x)16 b fq(=)g fk(h)8 b(::)f(y)16 b fq(parmi)g(ses)h(p)q(ost-)o(\023)-21 b(equations,)15 b(et)h fk(p')g fq(ob-)289 1489 y(ten)o(u)f(par)g(la)g (r)o(\022)-21 b(egle)15 b(\(i)q(i\).)g(leurs)h(propri)o(\023)-21 b(et)o(\023)g(es)14 b(son)o(t)7 b(:)314 1596 y fm(sat)p fq(\()p fk(p)p fq(\))61 b(:)49 b fm(p)581 1603 y fg(1)615 1596 y ff(\))16 b fq(\()p fm(p)717 1603 y fg(2)750 1596 y ff(^)f fm(v)817 1603 y fr(y)845 1596 y ff(\016)10 b fk(h)902 1579 y fq(-1)939 1596 y fq(=)16 b fm(v)1012 1603 y fr(x)1039 1596 y ff(\016)10 b fk(h)1096 1579 y fq(-1)1148 1596 y ff(^)16 b fm(\033)1220 1603 y fr(y)1248 1596 y ff(\016)9 b fk(h)1304 1579 y fq(-1)1342 1596 y fq(=)15 b fm(\033)1418 1603 y fr(x)1446 1596 y ff(\016)9 b fk(h)1502 1579 y fq(-1)558 1652 y ff(^)16 b fm(dom)p fq(\()p fm(v)730 1659 y fr(y)747 1652 y fq(\))f ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\))f ff(^)h fm(dom)p fq(\()p fm(v)1161 1659 y fr(x)1178 1652 y fq(\))g ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\)\))314 1708 y fm(sat)p fq(\()p fk(p)417 1692 y fx(0)428 1708 y fq(\))50 b(:)f fm(p)581 1715 y fg(1)615 1708 y ff(\))16 b fq(\()p fm(p)717 1715 y fg(2)735 1708 y ff(^)f fm(v)802 1715 y fr(y)835 1708 y fq(=)g fm(v)907 1715 y fr(x)940 1708 y ff(^)h fm(\033)1012 1715 y fr(y)1044 1708 y fq(=)g fm(\033)1121 1715 y fr(x)558 1765 y ff(^)g fm(dom)p fq(\()p fm(v)730 1772 y fr(x)747

1765 y fq(\))f ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\)\))289 1829 y(nous)h(a)o(v)o(ons)f(ici)i fm(sat)p fq(\()p fk(p)695 1813 y fx(0)707 1829 y fq(\))e ff(\))i fm(sat)p fq(\()p fk(p)p fq(\))f(car)f(\()p fm(v)1055 1836 y fr(y)1088 1829 y fq(=)i fm(v)1162 1836 y fr(x)1195 1829 y ff(^)f fm(\033)1267 1836 y fr(y)1301 1829 y fq(=)g fm(\033)1378 1836 y fr(x)1395 1829 y fq(\))g ff(\))g fq(\()p fm(v)1530 1836 y fr(y)1558 1829 y ff(\016)9 b fk(h)1614 1813 y fq(-1)1652 1829 y fq(=)16 b fm(v)1725 1836 y fr(x)1753 1829 y ff(\016)9 b fk(h)1809 1813 y fq(-1)289 1885 y ff(^)15 b fm(\033)360 1892 y fr(y)388 1885 y ff(\016)10 b fk(h)445 1869 y fq(-1)482 1885 y fq(=)15 b fm(\033)558 1892 y fr(x)586 1885 y ff(\016)10 b fk(h)643 1869 y fq(-1)680 1885 y fq(\).)131 1975 y(2.)22 b fo(r)o(\022)-25 b(egles)23 b(\023)-25 b(el)o(\023)g(emen)o(taires)8 b fq(:)21 b(il)h(faut)f (ensuite)g(\023)-21 b(etablir)22 b(la)f(preuv)o(e)h(p)q(our)f(c)o (hacune)i(des)e(r)o(\022)-21 b(egles)189 2032 y(list)o(\023)g(ees)15 b(dans)g(les)h(tables.)f(nous)h(n'en)f(mon)o(trons)f(ici)j(que)e (quelques-unes.)243 2121 y({)23 b(p)o(ar)11 b(exemple,)h(p)q(our)g(la)g (r)o(\022)-21 b(egle)12 b(\(5a\))e(nous)i(dev)o(ons)g(mon)o(trer)f(que) h(l')o(\023)-21 b(enonc)o(\023)g(e)11 b fk(p)g fq(a)o(y)o(an)o(t)f (parmi)289 2178 y(ses)15 b(p)q(ost-)o(\023)-21 b(equations)15 b fk(y)g fq(=)h fk(x)f fa(c)g fk(\(f1)10 b ff(\016)g fk(f2\))15 b fq(est)g(ra\016n)o(\023)-21 b(e)14 b(par)h(le)h(programme) e fk(p')h fq(a)o(y)o(an)o(t,)f(\022)-23 b(a)289 2234 y(la)15 b(place)h(de)g(cette)e(\023)-21 b(equation,)14 b(l')o(\023)-21 b(equation)15 b fk(y)g fq(=)h fk(\(x)23 b fa(c)h fk(f1\))15 b fa(c)h fk(f2)7 b fq(:)289 2353 y fm(sat)p fq(\()p fk(p)p fq(\))g(:)314 2408 y fm(p)337 2415 y fg(1)370 2408 y ff(\))50 b fq(\()p fm(p)506 2415 y fg(2)540 2408 y ff(^)15 b fq(\()p fm(v)625 2415 y fr(y)658 2408 y fq(=)g fm(v)730 2415 y fr(x)758 2408 y ff(\016)10 b fk(f1)f ff(\016)h fk(f2)p fq(\))15 b ff(^)g fq(\()p fm(\033)1051 2415 y fr(y)1084 2408 y fq(=)g fm(\033)1160 2415 y fr(x)1178 2408 y fq(\))f ff(^)465 2465 y fq(\()p fm(dom)p fq(\()p fm(v)609 2472 y fr(x)626 2465 y fq(\))h ff(\022)g fm(dom)p fq(\()p fm(\033)839 2472 y fr(x)856 2465 y fq(\))g ff(^)g fm(dom)p fq(\()p fm(v)1060 2472 y fr(y)1077 2465 y fq(\))g ff(\022)g fm(dom)p fq(\()p fm(\033)1290 2472 y fr(y)1307 2465 y fq(\))g ff(^)465 2521 y fm(dom)p fq(\()p fk(f1)10 b ff(\016)f fk(f2)p fq(\))15 b ff(\022)g fm(dom)p fq(\()p fm(\033)920 2528 y fr(x)937 2521 y fq(\))g ff(^)g fm(img)r fq(\()p fk(f1)9 b ff(\016)h fk(f2)p fq(\))k ff(\022)i fm(dom)p fq(\()p fm(v)1460 2528 y fr(x)1477 2521 y fq(\)\))p eop %%page: 39 41 39 40 bop 75 269 a fl(3.3.)29 b(calcul)16 b(de)f(raffinement)990 b fq(39)289 397 y fm(sat)p fq(\()p fk(p)392 380 y fx(0)404 397 y fq(\))7 b(:)314 452 y fm(p)337 459 y fg(1)370 452 y ff(\))50 b fq(\()p fm(p)506 459 y fg(2)540 452 y ff(^)15 b fq(\()p fm(v)625 459 y fr(y)658 452 y fq(=)g fm(v)730 459 y fr(x)758 452 y ff(\016)10 b fk(f1)f ff(\016)h fk(f2)p fq(\))15 b ff(^)g fq(\()p fm(\033)1051 459 y fr(y)1084 452 y fq(=)g fm(\033)1160 459 y fr(x)1178 452 y fq(\))f ff(^)465 508 y fq(\()p fm(dom)p fq(\()p fm(v)609 515 y fr(x)626 508 y fq(\))h ff(\022)g fm(dom)p fq(\()p fm(\033)839 515 y fr(x)856 508 y fq(\))g ff(^)g fm(dom)p fq(\()p fm(v)1060 515 y fr(y)1077 508 y fq(\))g ff(\022)g fm(dom)p fq(\()p fm(\033)1290 515 y fr(y)1307 508 y fq(\))g ff(^)465 565 y fm(dom)p fq(\()p fk(f1)p fq(\))f ff(\022)i fm(dom)p

fq(\()p fm(\033)830 572 y fr(x)847 565 y fq(\))e ff(^)i fm(dom)p fq(\()p fk(f2)p fq(\))e ff(\022)h fm(dom)p fq(\()p fm(\033)1289 572 y fr(x)1306 565 y fq(\))g ff(^)465 621 y fm(img)r fq(\()p fk(f1)p fq(\))f ff(\022)h fm(dom)p fq(\()p fm(v)819 628 y fr(x)836 621 y fq(\))g ff(^)g fm(img)r fq(\()p fk(f2)p fq(\))e ff(\022)j fm(dom)p fq(\()p fm(v)1268 628 y fr(x)1295 621 y ff(\016)10 b fk(f1)p fq(\)\))289 703 y(nous)15 b(dev)o(ons)g(donc)h(mon)o(trer)e(que)8 b(:)289 790 y(\(2\))362 718 y fx(\()416 762 y fm(img)r fq(\()p fk(f1)n fq(\))13 b ff(\022)g fm(dom)p fq(\()p fm(v)765 769 y fr(x)781 762 y fq(\))416 818 y fm(img)r fq(\()p fk(f2)n fq(\))g ff(\022)g fm(dom)p fq(\()p fm(v)765 825 y fr(x)792 818 y ff(\016)c fk(f1)p fq(\))961 790 y ff(\))43 b fq(\(1\))30 b ff(f)p fm(img)r fq(\()p fk(f1)8 b ff(\016)i fk(f2)o fq(\))j ff(\022)g fm(dom)p fq(\()p fm(v)1600 797 y fr(x)1617 790 y fq(\))289 907 y(de)i(la)g(d)o(\023)-21 b(e\014nition)16 b(de)g(la)f(comp)q(osition)9 b(:)15 b fm(img)r fq(\()p fk(f1)8 b ff(\016)i fk(f2)p fq(\))i ff(\022)h fm(img)r fq(\()p fk(f1)n fq(\))289 1001 y(donc)i(\(2\))f(se)i (r)o(\023)-21 b(e)o(\023)g(ecrit)6 b(:)686 929 y fx(\()740 972 y fm(img)r fq(\()p fk(f1)i ff(\016)i fk(f2)p fq(\))i ff(\022)h fm(img)r fq(\()p fk(f1)n fq(\))g ff(\022)g fm(dom)p fq(\()p fm(v)1402 979 y fr(x)1418 972 y fq(\))740 1029 y fm(img)r fq(\()p fk(f2)n fq(\))f ff(\022)h fm(dom)p fq(\()p fm(v)1088 1036 y fr(x)1115 1029 y ff(\016)d fk(f1)p fq(\))289 1094 y(et)15 b(on)g(a)g(\(2\))f ff(\))i fq(\(1\),)d(et)i (donc)h fm(sat)p fq(\()p fk(p')p fq(\))f ff(\))h fm(sat)p fq(\()p fk(p)p fq(\).)243 1160 y({)23 b(p)o(our)17 b(la)h(r)o(\022)-21 b(egle)17 b(\(7a\),)f(nous)i(mon)o(trons)e(la)i(relation)h(de)f (ra\016nemen)o(t)f(en)o(tre)g(l')o(\023)-21 b(enonc)o(\023)g(e)17 b fk(p)p fq(,)289 1216 y(a)o(y)o(an)o(t)c(parmi)j(ses)f(p)q(ost-)o (\023)-21 b(equations)15 b fk(y)g fq(=)g fk(\(f1)10 b ff(\016)g fk(f2\))d(::)g(x)p fq(,)15 b(et)g(sp)o(\023)-21 b(eci\014)o(\023)g(e)15 b(par)f fm(sat)p fq(\()p fk(p)p fq(\))8 b(:)314 1271 y fm(p)337 1278 y fg(1)370 1271 y ff(\))50 b fq(\()p fm(p)506 1278 y fg(2)540 1271 y ff(^)15 b fq(\()p fm(v)625 1278 y fr(y)658 1271 y fq(=)g fm(v)730 1278 y fr(x)758 1271 y ff(\016)10 b fk(f2)838 1254 y fq(-1)886 1271 y ff(\016)g fk(f1)966 1254 y fq(-1)1004 1271 y fq(\))15 b ff(^)g fq(\()p fm(\033)1126 1278 y fr(y)1158 1271 y fq(=)h fm(\033)1235 1278 y fr(x)1278 1271 y ff(\016)9 b fk(f2)1358 1254 y fq(-1)1406 1271 y ff(\016)g fk(f1)1486 1254 y fq(-1)1523 1271 y fq(\))15 b ff(^)465 1328 y fm(dom)p fq(\()p fm(v)591 1335 y fr(x)608 1328 y fq(\))g ff(\022)h fm(dom)p fq(\()p fm(\033)822 1335 y fr(x)838 1328 y fq(\))f ff(^)h fm(dom)p fq(\()p fm(v)1043 1335 y fr(y)1059 1328 y fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\033)1273 1335 y fr(y)1290 1328 y fq(\))e ff(^)465 1384 y fm(dom)p fq(\()p fm(v)591 1391 y fr(x)608 1384 y fq(\))h ff(\022)h fm(dom)p fq(\()p fk(f1)9 b ff(\016)h fk(f2)p fq(\)\))289 1466 y(et)k(l')o(\023)-21 b(enonc)o(\023)g(e)13 b fk(p')p fq(,)h(a)o(y)o(an)o(t)f(parmi)i(ses)f(p)q(ost-)o(\023)-21 b(equations)14 b fk(y)g fq(=)h fk(f1)7 b(::)h(\(f2)f(::)h(x\))p fq(,)14 b(sp)o(\023)-21 b(eci\014)o(\023)g(e)14 b(par)289 1522 y fm(sat)p fq(\()p fk(p)392 1506 y fx(0)404 1522 y fq(\))7 b(:)314 1577 y fm(p)337 1584 y fg(1)370 1577 y ff(\))50 b fq(\()p fm(p)506 1584 y fg(2)540 1577 y ff(^)15 b fq(\()p fm(v)625 1584 y fr(y)658 1577 y fq(=)g fm(v)730 1584 y fr(x)758 1577 y ff(\016)10 b fk(f2)838 1560 y fq(-1)886 1577 y ff(\016)g fk(f1)966 1560 y fq(-1)1004

1577 y fq(\))15 b ff(^)g fq(\()p fm(\033)1126 1584 y fr(y)1158 1577 y fq(=)h fm(\033)1235 1584 y fr(x)1278 1577 y ff(\016)9 b fk(f2)1358 1560 y fq(-1)1406 1577 y ff(\016)g fk(f1)1486 1560 y fq(-1)1523 1577 y fq(\))15 b ff(^)465 1634 y fm(dom)p fq(\()p fm(v)591 1641 y fr(x)608 1634 y fq(\))g ff(\022)h fm(dom)p fq(\()p fm(\033)822 1641 y fr(x)838 1634 y fq(\))f ff(^)h fm(dom)p fq(\()p fm(v)1043 1641 y fr(y)1059 1634 y fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\033)1273 1641 y fr(y)1290 1634 y fq(\))e ff(^)465 1690 y fm(dom)p fq(\()p fm(v)591 1697 y fr(x)618 1690 y ff(\016)c fk(f2)699 1673 y fq(-1)736 1690 y fq(\))15 b ff(\022)h fm(dom)p fq(\()p fk(f1)p fq(\))d ff(^)j fm(dom)p fq(\()p fm(v)1175 1697 y fr(x)1192 1690 y fq(\))f ff(\022)g fm(dom)p fq(\()p fk(f2)p fq(\)\))289 1772 y(il)h(reste)f(\022)-23 b(a)15 b(mon)o(trer)f(que)8 b(:)289 1859 y(\(2\))362 1787 y fx(\()416 1831 y fm(dom)p fq(\()p fm(v)542 1838 y fr(x)569 1831 y ff(\016)i fk(f2)650 1813 y fx(\000)p fg(1)694 1831 y fq(\))i ff(\022)h fm(dom)p fq(\()p fk(f1)o fq(\))416 1887 y fm(dom)p fq(\()p fm(v)542 1894 y fr(x)559 1887 y fq(\))f ff(\022)h fm(dom)p fq(\()p fk(f2)o fq(\))1012 1859 y ff(\))44 b fq(\(1\))29 b ff(f)p fm(dom)p fq(\()p fm(v)1338 1866 y fr(x)1355 1859 y fq(\))12 b ff(\022)h fm(dom)p fq(\()p fk(f1)c ff(\016)h fk(f2)p fq(\))289 1976 y(p)o(our)18 b(cela,)h(nous)f(utilisons)j(la)d(propri)o(\023)-21 b(et)o(\023)g(e)18 b(suiv)m(an)o(te)8 b(:)18 b(si)h fm(g)h fq(est)f(une)g(fonction)g(et)f fm(f)24 b fq(est)289 2033 y(une)15 b(bijection,)i(alors)e fm(dom)p fq(\()p fm(g)10 b ff(\016)g fm(f)878 2016 y fx(\000)p fg(1)923 2033 y fq(\))i(=)h fm(f)5 b fq(\()p fm(dom)p fq(\()p fm(g)r fq(\)\).)289 2093 y(comme)14 b fm(dom)p fq(\()p fm(v)581 2100 y fr(x)598 2093 y fq(\))h ff(\022)g fm(dom)p fq(\()p fk(f2)p fq(\))f(nous)i(a)o(v)o(ons)6 b(:)409 2149 y fm(dom)p fq(\()p fm(v)535 2156 y fr(x)562 2149 y ff(\016)k fk(f2)642 2131 y fq(-1)680 2149 y fq(\))k ff(\022)i fm(dom)p fq(\()p fk(f1)p fq(\))314 2205 y ff(\))50 b fk(f2)p fq(\()p fm(dom)p fq(\()p fm(v)601 2212 y fr(x)617 2205 y fq(\)\))15 b ff(\022)g fm(dom)p fq(\()p fk(f1)p fq(\))314 2261 y ff(\))50 b fk(f2)457 2244 y fq(-1)494 2261 y fq(\()p fk(f2)p fq(\()p fm(dom)p fq(\()p fm(v)704 2268 y fr(x)720 2261 y fq(\)\)\))14 b ff(\022)h fk(f2)886 2244 y fq(-1)924 2261 y fq(\()p fm(dom)p fq(\()p fk(f1)p fq(\)\))314 2318 y ff(,)50 b fm(dom)p fq(\()p fm(v)535 2325 y fr(x)552 2318 y fq(\))14 b ff(\022)i fm(dom)p fq(\()p fk(f1)9 b ff(\016)h fk(f2)p fq(\))289 2400 y(nous)15 b(a)o(v)o(ons)f(\(2\))g ff(\))i fq(\(1\),)e(et)h(donc)g fm(sat)p fq(\()p fk(p')p fq(\))g ff(\))h fm(sat)p fq(\()p fk(p)p fq(\).)146 2486 y fa(\003)p eop %%page: 40 42 40 41 bop 75 269 a fq(40)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)75 397 y ft(3.4)66 b(un)15 b(outil)i(visuel)f(p)r(our)g(la)f(conception)h(de)f(programmes) c(:)j fs(vpei)146 499 y fq(il)i(ressort)e(de)i(la)g(pr)o(\023)-21 b(esen)o(tation)14 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)14 b(que)i(le)g(placemen)o(t)g(g)o(\023)-21 b(eom)o(\023)g(etrique)14 b(des)i(donn)o(\023)-21 b(ees)15 b(est)75 555 y(l')o(\023)-21 b(el)o(\023)g(emen)o(t)12 b(d)o(\023)-21 b(eterminan)o(t)12 b(de)h(la)g(conception)h(d'un)f(programme.)f(la)h(d)o(\023)-21 b(e\014nition)13 b(de)g(l'op)o(\023)-21 b(eration)12 b(fonc-)75 612 y(tionnelle)19 b(imp)q(ose)f(en)f(e\013et)g(que)g(p)q (our)h(calculer)g(des)g(v)m(aleurs,)f(ces)h(v)m(aleurs)g(doiv)o(en)o(t)

f(se)g(trouv)o(er)f(en)75 668 y(un)21 b(m)o(^)-21 b(eme)20 b(p)q(oin)o(t)h(du)g(domaine)g(discret.)g(ce)f(c)o(hoix)h(doit)g(a)o(v) o(oir)f(une)h(in\015uence)i(fa)o(v)o(orable)d(sur)g(la)75 725 y(p)q(erformance)d(des)g(programmes,)e(car)i(il)h(oblige)f(le)h (programmeur)e(\022)-23 b(a)16 b(expliciter)j(les)f(d)o(\023)-21 b(eplacemen)o(ts)75 781 y(\(les)19 b(comm)o(unications)h(dans)f(une)h (mac)o(hine)g(parall)o(\022)-21 b(ele\))19 b(n)o(\023)-21 b(ecessaires)19 b(a)o(v)m(an)o(t)g(de)g(p)q(ouv)o(oir)h(e\013ectuer)75 838 y(un)d(calcul.)h(ce)e(c)o(hoix)h(est)g(similaire)h(\022)-23 b(a)16 b(celui)j(fait)d(dans)h(la)f(plupart)i(des)e(langages)h (data-parall)o(\022)-21 b(eles,)75 894 y(comme)20 b(c)267 878 y fx(\003)286 894 y fq([54)o(],)g(cm-f)l(ortan[55)m(],)g(mpl[38)o (],)f(hyp)q(er-c[45],)g fj(etc)p fq(,)h(qui)h(imp)q(osen)o(t)g(la)f(lo) q(calit)o(\023)-21 b(e)21 b(des)75 951 y(donn)o(\023)-21 b(ees)14 b(aux)g(pro)q(cesseurs)h(virtuels)g(qui)g(les)g(mo)q(di\014en) o(t.)g(en)g(rev)m(anc)o(he,)f(cette)g(d)o(\023)-21 b(emarc)o(he)14 b(demande)75 1007 y(un)19 b(e\013ort)e(suppl)o(\023)-21 b(emen)o(taire)20 b(au)e(programmeur)g(qui)h(doit)g(expliciter)h(le)g (placemen)o(t.)f(p)o(our)f(cela,)g(il)75 1064 y(nous)13 b(a)f(sem)o(bl)o(\023)-21 b(e)12 b(utile)i(de)e(prop)q(oser)h(un)g (outil)g(fournissan)o(t)f(une)h(aide)h(graphique)f(aux)f(manipulations) 75 1120 y(g)o(\023)-21 b(eom)o(\023)g(etriques)13 b(que)j(doit)f (e\013ectuer)h(l'utilisateur.)75 1300 y fw(3.4.1)55 b(pr)o(\023)-27 b(esen)n(tation)18 b(de)h(l'outil)146 1386 y fq(un)13 b(protot)o(yp)q(e)g(d'outil)h(nomm)o(\023)-21 b(e)12 b fn(vpei)840 1369 y fg(1)858 1386 y fq(,)h(a)f(\023)-21 b(et)o(\023)g(e)11 b(con\030)-20 b(cu)14 b(p)q(our)g(p)q(ermettre)f(la) h(conception)g(de)g(pro-)75 1442 y(grammes)e fn(pei)g fq(par)g(simple)i(manipulations)g(graphiques.)f(l'utilisateur)g(p)q (eut)g(cr)o(\023)-21 b(eer)12 b(des)h(c)o(hamps)f(de)75 1499 y(donn)o(\023)-21 b(ees)12 b(dans)g fe(z)-13 b fq(,)9 b fe(z)435 1482 y fg(2)463 1499 y fq(ou)j fe(z)556 1482 y fg(3)572 1499 y fq(,)f(et)h(leur)h(appliquer)h(ensuite)f(une)f(op)o (\023)-21 b(eration)11 b(fonctionnelle,)j(une)f(op)o(\023)-21 b(era-)75 1555 y(tion)13 b(de)h(routage)e(ou)h(de)g(c)o(hangemen)o(t)g (de)h(base.)e(p)o(our)h(appliquer)h(une)g(op)o(\023)-21 b(eration,)12 b(l'utilisateur)i(doit)75 1612 y(en)o(trer)k(au)g(cla)o (vier)i(la)e(fonction)h(et,)f(selon)h(l'op)o(\023)-21 b(eration)17 b(c)o(hoisie,)j(le)f(logiciel)i(p)q(ermet)d(de)h (visualiser)75 1668 y(sc)o(h)o(\023)-21 b(ematiquemen)o(t)15 b(les)h(e\013ets)f(de)h(cette)f(op)o(\023)-21 b(eration)14 b(en)i(mon)o(tran)o(t)e(l'asp)q(ect)i(du)g(c)o(hamp)f(de)h(donn)o(\023) -21 b(ees)75 1725 y(r)o(\023)g(esultan)o(t.)14 b(quand)j(le)f(c)o(hamp) g(obten)o(u)g(corresp)q(ond)g(aux)g(atten)o(tes,)e(il)j(p)q(eut)f(^)-21 b(etre)14 b(v)m(alid)o(\023)-21 b(e)17 b(et)f(devien)o(t)75 1781 y(un)g(c)o(hamp)f(du)h(programme)e(sur)i(lequel)h(on)e(p)q(eut)h (appliquer)h(de)f(nouv)o(elles)h(op)o(\023)-21 b(erations.)14 b(apr)o(\022)-21 b(es)14 b(v)m(a-)75 1838 y(lidation,)j(le)h(texte)e (de)h(l')o(\023)-21 b(equation)16 b(d)o(\023)-21 b(e\014nissan)o(t)16 b(ce)h(nouv)o(eau)f(c)o(hamp)g(est)h(a\016c)o(h)o(\023)-21 b(e)15 b(dans)h(une)h(fen)o(^)-21 b(etre)75 1894 y(texte.)18 b(l'outil)h(autorise)f(\023)-21 b(egalemen)o(t)17 b(certaines)i(d)o (\023)-21 b(e\014nitions)19 b(r)o(\023)-21 b(ecursiv)o(es)18 b(de)h(c)o(hamps)g(de)g(donn)o(\023)-21 b(ees)7 b(:)75 1951 y(apr)o(\022)-21 b(es)14 b(a)o(v)o(oir)g(d)o(\023)-21 b(e\014ni)15 b(un)h(pas)f(de)g(la)g(r)o(\023)-21 b(ecurrence,)15 b(p)q(ouv)m(an)o(t)f(^)-21 b(etre)14 b(constitu)o(\023)-21 b(e)14 b(de)h(plusieurs)i(op)o(\023)-21 b(erations,)75 2007 y(il)20 b(p)q(eut)g(r)o(\023)-21 b(eit)o(\023)g(erer)17 b(l'application)k(de)f(cette)f(suite)g(d'op)o(\023)-21

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00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000f4cccccccccccccccccc66f4cc cccccccccccccccc6600000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000f4cccccccccccccccccc66f466666666666666666666 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000f466666666666666666666 grestore showpage %%enddocument endtexfig 439 1490 a fn(fig.)15 b fq(3.2)f(-)31 b fj(construction)15 b(d'un)i(pr)n(o)n(gr)n(amme)f(ave)n(c)g fn(vpei)75 1633 y fq(sym)o(b)q(olis)o(\023)-21 b(es)17 b(par)f(des)h(p)q(oin)o(ts)g fj(\()-9 b(\()q fq(noircis)p fj(\))g(\))q fq(.)16 b(l'application)j(du) e(pro)q(duit)g(\022)-23 b(a)16 b(tous)g(les)i(couples)g(donne)f(le)75 1689 y(c)o(hamp)f fk(t)p fq(.)g(on)g(utilise)j(ensuite)e(une)g(r)o (\023)-21 b(ecurrence)16 b(p)q(our)g(faire)g(la)h(somme)f(de)g(tous)g (les)h(pro)q(duits.)f(un)75 1746 y(pas)k(de)g(r)o(\023)-21 b(ecurrence)19 b(est)g(d)o(\023)-21 b(ecrit)20 b(comme)f(la)h(somme)f (du)i(c)o(hamp)e fk(t)h fq(a)o(v)o(ec)f(lui-m)o(^)-21 b(eme)20 b(d)o(\023)-21 b(ecal)o(\023)g(e)19 b(d'une)75

1802 y(p)q(osition)g(dans)f(la)g(direction)h(\(0)p fm(;)8 b fq(0)p fm(;)g fq(1\).)14 b(cette)i(\023)-21 b(etap)q(e)17 b(est)g(repr)o(\023)-21 b(esen)o(t)o(\023)g(ee)17 b(dans)g(la)h(fen)o (^)-21 b(etre)17 b(du)h(milieu)75 1859 y(dans)e(la)g(\014gure)g(3.2.)f (ensuite,)h(l'utilisateur)i(p)q(eut)e(r)o(\023)-21 b(ep)o(\023)g(eter) 14 b(l')o(\023)-21 b(etap)q(e)16 b(de)g(r)o(\023)-21 b(ecurrence)16 b(pr)o(\023)-21 b(ec)o(\023)g(edemmen)o(t)75 1915 y(d)o(\023)g(e\014nie)16 b(en)g(appuy)o(an)o(t)e(sur)h(un)h(b)q (outon,)f(et)g(v)o(oir)g(\022)-23 b(a)15 b(c)o(haque)f(\023)-21 b(etap)q(e)15 b(l'asp)q(ect)h(des)f(c)o(hamps.)146 2031 y(une)f(fois)g(l'ensem)o(ble)i(des)e(c)o(hamps)g(cr)o(\023)-21 b(e)o(\023)g(es,)12 b(les)i(\023)-21 b(equations)13 b(en)o(tre)h(les)h (c)o(hamps)f(ainsi)h(que)f(les)h(d)o(\023)-21 b(e\014-)75 2087 y(nitions)16 b(des)g(fonctions)g(utilis)o(\023)-21 b(ees)16 b(p)q(euv)o(en)o(t)f(^)-21 b(etre)15 b(simplemen)o(t)h(rapp)q (el)o(\023)-21 b(ees)16 b(p)q(our)g(obtenir)g(le)g(texte)f(de)75 2144 y(programme)f fn(pei)p fq(.)g(notons)f(\023)-21 b(egalemen)o(t)13 b(qu'un)i(c)o(hamp)g(in)o(term)o(\023)-21 b(ediaire)15 b(in)o(utile)i(p)q(eut)d(^)-21 b(etre)13 b(supprim)o(\023)-21 b(e)75 2200 y(apr)o(\022)g(es)11 b(a)o(v)o(oir)h(substitu)o(\023)-21 b(e,)12 b(dans)g(les)i(d)o(\023)-21 b(e\014nitions)13 b(des)g(c)o(hamps)f(l'utilisan)o(t,)h(son)g(nom)f (par)g(sa)g(d)o(\023)-21 b(e\014nition.)75 2338 y fw(3.4.2)55 b(in)n(t)o(\023)-27 b(er)o(^)g(ets-limit)o(ations)146 2429 y fn(vpei)13 b fq(p)q(ermet)h(la)g(construction)f(pas-\022)-23 b(a-pas)14 b(du)g(programme)e(a)o(v)o(ec)h(un)h(con)o(tr^)-23 b(ole)14 b(visuel)h(sur)f(l'as-)75 2486 y(p)q(ect)20 b(de)f(c)o(haque)g(nouv)o(eau)h(c)o(hamp)f(cr)o(\023)-21 b(e)o(\023)g(e.)17 b(a)i(c)o(haque)g(fois)g(qu'un)h(nouv)o(eau)f(c)o (hamp)g(est)g(d)o(\023)-21 b(e\014ni,)19 b(le)75 2542 y(placemen)o(t)h(des)g(v)m(aleurs)h(est)f(calcul)o(\023)-21 b(e)20 b(sur)g(un)g(p)q(etit)g(nom)o(bre)g(de)g(p)q(oin)o(ts)g(\022)-23 b(a)19 b(l'aide)i(de)f(la)g(librairie)p eop %%page: 42 44 42 43 bop 75 269 a fq(42)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)75 397 y fq(p)o(olylib)22 b([63])e(et)h(du)g(logiciel)j fn(maple)p fq([11])c(p)q(our)h(les)h (comp)q(ositions)g(et)f(in)o(v)o(ersions)g(de)h(fonctions.)75 453 y(ce)d(dessin)h(d'un)g(nouv)o(eau)f(c)o(hamp,)g(m)o(^)-21 b(eme)18 b(s'il)i(est)f(partiel,)g(p)q(ermet)g(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t)18 b(de)h(d)o(\023)-21 b(etecter)75 510 y(imm)o(\023)g(ediatemen)o(t)18 b(si)h(l'op)o(\023)-21 b(eration)18 b(e\013ectu)o(\023)-21 b(ee)18 b(donne)h(le)g(r)o(\023)-21 b(esultat)17 b(pr)o(\023)-21 b(evu.)18 b(une)h(limitation)h(par)e(rap-) 75 566 y(p)q(ort)e(\022)-23 b(a)16 b(l'expressivit)o(\023)-21 b(e)18 b(du)f(langage)f fn(pei)g fq(r)o(\023)-21 b(eside)17 b(dans)f(l'obligation)i(de)f(construire)g(des)g(d)o(\023)-21 b(e\014nitions)75 623 y(explicites,)20 b(c'est-\022)-23 b(a-dire)19 b(assimilables)h(\022)-23 b(a)18 b(des)g(a\013ectations.)g (on)g(ne)h(p)q(eut)g(en)f(e\013et)g(appliquer)i(une)75 679 y(op)o(\023)-21 b(eration)15 b(que)i(sur)f(un)g(c)o(hamp)g(conn)o (u)h(\(pr)o(\023)-21 b(ec)o(\023)g(edemmen)o(t)14 b(d)o(\023)-21 b(e\014ni\))16 b(alors)g(que)h fn(pei)f fq(autorise)g(des)g(d)o(\023) -21 b(e-)75 736 y(\014nitions)16 b(implicites.)i(ce)d(probl)o(\022)-21 b(eme)15 b(se)g(r)o(\023)-21 b(ev)o(\022)g(ele)15 b(aussi)g(g)o(^)-21 b(enan)o(t)14 b(dans)h(les)h(d)o(\023)-21 b(e\014nitions)16 b(r)o(\023)-21 b(ecursiv)o(es.)146 795 y fn(vpei)18 b fq(ne)i(p)q(ermet)f(de)g(concev)o(oir)g(que)h(de)f(p)q(etits)h (programmes)e fn(pei)p fq(,)g(le)i(nom)o(bre)e(de)i(c)o(hamps)75 852 y(in)o(term)o(\023)-21 b(ediaires)22 b(v)o(enan)o(t)f(tr)o(\022)-21

b(es)20 b(vite)i(alourdir)g(les)g(manipulations,)h(et)e(ce)h(malgr)o (\023)-21 b(e)21 b(la)g(fonction)h(de)75 908 y(substitution)16 b(d)o(\023)-21 b(ecrite)15 b(pr)o(\023)-21 b(ec)o(\023)g(edemmen)o(t.) 146 968 y(finalemen)o(t,)18 b(l'exp)o(\023)-21 b(erience)18 b(de)g(ce)g(premier)g(protot)o(yp)q(e)f(\(en)o(viron)h(6000)e(lignes)j (de)f(langage)f(c\))75 1024 y(nous)j(fait)g(p)q(enser)h(que)f(l'a)s (jout)f(d'une)i(fen)o(^)-21 b(etre)19 b(in)o(terpr)o(\022)-21 b(ete)19 b(d'expressions)i fn(pei)f fq(rem)o(\023)-21 b(edierait)20 b(aux)75 1081 y(limitations)h(\023)-21 b(ev)o(o)q(qu)o(\023)g(ees.)18 b(cet)i(in)o(terpr)o(\022)-21 b(ete)20 b(devrait)f(^)-21 b(etre)20 b(capable)h(de)g(repr)o(\023)-21 b(esen)o(ter)19 b(graphiquemen)o(t)75 1137 y(une)c(\023)-21 b(equation)14 b(en)o(tr)o(\023)-21 b(ee)15 b(au)g(cla)o(vier.)75 1299 y ft(3.5)66 b(un)23 b(outil)h(p)r(our)f(le)f(ra\016nemen)n(t)h(de) g(programmes)146 1407 y fq(l'outil)13 b fn(pei)g fq(initialemen)o(t)i (d)o(\023)-21 b(ev)o(elopp)o(\023)g(e)13 b(par)f(e.)h(violard)h(dans)f (le)g(cadre)g(de)h(sa)e(th)o(\022)-21 b(ese)13 b(est)f(un)i(outil)75 1463 y(de)21 b(manipulation)g(de)g(programmes.)d(l'ensem)o(ble)k(des)e (r)o(\022)-21 b(egles)20 b(de)g(ra\016nemen)o(t)g(et)g(d'abstraction)75 1520 y(pr)o(\023)-21 b(esen)o(t)o(\023)g(ees)11 b(en)j(section)f(3.3)f (y)h(son)o(t)f(implan)o(t)o(\023)-21 b(ees)13 b(et)g(on)g(p)q(eut)h (transformer)d(un)j(programme)e(dans)h(cet)75 1576 y(en)o(vironnemen)o (t)j(par)g(application)h(des)f(r)o(\022)-21 b(egles)16 b(sur)f(c)o(hacune)i(des)e(\023)-21 b(equations.)15 b(les)h(seules)h (mo)q(di\014ca-)75 1633 y(tions)e(qu'on)f(p)q(eut)h(app)q(orter)f(au)g (programme)f(se)i(fon)o(t)e(par)h(ce)h(biais.)g(l'utilisation)h(de)f (cet)g(outil)g(est)75 1689 y(logiquemen)o(t)h(l')o(\023)-21 b(etap)q(e)14 b(suiv)m(an)o(te)h(d')o(\023)-21 b(ecriture)15 b(d'un)g(programme,)e(qui)i(a)g(pu)f(^)-21 b(etre)13 b(con\030)-20 b(cu)16 b(initialemen)o(t)75 1745 y(dans)f(l'en)o (vironnemen)o(t)i fn(vpei)d fq(par)h(exemple.)146 1862 y(l'en)o(vironnemen)o(t)h fn(pei)g fq(app)q(orte)g(une)g(aide)h(\022) -23 b(a)15 b(l'utilisateur)j(p)q(our)e(la)g(transformation)f(des)h (pro-)75 1918 y(grammes,)g(mais)h(ne)h(p)q(ermet)g(pas,)e(p)q(our)i (l'instan)o(t)f(de)h(transformations)e(compl)o(\022)-21 b(etemen)o(t)17 b(automa-)75 1974 y(tis)o(\023)-21 b(ees.)15 b(nous)h(esp)o(\023)-21 b(erons)15 b(p)q(ouv)o(oir)i(implan)o(ter)f(ce) h(t)o(yp)q(e)f(de)g(transformation)f(de)h(programmes)f(sur)h(la)75 2031 y(base)k(des)g(tra)o(v)m(aux)f(r)o(\023)-21 b(ealis)o(\023)g(es)19 b(concernan)o(t)h(les)g(strat)o(\023)-21 b(egies)19 b(de)h (transformation,)e(pr)o(\023)-21 b(esen)o(t)o(\023)g(es)18 b(dans)i(le)75 2087 y(c)o(hapitre)c(4.)146 2204 y(en)g(l')o(\023)-21 b(etat)14 b(actuel,)i(la)h(transformation)d(d'un)j(programme)e(se)h (fait)g(pas-\022)-23 b(a-pas)7 b(:)16 b(l'utilisateur)h(s)o(\023)-21 b(e-)75 2260 y(lectionne)19 b(une)e(expression)h(de)g(c)o(hamps)e(de)i (donn)o(\023)-21 b(ees)17 b(quelconque)h(dans)f(le)h(syst)o(\022)-21 b(eme)16 b(d')o(\023)-21 b(equations,)75 2316 y(et)16 b(p)q(eut)h(c)o(hoisir,)g(par)e(men)o(u,)i(le)g(ra\016nemen)o(t)e(ou)h (l'abstraction)g(de)h(cette)f(expression.)h(le)g(logiciel,)75 2373 y(s'il)22 b(p)q(eut)f(iden)o(ti\014er)h(cette)f(expression)g(\022) -23 b(a)21 b(un)g(mem)o(bre)g(de)g(l'une)g(des)h(r)o(\022)-21 b(egles,)19 b(remplace)j(dans)f(le)75 2429 y(programme)d(l'expression)j (initiale)g(par)f(l'expression)g(ra\016n)o(\023)-21 b(ee)18 b(ou)i(abstraite.)e(l'utilisateur)j(p)q(eut)74 2486 y(\023)-21 b(egalemen)o(t)14 b(r)o(\023)-21 b(e)o(\023)g(ecrire)14 b(une)i(expression)g(de)f(c)o(hamps)g(de)h(donn)o(\023)-21 b(ees)14 b(\(par)h(exemple)h(p)q(our)g(arriv)o(er)e(\022)-23 b(a)15 b(une)75 2542 y(forme)k(d'expression)h(qu'il)g(est)f(p)q (ossible)i(de)f(ra\016ner)f(o)q(\022)-24 b(u)19 b(d'abstraire\))7 b(;)19 b(le)h(syst)o(\022)-21 b(eme)18 b(v)o(\023)-21

b(eri\014e)19 b(alors)p eop %%page: 43 45 43 44 bop 75 269 a fl(3.5.)29 b(un)15 b(outil)i(pour)f(le)f (raffinement)g(de)g(pr)o(ogrammes)371 b fq(43)75 397 y(que)16 b(la)f(nouv)o(elle)i(expression)f(est)e(\023)-21 b(equiv)m(alen)o(te)16 b(\022)-23 b(a)15 b(la)g(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te.)146 453 y(p)o(our)17 b(plus)i(de)g(d)o(\023) -21 b(etails)18 b(sur)g(ce)h(logiciel,)h(on)e(p)q(ourra)g(se)g(r)o (\023)-21 b(ef)o(\023)g(erer)16 b(\022)-23 b(a)18 b([56)o(],)g(don)o(t) f(les)i(annexes)g(d)o(\023)-21 b(e-)75 510 y(criv)o(en)o(t)20 b(de)g(mani)o(\022)-21 b(ere)19 b(exhaustiv)o(e)i(les)f(fonctionnalit)o (\023)-21 b(es)20 b(de)g(base)g(du)g(logiciel.)i(le)e(mo)q(de)g(de)g (fonc-)75 566 y(tionnemen)o(t)d(reste)g(iden)o(tique)h(\022)-23 b(a)17 b(la)g(v)o(ersion)g(d)o(\023)-21 b(ev)o(elopp)o(\023)g(ee)16 b(initialemen)o(t,)j(bien)f(que)f(de)g(nom)o(breuses)75 623 y(mo)q(di\014cations)g(aien)o(t)d(\023)-21 b(et)o(\023)g(e)14 b(app)q(ort)o(\023)-21 b(ees.)14 b(notammen)o(t,)g(un)h(traducteur)h(p) q(ermettan)o(t)e(d'asso)q(cier)i(\022)-23 b(a)15 b(un)75 679 y(programme)20 b fn(pei)p fq(,)g(une)i(in)o(terpr)o(\023)-21 b(etation)20 b(fonctionnelle)k(\()p fj(cf.)c fq(section)i(2.3.5\))d (sous)h(la)i(forme)e(d'un)75 736 y(programme)13 b fn(caml)p fq(,)h(\022)-23 b(a)12 b(\023)-21 b(et)o(\023)g(e)12 b(a)s(jout)o(\023)-21 b(e)13 b(\022)-23 b(a)13 b(l'en)o(vironnemen)o (t.)i(le)g(programme)e fn(caml)h fq(g)o(\023)-21 b(en)o(\023)g(er)o (\023)g(e)11 b(p)q(eut)j(^)-21 b(etre)75 792 y(utilis)o(\023)g(e)19 b(p)q(our)g(v)o(\023)-21 b(eri\014er)18 b(l'exactitude)i(des)e(r)o (\023)-21 b(esultats)18 b(obten)o(us)g(sur)g(un)h(jeu)g(d'essai.)f (egalemen)o(t,)g(un)75 849 y(con)o(tr^)-23 b(oleur)15 b(de)g(t)o(yp)q(e)h(\022)-23 b(a)13 b(\023)-21 b(et)o(\023)g(e)13 b(d)o(\023)-21 b(ev)o(elopp)o(\023)g(e)15 b(p)q(our)g(les)h(programmes) e fn(pei)7 b fq(:)15 b(cet)g(outil,)h(utilisan)o(t)g(le)f(calcu-)75 905 y(lateur)h fn(omega)p fq([30)o(],)f(t)o(yp)q(e)g(c)o(hacune)i(des)e (expressions)h(de)g(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees)15 b(d'un)g(\023)-21 b(enonc)o(\023)g(e)14 b fn(pei)p fq(,)75 962 y(c'est-\022)-23 b(a-dire)19 b(calcule)h(le)f(domaine)f (des)h(v)m(aleurs)g(et)f(le)h(domaine)g(de)f(r)o(\023)-21 b(ef)o(\023)g(erence)17 b(de)i(c)o(hacune)g(de)f(ces)75 1018 y(expressions.)g(le)g(con)o(tr^)-23 b(ole)18 b(p)q(orte)g(alors)f (sur)h(la)g(coh)o(\023)-21 b(erence)17 b(des)h(expressions)h(vis-\022) -23 b(a-vis)19 b(de)f(la)g(d)o(\023)-21 b(e\014-)75 1074 y(nition)19 b(des)f(op)o(\023)-21 b(erations.)16 b(nous)i(v)o(errons)f (dans)h(le)h(c)o(hapitre)f(5,)f(que)h(les)h(r)o(\023)-21 b(esultats)16 b(pro)q(duits)j(par)e(ce)75 1131 y(con)o(tr^)-23 b(oleur)15 b(de)h(t)o(yp)q(e)f(p)q(euv)o(en)o(t)g(^)-21 b(etre)14 b(utilis)o(\023)-21 b(es)16 b(par)f(d'autres)f(outils.)p eop %%page: 44 46 44 45 bop 75 269 a fq(44)615 b fl(chapitre)16 b(3.)30 b(raffinement)15 b(et)f(abstra)o(ction)p eop %%page: 45 47 45 46 bop 1801 269 a fq(45)75 777 y fi(chapitre)32 b(4)75 1053 y fp(strat)m(\023)-56 b(egies)41 b(de)d(transformations)146 1344 y fq(il)21 b(a)d(\023)-21 b(et)o(\023)g(e)18 b(mon)o(tr)o(\023)-21 b(e)19 b(dans)h([56)o(,)f(46])g(que)i fn(pei)f fq(p)q(ermettait)g(de)g (transformer)f(un)h(programme)f(en)75 1401 y(utilisan)o(t)i(les)g(r)o (\023)-21 b(esultats)18 b(issus)j(des)f(tec)o(hniques)h(de)g(parall)o (\023)-21 b(elisation)21 b(automatique)7 b(:)20 b(l'analyse)g(des)75 1457 y(d)o(\023)-21 b(ep)q(endances)17 b(p)q(ermet)f(de)g(d)o(\023)-21 b(eterminer)16 b(un)g(c)o(hangemen)o(t)g(de)g(base)g(comp)q(os)o(\023) -21 b(e)15 b(d'une)i(fonction)f(d'or-)75 1514 y(donnancemen)o(t)f(et)f

(de)g(placemen)o(t)h(des)g(calculs.)g(ce)f(c)o(hangemen)o(t)g(de)h (base)f(est)g(appliqu)o(\023)-21 b(e)15 b(aux)f(deux)75 1570 y(mem)o(bres)20 b(de)g(c)o(hacune)h(des)e(\023)-21 b(equations)20 b(de)g(l')o(\023)-21 b(enonc)o(\023)g(e)19 b fn(pei)h fq(et)g(on)g(mon)o(tre)f(que)h(l'application)i(des)75 1627 y(r)o(\022)-21 b(egles)18 b(qui)i(on)o(t)e(p)q(our)h(e\013et)f (d'instiller)j(le)e(c)o(hangemen)o(t)g(de)g(base)g(au)g(co)q(eur)g(des) g(expressions,)g(est)75 1683 y(con\015uen)o(te.)f(cette)e(tec)o(hnique) j(de)e(transformation)f(est)h(main)o(tenan)o(t)g(classique)i(et)e(est)g (disp)q(onible)75 1740 y(dans)h(de)g(nom)o(breux)f(formalismes,)h (notammen)o(t)e fn(alpha)p fq(.)i(nous)f(rapp)q(elons)i(bri)o(\022)-21 b(ev)o(emen)o(t)17 b(dans)g(la)75 1796 y(premi)o(\022)-21 b(ere)15 b(section)h(ce)f(t)o(yp)q(e)h(de)f(transformation)f(sur)h (l'exemple)i(de)f(l')o(\023)-21 b(elimination)16 b(de)g(gauss.)146 1921 y(dans)j(la)i(suite)g(du)f(c)o(hapitre,)h(nous)f(nous)g(in)o(t)o (\023)-21 b(eressons)20 b(\022)-23 b(a)20 b(d'autres)f(applications)j (du)f(ra\016ne-)75 1978 y(men)o(t)e(de)g(programmes)e(\023)-21 b(ev)o(o)q(qu)o(\023)g(e)17 b(dans)i(le)g(c)o(hapitre)h(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t.)17 b(notammen)o(t,)g(nous)i(v)o(errons)f(que)75 2034 y(le)j(ra\016nemen)o(t)f(p)q(ermet)g(de)h(prendre)g(en)f(compte)g (des)h(relations)g(\014nes)f(en)o(tre)g(programmes,)f(que)75 2091 y(seule)j(l')o(\023)-21 b(equiv)m(alence)22 b(faible)g(p)q(ermet)e (de)h(d)o(\023)-21 b(ecrire.)20 b(les)h(strat)o(\023)-21 b(egies)19 b(que)i(nous)f(prop)q(osons)h(ici)g(son)o(t)75 2147 y(constitu)o(\023)-21 b(ees)19 b(de)h(l'application)i(successiv)o (e)f(de)f(plusieurs)h(r)o(\022)-21 b(egles)19 b(de)h(ra\016nemen)o(t,)f (et)g(constituen)o(t)75 2204 y(des)h(sc)o(h)o(\023)-21 b(emas)19 b(t)o(yp)q(es)h(de)g(transformation)f(d'une)i(forme)e(de)h (programme)f(v)o(ers)h(une)g(autre.)g(selon)75 2260 y(l'arc)o (hitecture)c(de)g(la)g(mac)o(hine)g(parall)o(\022)-21 b(ele,)16 b(les)g(primitiv)o(es)h(implan)o(t)o(\023)-21 b(ees)15 b(sur)h(le)g(mat)o(\023)-21 b(eriel)15 b(ou)g(dans)h(le)75 2316 y(langage)g(vis)o(\023)-21 b(e,)16 b(certaines)h(solutions)h(son)o (t)e(conn)o(ues)h(p)q(our)e(^)-21 b(etre)16 b(plus)h(e\016caces)g(que)g (d'autres.)f(nous)75 2373 y(mon)o(trons)e(ici,)h(par)f(le)i(biais)g(de) f(ces)f(strat)o(\023)-21 b(egies)13 b(de)i(transformation,)e(commen)o (t)h(passer)h(d'une)g(solu-)75 2429 y(tion)h(\022)-23 b(a)16 b(une)g(autre.)f(t)l(out)h(au)g(long)g(de)g(l'exp)q(os)o(\023) -21 b(e)16 b(des)g(strat)o(\023)-21 b(egies,)14 b(nous)i(illustrerons)i (les)e(tec)o(hniques)75 2486 y(d)o(\023)-21 b(ecrites)15 b(a)o(v)o(ec)f(l'exemple)i(du)f(pro)q(duit)g(matrice-v)o(ecteur.)g(cet) f(exp)q(os)o(\023)-21 b(e)15 b(des)g(strat)o(\023)-21 b(egies)13 b(de)i(transfor-)75 2542 y(mation)g(pr)o(\023)-21 b(ecise)15 b(les)h(id)o(\023)-21 b(ees)16 b(pr)o(\023)-21 b(esen)o(t)o(\023)g(ees)13 b(dans)j([22)o(].)p eop %%page: 46 48 46 47 bop 75 269 a fq(46)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)75 397 y ft(4.1)66 b(t)-6 b(ransformation)23 b(par)g(c)n (hangemen)n(t)f(de)h(base)146 508 y fq(le)16 b(programme)g fn(pei)f fq(suiv)m(an)o(t)i(traduit)g(l'algorithme)f(d')o(\023)-21 b(elimination)17 b(de)g(gauss.)e(l'algorithme)75 565 y(r)o(\023)-21 b(esout)12 b(le)j(syst)o(\022)-21 b(eme)12 b(lin)o(\023)-21 b(eaire)15 b fm(ax)e fq(=)g fm(b)g fq(o)q(\022)-24 b(u)13 b fm(a)h fq(est)f(une)i(matrice)f(de)g(dimension)h fm(n)7 b ff(\002)g fm(n)15 b fq(et)e fm(b)g fq(un)i(v)o(ecteur)75 621 y(de)i(taille)i fm(n)p fq(.)e(l'algorithme)g(pro)q(c)o(\022)-21 b(ede)17 b(par)g(une)g(triangularisation)h(de)g(la)f(matrice)g fm(a)g fq(p)q(our)h(ensuite)75 678 y(e\013ectuer)11 b(une)h (substitution)g(par)f(remon)o(t)o(\023)-21 b(ee)10 b(et)h(r)o(\023)-21

b(esoudre)11 b(le)h(syst)o(\022)-21 b(eme.)9 b(nous)j(ne)f(nous)h(in)o (t)o(\023)-21 b(eressons)10 b(ici)75 734 y(qu'\022)-23 b(a)14 b(la)h(phase)g(de)f(triangularisation.)h(le)g(programme)f fn(pei)g fq(suiv)m(an)o(t)h(prend)g(en)g(en)o(tr)o(\023)-21 b(ee)14 b(les)h(c)o(hamps)75 791 y fk(a)j fq(et)h fk(b)f fq(repr)o(\023)-21 b(esen)o(tan)o(t)17 b(resp)q(ectiv)o(emen)o(t)i(la)g (matrice)g(et)f(le)i(v)o(ecteur.)e(nous)g(plongeons)i(ensuite)f(ces)75 847 y(c)o(hamps)13 b(dans)g(un)g(espace)g(\022)-23 b(a)13 b(trois)f(dimensions)j(\()p fm(i;)8 b(j;)g(k)q fq(\))i(\022)-23 b(a)13 b(tra)o(v)o(ers)e(les)j(c)o(hamps)e fk(x)h fq(et)g fk(y)p fq(.)f(le)h(calcul)i(est)75 904 y(alors)g(sp)o(\023)-21 b(eci\014)o(\023)g(e)14 b(par)h(une)g(d)o(\023)-21 b(e\014nition)16 b(r)o(\023)-21 b(ecursiv)o(e)14 b(du)h(c)o(hamp)g fk(c)p fq(.)g(la)f(matrice)h(triangularis)o(\023)-21 b(ee)15 b(ainsi)h(que)75 960 y(le)d(v)o(ecteur)g(mo)q(di\014)o(\023)-21 b(e)13 b(se)f(trouv)o(en)o(t)g(sur)g(le)i(plan)f fm(i)f fq(=)h fm(k)h fq(et)e(son)o(t)g(repr)o(\023)-21 b(esen)o(t)o(\023)g(es) 11 b(par)h(les)h(c)o(hamps)g fk(xn)f fq(et)g fk(yn)p fq(.)146 1078 y(si)j(la)h(v)o(ersion)f(de)g(ce)g(programme)f(doit)g(^) -21 b(etre)14 b(in)o(terpr)o(\023)-21 b(et)o(\023)g(ee)14 b(op)o(\023)-21 b(erationnellemen)o(t,)15 b(en)g(c)o(hoisissan)o(t)75 1135 y(l'axe)c fm(k)h fq(comme)f(direction)i(d'ordonnancemen)o(t,)e (nous)g(constatons)f(que)i(le)g(programme)e(utilise)j(alors)75 1191 y(des)i(op)o(\023)-21 b(erations)13 b(g)o(\023)-21 b(eom)o(\023)g(etriques)13 b(de)h(t)o(yp)q(e)h(di\013usion,)g(comme)f (par)g(exemple)i(la)f(fonction)g fk(pivot)p fq(.)e(or,)75 1248 y(certaines)18 b(mac)o(hines)f(n'autorisen)o(t)g(pas)g(ce)g(t)o (yp)q(e)g(de)g(comm)o(unication)h(et)f(ne)g(p)q(ermetten)o(t)g(que)g (des)75 1304 y(comm)o(unications)12 b(d'un)h(pro)q(cesseur)f(v)o(ers)f (un)i(de)f(ses)g(v)o(oisins.)g(ce)g(son)o(t)f(par)h(exemple)h(les)f(h)o (yp)q(oth)o(\022)-21 b(eses)75 1360 y(prises)16 b(p)q(our)f(les)h(r)o (\023)-21 b(eseaux)14 b(systoliques.)h(cette)f(con)o(train)o(te)h(de)g (comm)o(unication)h(in\015ue)h(sur)d(l'ordon-)75 1417 y(nancemen)o(t)k(du)h(programme,)d(et)i(nous)g(c)o(herc)o(hons)g(une)h (nouv)o(elle)f(\023)-21 b(ecriture)18 b(du)g(programme)f(p)q(our)75 1473 y(en)d(tenir)h(compte.)f(le)g(nouv)o(eau)g(programme)f(est)h (obten)o(u)g(par)g(l'application)i(d'un)e(c)o(hangemen)o(t)g(de)75 1530 y(base)h(p)q(ermettan)o(t)g(d'obtenir)g(une)h(v)o(ersion)g (systolique)g(du)f(programme)g fk(gauss1)p fq(.)75 1648 y fk(gauss1)23 b(:)h(\(a,b\))14 b ff(7!)i fk(\(xn,yn\))75 1663 y fx(8)75 1700 y(>)75 1712 y(>)75 1725 y(>)75 1737 y(>)75 1750 y(>)75 1762 y(<)75 1837 y(>)75 1849 y(>)75 1862 y(>)75 1874 y(>)75 1887 y(>)75 1899 y(:)137 1698 y fk(a)f fq(=)g fk(alignx)7 b(::)g(x)137 1754 y(b)15 b fq(=)g fk(aligny)7 b(::)g(y)137 1811 y(c)15 b fq(=)g fk(el)g fa(b)h fk(\(x)23 b(/&/)8 b(y)24 b(/&/)7 b(\(c)24 b fa(c)g fk(pre\))f(/&/)8 b(\(c)23 b fa(c)i fk(cp)o(\))f(/&/)8 b(\(c)23 b fa(c)h fk(lp\))g(/&/)7 b(\(c)24 b fa(c)g fk(piv\)\))137 1867 y(xn)15 b fq(=)g fk(c)g fa(c)h fk(last1)137 1924 y(yn)f fq(=)g fk(c)g fa(c)h fk(last2)100 1996 y(alignx)49 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(j)p ff(\024)p fk(n)g(&)g(k)p fq(=)p fk(0\).\(i,j\))100 2053 y(aligny)49 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)6 b(&)i(j)p fq(=)p fk(n+1)g(&)g(k)p fq(=)p fk(0\).\(i\))100 2109 y(pre)121 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fm(<)p fk(i)p ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(j)p ff(\024)p fk(n+1)g(&)g(0)p fm(<)p fk(k\).\(i,j,k-1\))100 2165 y(piv)121 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fm(<)p fk(i)p

ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(j)p ff(\024)p fk(n+1)g(&)g(0)p fm(<)p fk(k\).\(k,k,k-1\))100 2222 y(cp)145 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fm(<)p fk(i)p ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(j)p ff(\024)p fk(n+1)g(&)g(0)p fm(<)p fk(k\).\(i,k,k-1\))100 2278 y(lp)145 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fm(<)p fk(i)p ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(j)p ff(\024)p fk(n+1)g(&)g(0)p fm(<)p fk(k\).\(k,j,k-1\))100 2335 y(el)145 b fq(=)50 b fk(id)10 b fq(#)g fm(\025)p fk(\(a)p fq(;)o fk(b)p fq(;)o fk(c)p fq(;)p fk(d\).\(a-b*c/d\))100 2391 y(last1)73 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)6 b(&)i(i)p ff(\024)p fk(j)p ff(\024)p fk(n)g(&)g(k)p fq(=)p fk(i-1\).\(i,j,k\))100 2448 y(last2)73 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)6 b(&)i(j)p fq(=)p fk(n+1)g(&)g(k)p fq(=)p fk(i-1\).\(i,j,k\))146 2542 y fq(l'analyse)20 b(des)g(d)o(\023)-21 b(ep)q(endances)21 b(selon)g(les)g(tec)o(hniques)g(systoliques)g(classiques)g(fournit)f (un)h(or-)p eop %%page: 47 49 47 48 bop 75 269 a fl(4.1.)29 b(transf)o(orma)l(tion)17 b(p)l(ar)f(changement)e(de)h(base)493 b fq(47)75 397 y(donnancemen)o(t)16 b(p)q(ossible)i(des)e(calculs)i(d)o(\023)-21 b(e\014ni)16 b(par)g(la)g(fonction)g fm(t)p fq(\()p fm(i;)8 b(j;)g(k)q fq(\))k(=)i fm(i)c fq(+)h fm(j)i fq(+)e fm(k)q fq(,)k(ainsi)i(qu'un)75 453 y(placemen)o(t)d(d)o(\023)-21 b(e\014ni)13 b(par)g fm(p)p fq(\()p fm(i;)8 b(j;)g(k)q fq(\))h(=)k(\()p fm(i;)8 b(k)q fq(\).)j(nous)i(p)q(ouv)o(ons)g (expliciter)i(ce)e(c)o(hoix)h(de)f(cadencemen)o(t)h(et)75 510 y(de)k(placemen)o(t)h(en)f(transforman)o(t)e(le)i(programme)f(pr)o (\023)-21 b(ec)o(\023)g(eden)o(t)16 b(par)i(l'application)i(du)e(c)o (hangemen)o(t)75 566 y(de)j(base)f fk(h)g fq(=)g fm(\025)p fk(\(i,j,k\).\(i,k,i+j+k\))d fq(aux)j(deux)h(mem)o(bres)f(de)h (certaines)f(des)g(\023)-21 b(equations)19 b(de)75 623 y fk(gauss1)p fq(.)g(sur)h(ce)h(t)o(yp)q(e)f(d'exemple,)h(il)h(est)e (in)o(t)o(\023)-21 b(eressan)o(t)19 b(d'appliquer)i(ce)g(c)o(hangemen)o (t)f(de)h(base)f(\022)-23 b(a)75 679 y(toutes)19 b(les)h(\023)-21 b(equations,)19 b(sauf)g(celles)j(qui)f(expliciten)o(t)h(un)e(c)o (hangemen)o(t)g(de)g(dimension)i(en)o(tres)d(v)m(a-)75 736 y(riables,)g(en)g(l'o)q(ccurrence)g(celles)h(d)o(\023)-21 b(e\014nissan)o(t)18 b fk(a)g fq(et)g fk(b)p fq(.)g(le)h(r)o(\023)-21 b(esultat)17 b(de)i(la)f(transformation)f(est)h(un)75 792 y(programme)h(don)o(t)h(les)g(c)o(hamps)g(son)o(t)g(exprim)o(\023) -21 b(es)20 b(dans)g(une)h(nouv)o(elle)g(base,)f(c'est-\022)-23 b(a-dire)20 b(que)h(les)75 849 y(expressions)16 b(fonction)g(d'un)f(c)o (hamp)g fk(x)g fq(deviennen)o(t)i(des)f(expressions)g(fonction)f(de)h fk(h)7 b(::)g(x)p fq(.)146 944 y(il)16 b(est)f(in)o(t)o(\023)-21 b(eressan)o(t)14 b(de)i(noter)f(que)g(cette)h(transformation)e(p)q(eut) g(^)-21 b(etre)15 b(automatis)o(\023)-21 b(ee)14 b(car)h(le)h(sous-)75 1000 y(ensem)o(ble)h(des)e(r)o(\022)-21 b(egles)15 b(\(1b\),)g(\(3b\),) f(\(3d\),)g(\(7b\),)h(\(11b\))f(augmen)o(t)o(\023)-21 b(e)14 b(de)i(la)g(r)o(\022)-21 b(egle)15 b fk(y)g fq(=)g fk(x)h ff(\000)-8 b(!)16 b fk(h)8 b(::)f(x)15 b fq(=)75 1057 y fk(h)7 b(::)h(x)16 b fq(est)g(con\015uen)o(t,)h(et)f(toute)g (expression)i(fonction)f(de)g fk(x)f fq(devien)o(t)i(une)f(expression)g (de)g(\()p fk(h)7 b(::)g(x)p fq(\).)75 1113 y(ce)19 b(t)o(yp)q(e)f(de)h (transformation)f(a)g(d'ailleurs)h(\023)-21 b(et)o(\023)g(e)17 b(implan)o(t)o(\023)-21 b(e)19 b(et)f(sera)g(in)o(t)o(\023)-21 b(egr)o(\023)g(e)17 b(pro)q(c)o(hainemen)o(t)i(dans)75 1170 y(l'en)o(vironnemen)o(t)e(de)f(transformation.)e(un)i(nom)o(bre)g

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b(1)11 b(:)22 b(simpli\014cation)j(des)d(comm)n(unications)146 499 y fq(la)f(premi)o(\022)-21 b(ere)21 b(strat)o(\023)-21 b(egie)19 b(de)j(transformation)e(que)h(nous)g(in)o(tro)q(duisons)h(a)f (p)q(our)g(but)h(de)f(sim-)75 556 y(pli\014er)f(les)f(op)o(\023)-21 b(erations)17 b(de)i(routage)e(d'un)i(programme.)e(ce)h(t)o(yp)q(e)h (de)f(simpli\014cation)j(pr)o(\023)-21 b(esen)o(te)18 b(au)75 612 y(moins)c(deux)g(in)o(t)o(\023)-21 b(er)o(^)g(ets.)11 b(il)k(p)q(ermet,)e(quand)h(la)g(simpli\014cation)i(est)d(p)q(ossible,) i(de)f(dimin)o(uer)h(le)f(v)o(olume)75 669 y(des)k(comm)o(unications)g (\022)-23 b(a)16 b(e\013ectuer.)h(egalemen)o(t,)g(cette)g(strat)o(\023) -21 b(egie)16 b(de)i(transformation)e(p)q(ermet)h(de)75 725 y(conclure)f(sur)f(l')o(\023)-21 b(equiv)m(alence)17 b(de)e(plusieurs)i(programmes)d(di\013)o(\023)-21 b(eran)o(t)14 b(par)g(leurs)i(comm)o(unications)g(et)75 782 y(nous)h(p)q(ensons)g (que)g(cet)g(asp)q(ect)f(concerne)i(particuli)o(\022)-21 b(eremen)o(t)17 b(les)g(algorithmes)g(systoliques,)g(c'est-)75 838 y(\022)-23 b(a-dire)15 b(les)h(algorithmes)e(parall)o(\022)-21 b(eles)15 b(qui)h(alternen)o(t)e(une)i(phase)e(de)h(calcul)i(a)o(v)o (ec)d(une)h(phase)g(de)g(com-)75 894 y(m)o(unication.)146 1008 y(la)20 b(simpli\014cation)j(est)e(p)q(ossible)h(lorsque)g(les)f (routages)f(son)o(t)g(uniformes)h(a)o(v)o(ec)f(la)h(forme)g(g)o(\023) -21 b(e-)75 1064 y(n)o(\023)g(erale)20 b(suiv)m(an)o(te)9 b(:)20 b fm(f)5 b fq(\()p fm(i)478 1071 y fg(1)496 1064 y fm(;)j(:)g(:)g(:)d(;)j(i)614 1071 y fw(m)p fx(\000)p fg(1)687 1064 y fm(;)g(i)724 1071 y fw(m)754 1064 y fq(\))21 b(=)h(\()p fm(i)884 1071 y fg(1)916 1064 y fq(+)14 b fm(a)989 1071 y fg(1)1008 1064 y fm(;)8 b(:)g(:)g(:)t(;)g(i)1125 1071 y fw(m)p fx(\000)p fg(1)1212 1064 y fq(+)14 b fm(a)1285 1071 y fw(m)p fx(\000)p fg(1)1359 1064 y fm(;)8 b(i)1396 1071 y fw(m)1440 1064 y ff(\000)14 b fq(1\))20 b(\(la)h(v)m(ariable)g fm(i)1815 1071 y fw(m)75 1121 y fq(p)q(eut)f(repr)o(\023)-21 b(esen)o(ter)19 b(la)h(direction)i(de)e(temps)g(par)f(exemple\).)i (bien)g(que)f(ces)g(conditions)h(sem)o(blen)o(t)75 1177 y(con)o(traignan)o(tes,)14 b(de)i(nom)o(breux)f(algorithmes)g (systoliques)h(utilisen)o(t)h(de)f(tels)f(routages.)75 1301 y fw(4.2.1)55 b(princip)r(e)146 1388 y fq(le)18 b(princip)q(e)j(de)d(la)g(simpli\014cation)j(rep)q(ose)d(sur)g (l'expression)h(de)g(la)f(fonction)g(utilis)o(\023)-21 b(ee)19 b(par)f(un)75 1444 y(routage,)13 b(sous)i(la)g(forme)f(d'une)h (comp)q(osition)g(de)h(fonctions.)e(dans)g(un)h(programme)f(o)q(\022) -24 b(u)15 b(un)g(c)o(hamp)75 1501 y(de)e(donn)o(\023)-21 b(ees)12 b(est)h(d)o(\023)-21 b(e\014ni)13 b(par)f fk(y)h fq(=)f fk(x)j fa(c)h fk(f)p fq(,)c(nous)h(c)o(herc)o(hons)g(\022)-23 b(a)12 b(r)o(\023)-21 b(e)o(\023)g(ecrire)11 b(la)i(fonction)g fk(f)f fq(sous)h(la)g(forme)f fk(f)75 1557 y fq(=)f fk(g)f ff(\016)g fk(f')p fq(,)h(o)q(\022)-24 b(u)11 b fk(f')g fq(est)g(une)g(fonction)h(plus)g fj(\()-9 b(\()q fq(simple)p fj(\))g(\))13 b fq(que)e fk(f)p fq(.)g(ensuite,)g(nous)h(c)o(herc)o (hons)f(une)h(d)o(\023)-21 b(e\014nition)74 1614 y(\023)g(equiv)m(alen) o(te)18 b(\022)-23 b(a)17 b(celle)h(de)g fk(y)p fq(,)e(utilisan)o(t)i (uniquemen)o(t)h(le)e(routage)f fk(f')p fq(.)h(p)o(our)f(cela,)h(il)i (faut)d(trouv)o(er)g(la)75 1670 y(bijection)g fk(h)d fq(d'un)i(c)o(hangemen)o(t)f(de)g(base,)g(nous)g(p)q(ermettan)o(t)g (d'obtenir)g(une)g(\023)-21 b(equation)13 b(\023)-21 b(equiv)m(alen)o(te)75 1727 y(de)20 b(la)h(forme)e fk(y')h fq(=)g fk(x)15 b fa(c)h fk(f')p fq(,)j(o)q(\022)-24 b(u)20 b fk(y')g fq(=)g fk(h)8 b(::)f(y)p fq(.)19 b(il)j(s'agit)d(donc)h(de)h (trouv)o(er)e(une)i(repr)o(\023)-21 b(esen)o(tation)75 1783 y(di\013)o(\023)g(eren)o(te)15 b(des)g(c)o(hamps)g(ne)h(n)o(\023) -21 b(ecessitan)o(t)14 b(plus)j(le)f(routage)e(des)h(v)m(aleurs)i(par)e

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/n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def end /$f2psbegin {$f2psdict begin /$f2psenteredstate save def} def /$f2psend {$f2psenteredstate restore end} def $f2psbegin 0 setlinecap 0 setlinejoin -9.0 391.0 translate 0.900 -0.900 scale 0.500 setlinewidth % polyline n 99 349 m 169 414 l 169 434 l 149 434 l 74 364 l 74 349 l 99 349 l gs col7 0.95 graycol fill gr gs col7 s gr % polyline n 184 129 m 254 194 l 254 214 l 234 214 l 159 144 l 159 129 l 184 129 l gs col7 0.95 graycol fill gr gs col7 s gr % polyline n 264 89 m 334 154 l 334 174 l 314 174 l 239 104 l 239 89 l 264 89 l gs col7 0.95 graycol fill gr gs col7 s gr [4.000000] 0 setdash % polyline n 174 309 m 249 379 l 249 339 l 219 309 l 174 309 l gs col7 0.95 graycol fill gr gs col7 s gr [] 0 setdash n 182.000 311.000 m 174.000 309.000 l 182.000 307.000 l gs 2 setlinejoin col7 s gr % polyline n 105 168 m 175 233 l 175 253 l 155 253 l 80 183 l 80 168 l 105 168 l gs col7 0.95 graycol fill gr gs col7 s gr % polyline n 279 269 m 329 314 l 329 284 l 314 269 l 279 269 l gs col7 0.95 graycol fill gr gs col7 s gr [4.000000] 0 setdash % polyline n 234 284 m 309 354 l 269 354 l 234 319 l 234 284 l gs col7 0.95 graycol fill gr gs col7 s gr [] 0 setdash n 232.000 292.000 m 234.000 284.000 l 236.000 292.000 l gs 2 setlinejoin col7 s gr

[4.000000] 0 setdash % polyline n 154 359 m 189 394 l 164 394 l 154 384 l 154 359 l gs col7 0.95 graycol fill gr gs col7 s gr [] 0 setdash n 152.000 367.000 m 154.000 359.000 l 156.000 367.000 l gs 2 setlinejoin col7 s gr n 211.000 156.000 m 209.000 164.000 l 207.000 156.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 224.000 164.000 15.000 0.000 180.000 arcn gs col-1 s gr [] 0 setdash n 181.000 131.000 m 179.000 139.000 l 177.000 131.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash n 194.000 139.000 15.000 0.000 180.000 arcn gs col-1 s gr [] 0 setdash [4.000000] 0 setdash % polyline n 124 364 m 194 329 l gs col-1 s gr [] 0 setdash n 185.950 330.789 m 194.000 329.000 l 187.739 334.367 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % polyline n 157 205 m 237 165 l gs col-1 s gr [] 0 setdash [4.000000] 0 setdash % polyline n 129 184 m 209 144 l gs col-1 s gr [] 0 setdash [4.000000] 0 setdash % interpolated spline n 98 243 m 87.293 248.999 84.043 252.249 85 256 curveto 86.229 260.821 97.406 261.258 102 261 curveto 128.581 259.508 160.081 246.258 228 208 curveto gs col-1 s gr [] 0 setdash n 220.048 210.184 m 228.000 208.000 l 222.011 213.669 l gs 2 setlinejoin col-1 s gr % polyline n 174 254 m 334 174 l gs col-1 s gr % polyline n 254 214 m 254 129 l 159 129 l 159 214 l clp gs col-1 s gr % polyline n 174 254 m 174 169 l 79 169 l 79 254 l clp gs col-1 s gr [4.000000] 0 setdash % polyline n 159 394 m 229 359 l gs col-1 s gr [] 0 setdash n 220.950 360.789 m 229.000 359.000 l 222.739 364.367 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % polyline n 89 419 m 159 384 l gs col-1 s gr [] 0 setdash n 150.950 385.789 m 159.000 384.000 l 152.739 389.367 l gs 2 setlinejoin col-1 s

gr % polyline n 169 434 m 329 354 l gs col-1 s gr % polyline n 74 349 m 234 269 l gs col-1 s gr % polyline n 19 139 m 19 164 l gs col-1 s gr n 21.000 156.000 m 19.000 164.000 l 17.000 156.000 l gs 2 setlinejoin % polyline n 249 394 m 249 309 l 154 309 l 154 394 l clp gs col-1 s gr n 174 169 m 174 169 l gs col-1 s gr % polyline n 129 364 m 124 364 l 124 359 l 129 359 l clp gs 0.50 setgray fill gs col-1 s gr % polyline n 89 424 m 84 424 l 84 419 l 89 419 l clp gs 0.50 setgray fill gr gs col-1 s gr % polyline n 134 184 m 129 184 l 129 179 l 134 179 l clp gs 0.50 setgray fill gs col-1 s gr n 19 139 m 19 139 l gs col-1 s gr % polyline n 79 169 m 239 89 l gs col-1 s gr % polyline n 164 394 m 159 394 l 159 389 l 164 389 l clp gs 0.50 setgray fill gs col-1 s gr % polyline n 19 139 m 54 139 l gs col-1 s gr n 46.000 137.000 m 54.000 139.000 l 46.000 141.000 l gs 2 setlinejoin n 174 169 m 174 169 l gs col-1 s gr n 174 169 m 174 169 l gs col-1 s gr n 174 169 m 174 169 l gs col-1 s gr % polyline n 159 209 m 154 209 l 154 204 l 159 204 l clp gs 0.50 setgray fill gs col-1 s gr % polyline n 169 434 m 169 349 l 74 349 l 74 434 l clp gs col-1 s gr % polyline n 19 139 m 49 119 l gs col-1 s gr n 41.234 121.774 m 49.000 119.000 l 43.453 125.102 l gs 2 setlinejoin % polyline n 239 129 m 239 89 l 334 89 l 334 174 l 254 174 l gs col-1 s gr % polyline n 234 309 m 234 269 l 329 269 l 329 354 l 249 354 l gs col-1 s gr % polyline n 99 244 m 94 244 l 94 239 l 99 239 l clp gs 0.50 setgray fill gr gs col-1 s gr $f2psend %%enddocument @endspecial 356 707 a fh(a)378 713 y fg(0)p fw(;)p fg(0)423 707 y fh(b)441 713 y fg(0)581 914 y fh(a)603 920 y fg(2)p fw(;)p fg(2)648 914 y fh(b)666 920 y fg(2)656 558 y fh(a)678 564 y fg(0)p fw(;)p fg(1)723 558 y fh(b)741 564 y fg(1)769 670 y fh(a)791 676 y fg(1)p fw(;)p fg(2)836 670 y fh(b)854 676 y fg(2)881 782 y fh(a)903 788 y fg(2)p fw(;)p fg(0)948 782 y fh(b)966 788 y fg(0)956 408 y fh(a)978 414 y fg(0)p fw(;)p fg(2)1023 408 y fh(b)1041 414 y fg(2)1181 632 y fh(a)1203 638 y fg(2)p fw(;)p fg(1)1248 632 y fh(b)1266 638 y fg(1)337 1382 y fh(a)359 1388 y fg(0)p fw(;)p fg(0)404 1382 y fh(b)422 1388 y fg(0)1162 1082 y fh(a)1184 1088

col-1 s gr

gr

gr

gr

col-1 s gr

gr

col-1 s gr

y fg(0)p fw(;)p fg(2)1229 1082 y fh(b)1247 1088 y fg(2)637 1457 y fh(a)659 1463 y fg(2)p fw(;)p fg(0)704 1457 y fh(b)722 1463 y fg(0)75 595 y ff(i)95 b(j)131 483 y(k)562 1589 y fh(a)584 1595 y fg(2)p fw(;)p fg(2)629 1589 y fh(b)647 1595 y fg(2)1069 520 y fh(a)1091 526 y fg(1)p fw(;)p fg(0)1135 520 y fh(b)1153 526 y fg(0)937 1195 y fh(a)959 1201 y fg(1)p fw(;)p fg(0)1004 1195 y fh(b)1022 1201 y fg(0)1050 1307 y fh(a)1072 1313 y fg(2)p fw(;)p fg(1)1117 1307 y fh(b)1135 1313 y fg(1)750 1232 y fh(a)772 1238 y fg(0)p fw(;)p fg(1)817 1232 y fh(b)835 1238 y fg(1)450 1476 y fh(a)472 1482 y fg(1)p fw(;)p fg(1)517 1476 y fh(b)535 1482 y fg(1)469 801 y fh(a)491 807 y fg(1)p fw(;)p fg(1)536 801 y fh(b)554 807 y fg(1)862 1326 y fh(a)884 1332 y fg(1)p fw(;)p fg(2)929 1326 y fh(b)947 1332 y fg(2)311 1743 y fn(fig.)15 b fq(4.1)f(-)31 b fj(t)m(r)n(ansformation)15 b(de)h fk(matvect1)f fj(en)h fk(matvect2)f fj(p)n(our)i fm(n)c fq(=)g(3)75 1894 y fo(t)l(ransformation)18 b(de)f fk(matvect1)146 1993 y fq(nous)i(p)q(ouv)o(ons)i(main)o(tenan)o(t)e(appliquer)j(la)e(strat)o (\023)-21 b(egie)19 b(de)h(transformation)e(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)19 b(p)q(our)75 2050 y(obtenir)14 b(de)f(nouv)o(elles)i(d)o(\023)-21 b(e\014nitions)14 b(des)f(c)o(hamps)g fk(a)g fq(et)g fk(b)p fq(,)g(don)o(t)g(les)g (routages)g(ne)g(comp)q(orten)o(t)g(plus)h(le)75 2106 y(facteur)e fk(g)p fq(.)h(ensuite,)g(il)h(faut)e(transformer)f(c)o (haque)h(\023)-21 b(equation)12 b(faisan)o(t)h(r)o(\023)-21 b(ef)o(\023)g(erence)11 b(\022)-23 b(a)13 b fk(a)f fq(et)h fk(b)f fq(p)q(our)h(tenir)75 2163 y(compte)h(de)g(leurs)h(nouv)o(elles) g(d)o(\023)-21 b(e\014nitions.)15 b(la)f(transformation)e(est)i(r)o (\023)-21 b(ealis)o(\023)g(ee)13 b(par)h(le)g(c)o(hangemen)o(t)g(de)75 2219 y(base)h fk(h)p fq(.)g(dans)g(notre)f(exemple,)j(nous)e(d)o(\023) -21 b(eduisons)16 b(du)f(facteur)506 2342 y fk(g)d fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(k)p fm(<)p fk(n\).\(i,\(j+1\))21 b(mod)i(n,k\))75 2465 y fq(la)16 b(bijection)9 b(:)506 2542 y fk(h)j fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(k)p fm(<)p fk(n\).\(i,\(j+k\))21 b(mod)i(n,k\))p eop %%page: 53 55 53 54 bop 75 269 a fl(4.2.)29 b(stra)l(t)336 257 y(\023)330 269 y(egie)16 b(1)7 b(:)15 b(simplifica)l(tion)i(des)e(communica)l (tions)333 b fq(53)75 397 y(don)o(t)15 b(l'in)o(v)o(erse)h(est)f fk(h)451 380 y fq(-1)503 397 y fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(k
b(nous)e(dev)o(ons)g(main)o(tenan)o(t)f(exprimer)i(les)g(c)o(hamps)f fk(m)g fq(et)g fk(v)f fq(qui)i(d)o(\023)-21 b(ep)q(enden)o(t)22 b(de)f fk(a0)g fq(et)g fk(b0)189 978 y fq(resp)q(ectiv)o(emen)o(t,)c (en)h(fonction)g(de)f(\()p fk(h)8 b(::)f(a0)p fq(\))17 b(et)g(\()p fk(h)7 b(::)g(b0)p fq(\))17 b(.)g(p)o(our)f(cela,)i(nous)g (in)o(tro)q(duisons)189 1035 y(ces)13 b(derni)o(\022)-21 b(eres)13 b(expressions)h(dans)g(les)f(d)o(\023)-21 b(e\014nitions)9 b(:)k fk(m)g fq(=)g fk(aligna)7 b(::)g(a0)p fq(,)12 b(et)h fk(v)g fq(=)h fk(alignb)7 b(::)g(b0)189 1091 y fq(sous)15 b(la)g(forme)g(suiv)m(an)o(te)8 b(:)214 1218 y fk(m)15 b fq(=)g fk(\(aligna)9 b ff(\016)h fk(h)537 1201 y fq(-1)584 1218 y ff(\016)g fk(h\))d(::)h(a0)214 1274 y(v)15 b fq(=)g fk(\(alignb)9 b ff(\016)h fk(h)537 1257 y fq(-1)584 1274 y ff(\016)g fk(h\))d(::)h(b0)189 1385 y fq(l'application)16 b(de)e(la)h(r)o(\022)-21 b(egle)14 b(\(7a\))f(m)o(\022)-21 b(ene)14 b(\022)-23 b(a)14 b(une)g(r)o(\023)-21 b(e)o(\023)g(ecriture) 14 b(des)g(alignemen)o(ts)h(de)g fk(a0)f fq(et)g fk(b0)g fq(en)8 b(:)214 1511 y fk(m)15 b fq(=)g fk(aligna)9 b ff(\016)h fk(h)513 1494 y fq(-1)558 1511 y fk(::)d(\(h)h(::)g(a0)o(\)) 214 1567 y(v)15 b fq(=)g fk(alignb)9 b ff(\016)h fk(h)513 1551 y fq(-1)558 1567 y fk(::)d(\(h)h(::)g(b0)o(\))189 1678 y fq(le)20 b(calcul)h(de)g fk(aligna)9 b ff(\016)h fk(h)666 1661 y fq(-1)723 1678 y fq(donne)20 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(k)p fq(=)p fk(0\).\(i,j\))19 b fq(=)h fk(aligna)f fq(et)189 1735 y(celui)e(de)e fk(alignb)10 b ff(\016)f fk(h)565 1717 y fq(-1)618 1735 y fq(donne)16 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)h(i)p fq(=)p fk(j)i(&)e(k)p fq(=)p fk(0\).\(i\))15 b fq(=)g fk(alignb)p fq(.)143 1843 y({)23 b(egalemen)o(t,)10 b(nous)g(in)o(tro)q(duisons)i(les)f(nouv)o(elles)h(expressions)f fk(h)c(::)h(a)i fq(et)g fk(h)e(::)f(b)j fq(dans)h(l')o(\023)-21 b(equation)189 1900 y(d)o(\023)g(e\014nissan)o(t)13 b fk(p)p fq(,)g(en)g(appliquan)o(t)i(le)f(c)o(hangemen)o(t)f(de)h(base)f (aux)g(deux)h(mem)o(bres)g(de)f(l')o(\023)-21 b(equation.)189 1956 y(l')o(\023)g(equation)14 b fk(h)8 b(::)f(p)15 b fq(=)h fk(h)7 b(::)g(\(prod)23 b fa(b)i fk(\(\(a)e(/&/)8 b(b\))23 b fa(c)h fk(diag\)\))15 b fq(devien)o(t)8 b(:)222 2069 y fg(\(1)p fw(b)p fg(\))214 2097 y ff(\000)-7 b(!)50 b fk(h)7 b(::)h(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(h)8 b(::)g(\(\(a)23 b(/&/)8 b(b\))23 b fa(c)i fk(diag)o(\)\))222 2140 y fg(\(3)p fw(b)p fg(\))214 2169 y ff(\000)-7 b(!)50 b fk(h)7 b(::)h(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(\(h)8 b(::)g(\(a)23 b(/&/)8 b(b\))o(\))24 b fa(c)g fk(diag'\))337 2225 y fq(car)15 b fm(dom)p fq(\()p fk(diag)o fq(\))9 b ff([)i fm(img)r fq(\()p fk(diag)n fq(\))h ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))h(et)337 2282 y(o)q(\022)-24 b(u)15 b fk(diag')g fq(=)g fk(h)10 b ff(\016)g fk(diag)f ff(\016)h fk(h)814 2265 y fq(-1)867 2282 y fq(=)15 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(\(j-k\))22 b(mod)h(n\))214 2336 y fg(\(11)p fw(b)p fg(\))214 2365 y ff(\000)-7 b(!)50 b fk(h)7 b(::)h(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(\(\(h)8 b(::)f(a\))24 b(/&/)8 b(\(h)f(::)h(b\)\))23 b fa(c)i fk(diag')o(\))75 2486 y fq(t)l(outes)15 b(ces)f(\023)-21 b(equations)15 b(formen)o(t)f(le)i(programme)e fk(matvect1')7 b fq(:)p eop %%page: 54 56 54 55 bop 75 269 a fq(54)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)75 397 y fk(matvect1')23 b(:)g(\(m,v\))15 b ff(7!)g

fk(p)75 407 y fx(8)75 445 y(>)75 457 y(>)75 470 y(>)75 482 y(>)75 495 y(>)75 507 y(>)75 520 y(>)75 532 y(>)75 544 y(>)75 557 y(>)75 569 y(<)75 644 y(>)75 657 y(>)75 669 y(>)75 681 y(>)75 694 y(>)75 706 y(>)75 719 y(>)75 731 y(>)75 744 y(>)75 756 y(>)75 769 y(:)137 449 y fk(h)7 b(::)h(a0)14 b fq(=)i fk(cube)7 b(::)g(\(h)h(::)g(a0)o(\))137 505 y(h)f(::)h(b0)14 b fq(=)i fk(cube)7 b(::)g(\(h)h(::)g(b0)o(\))137 562 y(m)15 b fq(=)g fk(aligna)7 b(::)g(\(h)h(::)g(a0)o(\))137 618 y(v)15 b fq(=)g fk(alignb)7 b(::)g(\(h)h(::)g(b0)o(\))137 674 y(h)f(::)h(a)15 b fq(=)g fk(h)8 b(::)f(a0)15 b(/&/)7 b fq(\()p fk(\(h)h(::)f(a\))15 b fa(c)h fk(prev)o fq(\))137 731 y fk(h)7 b(::)h(b)15 b fq(=)g fk(h)8 b(::)f(b0)15 b(/&/)7 b fq(\()p fk(\(h)h(::)f(b\))15 b fa(c)h fk(up)9 b ff(\016)h fk(prev)p fq(\))137 787 y fk(h)d(::)h(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(\(\(h)8 b(::)f(a\))24 b(/&/)7 b(\(h)h(::)g(b\)\))23 b fa(c)h fk(diag'\))96 850 y(prev)52 b fq(=)42 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,k-1\))96 906 y(up)100 b fq(=)42 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(\(i+1\))21 b(mod)j(n,j,k\))96 963 y(diag)192 946 y fx(0)244 963 y fq(=)42 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(\(j-k\))22 b(mod)h(n\))75 1048 y fq(ce)e(programme)g(p)q(eut)f(^)-21 b(etre)21 b(simpli\014)o(\023)-21 b(e)22 b(en)g(utilisan)o(t)g(la)g(propri)o (\023)-21 b(et)o(\023)g(e)20 b(3,)g(car)h(tous)g(les)h(c)o(hamps)f(on)o (t)75 1104 y(leur)h(domaines)g(de)g(r)o(\023)-21 b(ef)o(\023)g(erence) 20 b(inclus)j(dans)e fm(dom)p fq(\()p fk(h)p fq(\))f(et)h(on)g(v)o (\023)-21 b(eri\014e)22 b fm(img)r fq(\()p fk(h)p fq(\))f ff(\022)i fm(dom)p fq(\()p fk(h)p fq(\).)c(egale-)75 1161 y(men)o(t,)12 b(les)g(deux)h(premi)o(\022)-21 b(eres)11 b(\023)-21 b(equations)12 b(p)q(euv)o(en)o(t)g(^)-21 b(etre)11 b(supprim)o(\023)-21 b(ees)12 b(car)g fk(h)e ff(\016)g fk(cube)23 b(=)h(h)p fq(,)11 b(et)h(on)h(v)o(\023)-21 b(eri\014e)75 1217 y fm(dom)p fq(\()p fm(v)201 1224 y fr(a0)233 1217 y fq(\))13 b ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\).)g(nous)i(obtenons)g(ainsi)i(le)f(programme)e fk(matvect2)p fq(,)g(ra\016nan)o(t)g fk(matvect1)6 b fq(:)75 1338 y fk(matvect2)23 b(:)g(\(m,v\))15 b ff(7!)h fk(p)75 1353 y fx(8)75 1390 y(>)75 1402 y(>)75 1415 y(>)75 1427 y(>)75 1440 y(>)75 1452 y(<)75 1527 y(>)75 1539 y(>)75 1552 y(>)75 1564 y(>)75 1577 y(>)75 1589 y(:)137 1388 y fk(m)f fq(=)g fk(aligna)7 b(::)g(a0)137 1444 y(v)15 b fq(=)g fk(alignb)7 b(::)g(b0)137 1501 y(a)15 b fq(=)g fk(a0)24 b(/&/)7 b(\(a)24 b fa(c)g fk(prev\))137 1557 y(b)15 b fq(=)g fk(b0)24 b(/&/)7 b(\(b)24 b fa(c)g fk(up)10 b ff(\016)g fk(prev)o(\))137 1614 y(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(a/&/)8 b(b\))23 b fa(c)i fk(diag')96 1674 y(prev)41 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)h(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,k-1\))96 1731 y(up)89 b fq(=)42 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(\(i+1\))20 b(mod)k(n,j,k\))75 1823 y fq(en)17 b(fait,)g(en)h(utilisan)o(t)g(la)g(m)o(^)-21 b(eme)16 b(tec)o(hnique,)i(on)g(p)q(eut)g(mon)o(trer)e(que)i(les)g (programmes)e fk(matvect1)75 1880 y fq(et)f fk(matvect2)f fq(son)o(t)g(\023)-21 b(equiv)m(alen)o(ts.)75 2046 y fw(4.2.4)55 b(application)18 b(au)h(pro)r(duit)f(de)h(matrices)d(de)i (cannon)146 2147 y fq(un)12 b(exemple)i(similaire)h(est)d(celui)j(de)e (l'algorithme)g(de)g(m)o(ultiplication)i(de)e(matrices)g(de)g(cannon)75

2204 y([8)o(].)f(l'algorithme)g(\023)-21 b(evite)13 b(la)g(r)o(\023)-21 b(eplication)13 b(des)g(co)q(e\016cien)o(ts)h(en)f(routan)o(t)f(les)h (v)m(aleurs)h(n)o(\023)-21 b(ecessaires)12 b(p)q(our)75 2260 y(c)o(hacune)20 b(des)f(m)o(ultiplications.)i(l'encom)o(bremen)o (t)e(m)o(\023)-21 b(emoire)18 b(induit)j(par)d(cet)h(algorithme)g(est)g (de)75 2316 y(fait)d(r)o(\023)-21 b(eduit)15 b([34)o(].)g(comme)g(le)h (mon)o(tre)f(la)h(\014gure)g(4.2,)e(apr)o(\022)-21 b(es)15 b(l'alignemen)o(t)i(initial)h(des)d(matrices,)h(et)75 2373 y(le)f(pro)q(duit)h(des)f(co)q(e\016cien)o(ts)g(appari)o(\023)-21 b(es,)14 b(le)h(routage)f(consiste)h(en)g(une)h(rotation)d(de)i(droite) g(\022)-23 b(a)15 b(gauc)o(he)75 2429 y(des)e(v)m(aleurs)h(de)g(la)f (premi)o(\022)-21 b(ere)13 b(matrice,)g(et)g(du)h(bas)f(v)o(ers)f(le)i (haut)f(de)h(celles)g(de)g(la)f(deuxi)o(\022)-21 b(eme)14 b(matrice.)75 2486 y(l'algorithme)h(p)q(eut)h(s')o(\023)-21 b(ecrire)7 b(:)p eop %%page: 55 57 55 56 bop 75 269 a fl(4.2.)29 b(stra)l(t)336 257 y(\023)330 269 y(egie)16 b(1)7 b(:)15 b(simplifica)l(tion)i(des)e(communica)l (tions)333 b fq(55)517 1042 y @beginspecial @setspecial %%begindocument: cannon.pstex /$f2psdict 200 dict def $f2psdict begin $f2psdict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def end /$f2psbegin {$f2psdict begin /$f2psenteredstate save def} def /$f2psend {$f2psenteredstate restore end} def $f2psbegin 0 setlinecap 0 setlinejoin -92.0 279.0 translate 0.900 -0.900 scale 0.500 setlinewidth % polyline n 164 149 m 164 310 l gs col-1 s gr % polyline

n % n % n n % n % n % n % n n

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l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 159 159 m 179 159 l gs col-1 s gr [] 0 setdash 0 setlinecap n 222.000 161.000 m 214.000 159.000 l 222.000 157.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 214 159 m 234 159 l gs col-1 s gr [] 0 setdash 0 setlinecap n 272.000 161.000 m 264.000 159.000 l 272.000 157.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 264 159 m 284 159 l gs col-1 s gr [] 0 setdash 0 setlinecap n 162.000 201.000 m 154.000 199.000 l 162.000 197.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 154 199 m 174 199 l gs col-1 s gr [] 0 setdash 0 setlinecap n 217.000 201.000 m 209.000 199.000 l 217.000 197.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 209 199 m 229 199 l gs col-1 s gr [] 0 setdash 0 setlinecap n 277.000 201.000 m 269.000 199.000 l 277.000 197.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 269 199 m 289 199 l gs col-1 s gr [] 0 setdash 0 setlinecap n 277.000 241.000 m 269.000 239.000 l 277.000 237.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 269 239 m 289 239 l gs col-1 s gr [] 0 setdash 0 setlinecap n 217.000 241.000 m 209.000 239.000 l 217.000 237.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 209 239 m 229 239 l gs col-1 s gr [] 0 setdash 0 setlinecap n 162.000 241.000 m 154.000 239.000 l 162.000 237.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 154 239 m 174 239 l gs col-1 s gr [] 0 setdash 0 setlinecap n 222.000 281.000 m 214.000 279.000 l 222.000 277.000 gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 214 279 m 234 279 l gs col-1 s gr [] 0 setdash 0 setlinecap n 277.000 281.000 m 269.000 279.000 l 277.000 277.000 gr

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

l gs 2 setlinejoin col-1 s

1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 269 279 m 289 279 l gs col-1 s gr [] 0 setdash 0 setlinecap % polyline n 114 149 m 114 310 l gs col-1 s gr n 112.000 281.000 m 104.000 279.000 l 112.000 277.000 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 104 279 m 124 279 l gs col-1 s gr [] 0 setdash 0 setlinecap n 162.000 281.000 m 154.000 279.000 l 162.000 277.000 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % polyline n 154 279 m 174 279 l gs col-1 s gr [] 0 setdash 0 setlinecap 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 119 179 m 108.272 169.788 104.522 164.788 104 159 curveto 103.533 153.823 106.033 148.823 114 139 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 107.407 143.953 m 114.000 139.000 l 110.514 146.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 284 179 m 273.272 169.788 269.522 164.788 269 159 curveto 268.533 153.823 271.033 148.823 279 139 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 272.407 143.953 m 279.000 139.000 l 275.514 146.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 229 299 m 218.272 289.788 214.522 284.788 214 279 curveto 213.533 273.823 216.033 268.823 224 259 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 217.407 263.953 m 224.000 259.000 l 220.514 266.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 227 260 m 216.272 250.788 212.522 245.788 212 240 curveto 211.533 234.823 214.033 229.823 222 220 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 215.407 224.953 m 222.000 220.000 l 218.514 227.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 224 214 m 213.272 204.788 209.522 199.788 209 194 curveto 208.533 188.823 211.033 183.823 219 174 curveto gs col-1 s gr

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

[] 0 setdash 0 setlinecap n 212.407 178.953 m 219.000 174.000 l 215.514 181.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 284 219 m 273.272 209.788 269.522 204.788 269 199 curveto 268.533 193.823 271.033 188.823 279 179 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 272.407 183.953 m 279.000 179.000 l 275.514 186.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 284 259 m 273.272 249.788 269.522 244.788 269 239 curveto 268.533 233.823 271.033 228.823 279 219 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 272.407 223.953 m 279.000 219.000 l 275.514 226.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 284 299 m 273.272 289.788 269.522 284.788 269 279 curveto 268.533 273.823 271.033 268.823 279 259 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 272.407 263.953 m 279.000 259.000 l 275.514 266.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 174 179 m 163.272 169.788 159.522 164.788 159 159 curveto 158.533 153.823 161.033 148.823 169 139 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 162.407 143.953 m 169.000 139.000 l 165.514 146.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 117 220 m 106.272 210.788 102.522 205.788 102 200 curveto 101.533 194.823 104.033 189.823 112 180 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 105.407 184.953 m 112.000 180.000 l 108.514 187.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 172 216 m 161.272 206.788 157.522 201.788 157 196 curveto 156.533 190.823 159.033 185.823 167 176 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 160.407 180.953 m 167.000 176.000 l 163.514 183.473 l gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 169 254 m

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

gs 2 setlinejoin col-1 s

158.272 244.788 154.522 239.788 154 234 curveto 153.533 228.823 156.033 223.823 164 214 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 157.407 218.953 m 164.000 214.000 l 160.514 221.473 l gs 2 setlinejoin gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 169 294 m 158.272 284.788 154.522 279.788 154 274 curveto 153.533 268.823 156.033 263.823 164 254 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 157.407 258.953 m 164.000 254.000 l 160.514 261.473 l gs 2 setlinejoin gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 119 259 m 108.272 249.788 104.522 244.788 104 239 curveto 103.533 233.823 106.033 228.823 114 219 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 107.407 223.953 m 114.000 219.000 l 110.514 226.473 l gs 2 setlinejoin gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 122 300 m 111.272 290.788 107.522 285.788 107 280 curveto 106.533 274.823 109.033 269.823 117 260 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 110.407 264.953 m 117.000 260.000 l 113.514 267.473 l gs 2 setlinejoin gr 1 setlinecap [1 3.000000] 3.000000 setdash % interpolated spline n 229 179 m 218.272 169.788 214.522 164.788 214 159 curveto 213.533 153.823 216.033 148.823 224 139 curveto gs col-1 s gr [] 0 setdash 0 setlinecap n 217.407 143.953 m 224.000 139.000 l 220.514 146.473 l gs 2 setlinejoin gr $f2psend %%enddocument @endspecial 580 476 a ff(a)606 483 y fs(0)p fe(;)p fs(0)580 551 y ff(b)601 558 y fs(0)p fe(;)p fs(0)786 476 y ff(a)812 483 y fs(0)p fe(;)p fs(1)786 551 y ff(b)807 558 y fs(1)p fe(;)p fs(1)1199 551 y ff(b)1220 558 y fs(3)p fe(;)p fs(3)1199 476 y ff(a)1225 483 y fs(0)p fe(;)p fs(3)993 476 y ff(a)1019 483 y fs(0)p fe(;)p fs(2)993 551 y ff(b)1014 558 y fs(2)p fe(;)p fs(2)580 626 y ff(a)606 633 y fs(1)p fe(;)p fs(1)580 701 y ff(b)601 708 y fs(1)p fe(;)p fs(0)580 1001 y ff(b)601 1008 y fs(3)p fe(;)p fs(0)580 926 y ff(a)606 933 y fs(3)p fe(;)p fs(3)786 1001 y ff(b)807 1008 y fs(0)p fe(;)p fs(1)786 626 y ff(a)812 633 y fs(1)p fe(;)p fs(2)786 776 y ff(a)812 783 y fs(2)p fe(;)p fs(3)786 851 y ff(b)807 858 y fs(3)p fe(;)p fs(1)786 926 y ff(a)812 933 y fs(3)p fe(;)p fs(0)993 626 y ff(a)1019 633 y fs(1)p fe(;)p fs(3)993 701 y ff(b)1014 708 y fs(3)p fe(;)p fs(2)1199 701 y ff(b)1220 708 y fs(0)p fe(;)p fs(3)1199 626 y ff(a)1225 633 y fs(1)p

col-1 s

col-1 s

col-1 s

col-1 s

col-1 s

fe(;)p fs(0)993 776 y ff(a)1019 783 y fs(2)p fe(;)p fs(0)1199 776 y ff(a)1225 783 y fs(2)p fe(;)p fs(1)1199 851 y ff(b)1220 858 y fs(1)p fe(;)p fs(3)993 851 y ff(b)1014 858 y fs(0)p fe(;)p fs(2)993 1001 y ff(b)1014 1008 y fs(1)p fe(;)p fs(2)1199 1001 y ff(b)1220 1008 y fs(2)p fe(;)p fs(3)993 926 y ff(a)1019 933 y fs(3)p fe(;)p fs(1)1199 926 y ff(a)1225 933 y fs(3)p fe(;)p fs(2)580 776 y ff(a)606 783 y fs(2)p fe(;)p fs(2)580 851 y ff(b)601 858 y fs(2)p fe(;)p fs(0)786 701 y ff(b)807 708 y fs(2)p fe(;)p fs(1)75 1186 y fn(fig.)22 b fq(4.2)f(-)44 b fj(pr)n(o)n(duit)22 b(de)h(matric)n(es)f(de)h(cannon) e fq(\()p fm(n)p fq(=4\))8 b fj(:)22 b(alignement)f(initial)h(des)g (matric)n(es)h(et)75 1242 y(c)n(ommunic)n(ations)75 1573 y fk(cannon)g(:)h(\(ma,mb\))14 b ff(7!)i fk(c)75 1591 y fx(8)75 1628 y(>)75 1641 y(>)75 1653 y(>)75 1666 y(>)75 1678 y(>)75 1690 y(>)75 1703 y(>)75 1715 y(<)75 1790 y(>)75 1802 y(>)75 1815 y(>)75 1827 y(>)75 1840 y(>)75 1852 y(>)75 1865 y(>)75 1877 y(:)137 1623 y fk(ma)f fq(=)g fk(align1)7 b(::)g(a0)137 1679 y(mb)15 b fq(=)g fk(align2)7 b(::)g(b0)137 1736 y(a)15 b fq(=)g fk(a0)24 b(/&/)7 b(\(a)24 b fa(c)g fk(left\))137 1792 y(b)15 b fq(=)g fk(b0)24 b(/&/)7 b(\(b)24 b fa(c)g fk(up\))137 1849 y(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 1905 y(c)15 b fq(=)g fk(add)g fa(b)h fk(\(p)23 b(/&/)8 b(c)23 b fa(c)i fk(prev)o(\))100 1978 y(align1)49 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p ff(\024)p fk(n)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n)g(&)g(k)p fq(=)p fk(0\).\(i,\(i+j\))23 b(mod)g(n\))100 2034 y(align2)49 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p ff(\024)p fk(n)6 b(&)i(k)p fq(=)p fk(0\).\(\(i+j\))23 b(mod)g(n,j\))100 2090 y(left)97 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,\(j+1\))23 b(mod)g(n,k-1\))100 2147 y(up)145 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(\(i+1\))23 b(mod)g(n,j,k-1\))100 2203 y(prev)97 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,k-1\))100 2260 y(prod)97 b fq(=)50 b fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+b\))100 2316 y(add)121 b fq(=)50 b fk(id)10 b fq(#)g fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+b\))146 2486 y fq(nous)h(p)q(ouv)o(ons)h(l\022)-23 b(a)12 b(aussi,)g(trouv)o (er)f(des)h(programmes)d(\023)-21 b(equiv)m(alen)o(ts.)12 b(p)o(ar)f(exemple,)i(en)f(r)o(\023)-21 b(e)o(\023)g(ecriv)m(an)o(t)75 2542 y(les)16 b(fonctions)f(de)h(routages)e fk(left)h fq(et)g fk(up)g fq(de)g(la)h(mani)o(\022)-21 b(ere)15 b(suiv)m(an)o(te)8 b(:)p eop %%page: 56 58 56 57 bop 75 269 a fq(56)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)207 502 y fk(left)41 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(k)p fm(<)p fk(n\).\(\(i+1\))21 b(mod)i(n,j,k\))10 b ff(\016)421 558 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(j)p fm(<)p fk(n)c(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(\(i-1\))23 b(mod)g(n,\(j+1\))g(mod)h (n,k-1\))344 615 y fq(=)42 b fk(g)10 b ff(\016)f fk(downleft)207 671 y(up)89 b fq(=)42 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(k)p fm(<)p fk(n\).\(\(i+1\))21 b(mod)i(n,j,k\))10 b ff(\016)421 728 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)c(&)h(0)p

fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,k-1\))344 784 y fq(=)42 b fk(g)10 b ff(\016)f fk(prev)146 906 y fq(la)18 b(bijection)i fk(h)f fq(=)g fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(k)p fm(<)p fk(n\).\(\(i+k\))i(mod)i(n,j,k\))p fq(,)18 b(obten)o(ue)h(\022)-23 b(a)18 b(partir)h(du)g(fac-)75 963 y(teur)13 b(comm)o(un)h fk(g)f fq(est)h(utilis)o(\023)-21 b(ee)14 b(p)q(our)g(e\013ectuer)g(un)g(c)o(hangemen)o(t)f(de)h(base)g (et)f(obtenir)h(le)h(programme)75 1019 y(suiv)m(an)o(t,)g(selon)h(le)g (m)o(^)-21 b(eme)15 b(pro)q(c)o(\023)-21 b(ed)o(\023)g(e)6 b(:)75 1137 y fk(cannon2)23 b(:)h(\(ma,mb\))14 b ff(7!)h fk(c)75 1155 y fx(8)75 1192 y(>)75 1205 y(>)75 1217 y(>)75 1230 y(>)75 1242 y(>)75 1255 y(>)75 1267 y(>)75 1280 y(<)75 1354 y(>)75 1367 y(>)75 1379 y(>)75 1392 y(>)75 1404 y(>)75 1417 y(>)75 1429 y(>)75 1442 y(:)137 1187 y fk(ma)g fq(=)g fk(align1)7 b(::)g(a0)137 1244 y(mb)15 b fq(=)g fk(align2)7 b(::)g(b0)137 1300 y(a)15 b fq(=)g fk(a0)24 b(/&/)7 b(\(a)24 b fa(c)g fk(downleft)o(\))137 1357 y(b)15 b fq(=)g fk(b0)24 b(/&/)7 b(\(b)24 b fa(c)g fk(prev\))137 1413 y(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 1469 y(c)15 b fq(=)g fk(add)g fa(b)h fk(\(p)23 b(/&/)8 b(c)23 b fa(c)i fk(prev)o(\))100 1542 y(downleft)49 b fq(=)h fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(\(i-1\))23 b(mod)g(n,\(j+1\))g(mod)h(n,k-1\))100 1598 y(prev)145 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,k-1\))146 1693 y fq(dans)19 b(ce)g(nouv)o(eau)h(programme,)e(les)i(co)q(e\016cien)o(ts)g(de)g(la)g (matrice)f fm(a)h fq(son)o(t)e(d)o(\023)-21 b(eplac)o(\023)g(es)19 b(succesi-)75 1749 y(v)o(emen)o(t)i(sur)g(la)h(ligne)h(suiv)m(an)o(te)f (et)f(la)h(colonne)g(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te,)20 b(du)i(plan)h(suiv)m(an)o(t,)e(tandis)h(que)g(les)75 1806 y(co)q(e\016cien)o(ts)16 b(de)g fm(b)i fq(son)o(t)c(simplemen)o(t) i(d)o(\023)-21 b(eplac)o(\023)g(es)15 b(sur)g(le)h(plan)g(suiv)m(an)o (t.)75 1979 y ft(4.3)66 b(strat)o(\023)-32 b(egie)22 b(2)11 b(:)22 b(in)n(tro)r(duction)j(d'une)e(di\013usion)146 2091 y fq(le)12 b(programme)e(du)j(pro)q(duit)f(matrice-v)o(ecteur)g fk(matvect1)p fq(,)e(est)i(un)g(algorithme)g(que)g(nous)g(a)o(v)o(ons) 75 2147 y(quali\014)o(\023)-21 b(e)16 b(de)f(systolique)h(car)f(ses)g (routages)f(repr)o(\023)-21 b(esen)o(ten)o(t)14 b(des)h(comm)o (unications)h(p)q(oin)o(t-\022)-23 b(a-p)q(oin)o(t.)15 b(en)75 2204 y(supp)q(osan)o(t)f(que)g(l'on)g(disp)q(ose)h(d'une)f (primitiv)o(e)h(de)f(di\013usion,)h(caract)o(\023)-21 b(eristique)13 b(du)i(mo)q(d)o(\022)-21 b(ele)14 b(de)g(pro-)75 2260 y(grammation)d(data-parall)o(\022)-21 b(ele,)12 b(ces)h(comm)o(unications)g(p)q(euv)o(en)o(t)e(^)-21 b(etre)12 b(remplac)o(\023)-21 b(ees)12 b(par)g(cet)g(op)o(\023)-21 b(erateur)75 2316 y(de)16 b(di\013usion.)g(cela)g(signi\014e)h(qu'une)e (d)o(\023)-21 b(e\014nition)17 b(r)o(\023)-21 b(ecursiv)o(e)15 b(de)h(c)o(hamp)f(de)h(donn)o(\023)-21 b(ees)15 b(\(celle)i(de)e fk(b)h fq(sur)75 2373 y(l'exemple\))k(est)f(transform)o(\023)-21 b(ee)18 b(en)h(une)h(d)o(\023)-21 b(e\014nition)20 b(simple,)g(faisan)o (t)f(in)o(terv)o(enir)h(une)g(op)o(\023)-21 b(eration)18 b(de)75 2429 y(routage,)e(elle-m)o(^)-21 b(eme)17 b(d)o(\023)-21 b(e\014nie)18 b(r)o(\023)-21 b(ecursiv)o(emen)o(t.)16 b(une)h(telle)i(transformation)c(rep)q(ose)i(sur)g(le)h(r)o(\023)-21 b(esultat)75 2486 y(suiv)m(an)o(t,)15 b(qui)h(d)o(\023)-21 b(ecrit)15 b(une)h(autre)f(strat)o(\023)-21 b(egie)14 b(de)h(transformation)f(de)i(programmes)6 b(:)p eop %%page: 57 59

57 58 bop 75 269 a fl(4.3.)29 b(stra)l(t)336 257 y(\023)330 269 y(egie)16 b(2)7 b(:)15 b(intr)o(oduction)h(d'une)f(diffusion)480 b fq(57)75 397 y fo(th)o(\023)-25 b(eor)o(\022)g(eme)17 b(2)49 b fj(soit)16 b fk(g)g fj(une)h(fonction)f(d)o(\023)-22 b(e\014nie)15 b(r)o(\023)-22 b(e)n(cursivement)16 b(p)n(ar)h fk(g)24 b(=)f(i)10 b fq(#)g fk(\(g)24 b ff(\016)f fk(s\))p fj(,)17 b(o)q(\022)-24 b(u)17 b fk(i)f fj(est)75 453 y(l'identit)o(\023)-22 b(e)20 b(d)o(\023)-22 b(e\014nie)20 b(sur)h(un)g(sous-domaine,)g(et)f fk(s)h fj(est)f(une)h(fonction.)f(l') o(\023)-22 b(e)n(quation)20 b fk(y)h fj(=)g fk(x)16 b fa(c)h fk(g)j fj(se)75 510 y(r)o(\023)-22 b(e)o(\023)g(e)n(crit)16 b(alors)g(en)7 b(:)33 b fk(y)24 b(=)f(x)16 b fa(c)h fk(i)f(/&/)8 b(y)16 b fa(c)g fk(s)75 604 y fj(pr)n(euve.)f fq(p)o(ar)f(application)j (des)f(r)o(\022)-21 b(egles)14 b(de)i(ra\016nemen)o(t,)e(nous)i(a)o(v)o (ons)6 b(:)740 711 y fk(y)15 b fq(=)g fk(x)g fa(c)h fk(g)623 752 y fw(def)q(:)616 779 y ff(\000)-7 b(!)51 b fk(y)15 b fq(=)g fk(x)g fa(c)h fk(\(i)9 b fq(#)h fk(\(g)g ff(\016)g fk(s\)\))623 822 y fg(\(9)p fw(a)p fg(\))616 850 y ff(\000)-7 b(!)51 b fk(y)15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(x)15 b fa(c)h fk(\(g)9 b ff(\016)h fk(s\))623 893 y fg(\(5)p fw(a)p fg(\))616 922 y ff(\000)-7 b(!)51 b fk(y)15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(\(x)24 b fa(c)g fk(g\))15 b fa(c)g fk(s)75 1009 y fq(en)g(r)o(\023)-21 b(eit)o(\023)g(eran)o(t)13 b(le)j(calcul)9 b(:)421 1105 y fk(y)15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(\(x)24 b fa(c)g fk(\(i)10 b fq(#)g fk(\(g)g ff(\016)g fk(s\)\))o(\))15 b fa(c)h fk(s)298 1161 y ff(\000)-7 b(!)50 b fk(y)15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(\(x)24 b fa(c)g fk(i\))15 b fa(c)h fk(s)f(/&/)7 b(\(x)24 b fa(c)g fk(g)g ff(\016)f fk(s\))15 b fa(c)h fk(s)298 1218 y ff(\000)-7 b(!)50 b fk(y)15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(\(x)24 b fa(c)g fk(i\))15 b fa(c)h fk(s)f(/&/)7 b(\(\(x)24 b fa(c)g fk(i\))f fa(c)i fk(s\))15 b fa(c)g fk(s)g(/&/)8 b fq(..)75 1404 y(d)o(\023)-21 b(eplions)15 b(main)o(tenan)o(t)g(la)h (d)o(\023)-21 b(e\014nition)16 b fk(y')24 b(=)f(x)15 b fa(c)h fk(i)f(/&/)7 b(y')15 b fa(c)h fk(s)7 b fq(:)409 1509 y fk(y')15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(y')15 b fa(c)h fk(s)296 1550 y fw(def)286 1577 y ff(\000)-7 b(!)50 b fk(y')15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(\(x)24 b fa(c)g fk(i\))15 b fa(c)h fk(s)f(/&/)7 b(\(y')24 b fa(c)g fk(s\))15 b fa(c)g fk(s)296 1618 y fw(def)286 1645 y ff(\000)-7 b(!)50 b fk(y')15 b fq(=)g fk(x)g fa(c)h fk(i)f(/&/)7 b(\(x)24 b fa(c)g fk(i\))15 b fa(c)h fk(s)f(/&/)7 b(\(\(x)24 b fa(c)g fk(i\))f fa(c)i fk(s)o(\))15 b fa(c)h fk(s)f(/&/)8 b fq(..)75 1806 y(p)o(our)14 b(toute)g(\023)-21 b(etap)q(e)14 b fm(n)i fq(de)f(la)g(r)o(\023)-21 b(ecurrence,)15 b(les)h fm(n)f fq(premiers)h(termes)e(des)i(d)o(\023)-21 b(e\014nitions)15 b(de)h fk(y)f fq(et)g fk(y')f fq(son)o(t)74 1863 y(\023)-21 b(egaux,)14 b(donc)h fk(y)g fq(=)h fk(y')p fq(.)e(finalemen)o(t)i(nous)g(a)o(v)o(ons)6 b(:)15 b fk(y)g fq(=)h fk(x)f fa(c)g fk(g)g ff(\000)-7 b(!)15 b fk(y)h fq(=)f fk(\(x)24 b fa(c)g fk(i\))f(/&/)8 b(\(y)24 b fa(c)g fk(s\))75 1919 y fa(\003)146 2034 y fq(les)16 b(d)o(\023)-21 b(e\014nitions)16 b(des)g(c)o(hamps)f(de)h(donn)o(\023) -21 b(ees)16 b fk(a)f fq(et)g fk(b)h fq(dans)f(le)i(programme)d fk(matvect2)h fq(son)o(t)f(sem-)75 2091 y(blables)22 b(\022)-23 b(a)20 b(l'expression)i(ra\016n)o(\023)-21 b(ee)19 b(du)i(th)o(\023)-21 b(eor)o(\022)g(eme)19 b(2.)g(in)o(tuitiv)o (emen)o(t,)i(on)g(constate)e(que)i(le)h(c)o(hamp)75 2147 y fk(a)p fq(,)17 b(par)h(exemple,)h(est)e(constitu)o(\023)-21

b(e)18 b(des)g(v)m(aleurs)h(de)f fk(a0)f fq(qui)i(on)o(t)d(\023)-21 b(et)o(\023)g(e)16 b(successiv)o(emen)o(t)j(recopi)o(\023)-21 b(ees)18 b(sur)75 2204 y(c)o(hacun)k(des)g(plans)g fm(k)g fq(de)g fk(a)p fq(.)e(on)i(d)o(\023)-21 b(eduit)22 b(facilemen)o(t)g (de)g(cette)f(observ)m(ation,)g(une)h(fonction)g(qui)75 2260 y(exprime)i(qu'une)f(v)m(aleur)i(plac)o(\023)-21 b(ee)23 b(au)g(p)q(oin)o(t)g(\()p fm(i;)8 b(j;)g(k)q fq(\))20 b(du)k(c)o(hamp)f fk(a)g fq(pro)o(vien)o(t)g(du)g(p)q(oin)o(t) h(\()p fm(i;)8 b(j;)g fq(0\))75 2316 y(du)19 b(c)o(hamp)g fk(a0)p fq(.)f(nous)g(\023)-21 b(ecriv)o(ons)18 b(donc)h(en)g fn(pei)f fq(l')o(\023)-21 b(equation)19 b fk(a)f fq(=)h fk(a0)c fa(c)h fk(spreada)o fq(,)i(o)q(\022)-24 b(u)19 b fk(spreada)e fq(=)75 2373 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n\).\(i,j,0\))g fq(est)k(l'op)o(\023)-21 b(eration)20 b(de)i(routage)e(traduisan)o(t)g (cette)h(di\013usion.)75 2429 y(apr)o(\022)-21 b(es)16 b(a)o(v)o(oir)h(d)o(\023)-21 b(e\014ni)18 b fk(b)f fq(de)g(mani)o(\022) -21 b(ere)17 b(similaire,)i(nous)f(trouv)o(ons)e(le)i(programme)e(suiv) m(an)o(t,)i(utilisan)o(t)75 2486 y(uniquemen)o(t)e(des)g(routages)e(de) i(t)o(yp)q(e)f(di\013usion.)p eop %%page: 58 60 58 59 bop 75 269 a fq(58)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)75 397 y fk(matvect3)23 b(:)g(\(m,v\))15 b ff(7!)h fk(p)75 412 y fx(8)75 449 y(>)75 461 y(>)75 474 y(>)75 486 y(>)75 499 y(>)75 511 y(<)75 586 y(>)75 598 y(>)75 611 y(>)75 623 y(>)75 636 y(>)75 648 y(:)137 447 y fk(m)f fq(=)g fk(aligna)7 b(::)g(a0)137 503 y(v)15 b fq(=)g fk(alignb)7 b(::)g(b0)137 560 y(a)15 b fq(=)g fk(a0)g fa(c)h fk(spreada)137 616 y(b)f fq(=)g fk(b0)g fa(c)h fk(spreadb)137 673 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(\(a/&/)8 b(b\))23 b fa(c)h fk(diag'\))100 733 y(spreada)14 b fq(=)i fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n\).\(i,j,)o(0\))100 790 y(spreadb)e fq(=)i fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n\).\(\(i+k)o(\))21 b(mod)j(n,j,0\))75 876 y fq(il)17 b(s'agit)f(main)o(tenan)o(t)g(de)h(v)o(\023)-21 b(eri\014er)16 b(que)h(ce)f(programme)f(est)i(ra\016n)o(\023)-21 b(e)15 b(par)h(le)h(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t.)15 b(p)o(our)g(cela,)75 933 y(nous)g(appliquons)i(le)f(th)o(\023)-21 b(eor)o(\022)g(eme)13 b(2.)75 1048 y fj(pr)n(euve.)j fq(d'apr)o(\022)-21 b(es)14 b(l')o(\023)-21 b(enonc)o(\023)g(e)15 b(du)h(th)o(\023)-21 b(eor)o(\022)g(eme)14 b(2,)h(il)i(faut)f(mon)o (trer)f(qu'on)g(p)q(eut)i(r)o(\023)-21 b(e)o(\023)g(ecrire)15 b(les)h(fonctions)75 1104 y fk(spreada)f fq(et)g fk(spreadb)g fq(sous)h(une)g(forme)g(r)o(\023)-21 b(ecursiv)o(e)15 b(iden)o(tique)j(\022)-23 b(a)15 b(la)i(d)o(\023)-21 b(e\014nition)16 b(de)g fk(g)g fq(dans)g(le)h(th)o(\023)-21 b(eo-)75 1160 y(r)o(\022)g(eme.)14 b(en)h(e\013et,)g fk(spreada)f fq(et)h fk(spreadb)f fq(p)q(euv)o(en)o(t)i(se)f(r)o(\023) -21 b(e)o(\023)g(ecrire)14 b(de)i(la)f(mani)o(\022)-21 b(ere)15 b(suiv)m(an)o(te)8 b(:)170 1262 y fk(spreada)49 b fq(=)58 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n\).\(i,j,0\))387 1319 y fq(=)g fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fq(=)p fk(0\))23 b fq(#)10 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)c(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,0\))387 1375 y fq(=)58 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fq(=)p fk(0\))23 b fq(#)10 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p

fk(i,j,k)p fm(<)p fk(n\).\(i,j,0\))397 1432 y ff(\016)60 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(i,j,k-1\))387 1488 y fq(=)58 b fk(i)10 b fq(#)g fk(spreada)f ff(\016)h fk(prev)170 1555 y(spreadb)49 b fq(=)58 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n\).\(\(i+k\))21 b(mod)i(n,j,0\))387 1611 y fq(=)58 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fq(=)p fk(0\).\(\(i+k\))21 b(mod)i(n,j,0\))i fq(#)480 1668 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(\(i+k\))23 b(mod)g(n,j,0\))387 1724 y fq(=)58 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(k)p fq(=)p fk(0\))23 b fq(#)480 1781 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n\).\(\(i+k\))e(mod)i(n,j,0\))397 1837 y ff(\016)60 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(0)p fm(<)p fk(k)p fm(<)p fk(n\).\(\(i+1\))23 b(mod)g(n,j,k-1\))387 1893 y fq(=)58 b fk(i)10 b fq(#)g fk(spreadb)f ff(\016)h fk(up)g ff(\016)g fk(prev)75 1945 y fa(\003)75 2155 y ft(4.4)66 b(repr)o(\023)-32 b(esen)n(tation)22 b(du)h(probl)o(\022)-32 b(eme)23 b(en)f(deux)i(dimensions)146 2260 y fq(nous)17 b(p)q(ouv)o(ons)h(constater,)f(d'apr)o(\022)-21 b(es)17 b(la)h(d)o(\023)-21 b(e\014nition)18 b(du)h(c)o(hamp)e fk(p)h fq(du)g(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t)17 b(programme,)75 2316 y(que)k(ses)g(v)m(aleurs)h(son)o(t)e(au)g(nom)o (bre)h(de)g fm(n)834 2300 y fg(2)853 2316 y fq(,)f(car)h(les)g(pro)q (duits)h(ne)f(s'e\013ectuen)o(t)f(que)i(sur)e(le)i(plan)75 2373 y(diagonal)17 b(du)g(cub)q(e.)g(on)g(p)q(eut)g(donc)g(souhaiter)f fj(\()-9 b(\()q fq(compacter)p fj(\))g(\))16 b fq(les)h(v)m(aleurs)g (plac)o(\023)-21 b(ees)17 b(par)f(ce)h(c)o(hamp)75 2429 y(dans)k(un)g(espace)g(carr)o(\023)-21 b(e)19 b(\022)-23 b(a)21 b(deux)g(dimensions,)h(par)f(exemple,)g(comme)g(le)g(mon)o(tre)f (la)h(\014gure)g(4.3.)75 2486 y(ce)d(compactage)f(illustre)i(une)f (application)i(de)e(l')o(\023)-21 b(equiv)m(alence)19 b(faible)9 b(:)18 b(nous)f(ne)i(nous)e(in)o(t)o(\023)-21 b(eressons)75 2542 y(plus)22 b(\022)-23 b(a)21 b(tous)g(les)h(p)q(oin)o (ts)g(de)f(notre)g(domaine)h(de)g(r)o(\023)-21 b(ef)o(\023)g(erence,)20 b(mais)h(seulemen)o(t)h(au)f(domaine)h(des)p eop %%page: 59 61 59 60 bop 75 269 a fl(4.4.)29 b(repr)310 257 y(\023)304 269 y(esent)l(a)l(tion)17 b(du)e(pr)o(obl)867 257 y(\022)861 269 y(eme)h(en)f(deux)g(dimensions)298 b fq(59)75 397 y(v)m(aleurs.)17 b(nous)g(pro)q(c)o(\023)-21 b(edons)16 b(en)h(deux)g(\023)-21 b(etap)q(es)16 b(p)q(our)h(mon)o(trer)e(la)i (relation)h(en)o(tre)e(le)i(programme)d(en)75 453 y(deux)k(dimensions)g (et)f(celui)i(en)e(trois)g(dimensions)9 b(:)18 b(d'ab)q(ord)f(nous)h (transformons)f(le)i(programme)75 510 y fk(matvect3')h fq(a\014n)h(que)g(les)h(v)m(aleurs)g(de)g(tous)e(ses)h(c)o(hamps)g (soien)o(t)g(plac)o(\023)-21 b(ees)22 b(sur)f(le)g(premier)h(plan.)75 566 y(ensuite)h(nous)f(mon)o(trons)f(que)h(ce)g(nouv)o(eau)g(programme) f(est)h(un)g(ra\016nemen)o(t)f(du)i(programme)75 623 y(don)o(t)15 b(les)h(c)o(hamps)f(appartiennen)o(t)h(\022)-23 b(a)15 b fe(z)772 606 y fg(2)787 623 y fq(.)75 751 y fw(4.4.1)55 b(regroup)r(emen)n(t)16 b(des)j(v)m(aleurs)e(sur)i(le)f (premier)e(plan)75 1574 y @beginspecial @setspecial %%begindocument: transf2.pstex /$f2psdict 200 dict def $f2psdict begin $f2psdict /mtrx matrix put

/l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def end /$f2psbegin {$f2psdict begin /$f2psenteredstate save def} def /$f2psend {$f2psenteredstate restore end} def $f2psbegin 0 setlinecap 0 setlinejoin -103.0 319.0 translate 0.900 -0.900 scale 0.500 setlinewidth % polyline n 319 159 m 369 204 l 369 174 l 354 159 l 319 159 l gs col7 0.95 graycol fill gr gs col7 s gr [4.000000] 0 setdash % polyline n 274 174 m 349 244 l 309 244 l 274 209 l 274 174 l gs col7 0.95 graycol fill gr gs col7 s gr [] 0 setdash n 272.000 182.000 m 274.000 174.000 l 276.000 182.000 l gs 2 setlinejoin col7 s gr [4.000000] 0 setdash % polyline n 194 249 m 229 284 l 204 284 l 194 274 l 194 249 l gs col7 0.95 graycol fill gr gs col7 s gr [] 0 setdash n 192.000 257.000 m 194.000 249.000 l 196.000 257.000 l gs 2 setlinejoin col7 s gr % polyline n 139 239 m 209 304 l 209 324 l 189 324 l 114 254 l 114 239 l 139 239 l gs col7 0.95 graycol fill gr gs col7 s gr [4.000000] 0 setdash % polyline n 214 199 m 289 269 l 289 229 l 259 199 l 214 199 l gs col7 0.95 graycol fill gr gs col7 s gr [] 0 setdash n 222.000 201.000 m 214.000 199.000 l 222.000 197.000 l gs 2 setlinejoin col7 s

gr % polyline n 209 324 m 369 244 l gs col-1 s gr % polyline n 114 239 m 274 159 l gs col-1 s gr % polyline n 369 329 m 369 354 l gs col-1 s gr n 371.000 346.000 m 369.000 354.000 l 367.000 346.000 l gs 2 setlinejoin col-1 s gr % polyline n 369 329 m 399 309 l gs col-1 s gr n 391.234 311.774 m 399.000 309.000 l 393.453 315.102 l gs 2 setlinejoin col-1 s gr % polyline n 369 329 m 404 329 l gs col-1 s gr n 396.000 327.000 m 404.000 329.000 l 396.000 331.000 l gs 2 setlinejoin col-1 s gr % polyline n 209 324 m 209 239 l 114 239 l 114 324 l clp gs col-1 s gr n 369 329 m 369 329 l gs col-1 s gr % polyline n 289 284 m 289 199 l 194 199 l 194 284 l clp gs col-1 s gr % polyline n 544 284 m 544 179 l 429 179 l 429 284 l clp gs col-1 s gr % polyline n 274 199 m 274 159 l 369 159 l 369 244 l 289 244 l gs col-1 s gr $f2psend %%enddocument @endspecial 1106 1537 a ff(j)-132 b(i)1050 1424 y(k)1425 1124 y fh(a)1447 1130 y fg(1)p fw(;)p fg(1)1492 1124 y fh(b)1510 1130 y fg(1)1275 1124 y fh(a)1297 1130 y fg(1)p fw(;)p fg(0)1342 1124 y fh(b)1360 1130 y fg(0)1556 1124 y fh(a)1578 1130 y fg(1)p fw(;)p fg(2)1623 1124 y fh(b)1641 1130 y fg(2)1425 1012 y fh(a)1447 1018 y fg(0)p fw(;)p fg(1)1492 1012 y fh(b)1510 1018 y fg(1)1556 1012 y fh(a)1578 1018 y fg(0)p fw(;)p fg(2)1623 1012 y fh(b)1641 1018 y fg(2)1275 1237 y fh(a)1297 1243 y fg(2)p fw(;)p fg(0)1342 1237 y fh(b)1360 1243 y fg(0)1425 1237 y fh(a)1447 1243 y fg(2)p fw(;)p fg(1)1492 1237 y fh(b)1510 1243 y fg(1)1556 1237 y fh(a)1578 1243 y fg(2)p fw(;)p fg(2)1623 1237 y fh(b)1641 1243 y fg(2)1275 1012 y fh(a)1297 1018 y fg(0)p fw(;)p fg(0)1342 1012 y fh(b)1360 1018 y fg(0)94 1199 y fh(a)116 1205 y fg(0)p fw(;)p fg(0)161 1199 y fh(b)179 1205 y fg(0)919 900 y fh(a)941 906 y fg(0)p fw(;)p fg(2)986 900 y fh(b)1004 906 y fg(2)394 1274 y fh(a)416 1280 y fg(2)p fw(;)p fg(0)461 1274 y fh(b)479 1280 y fg(0)319 1406 y fh(a)341 1412 y fg(2)p fw(;)p fg(2)386 1406 y fh(b)404 1412 y fg(2)694 1012 y fh(a)716 1018 y fg(1)p fw(;)p fg(0)761 1012 y fh(b)779 1018 y fg(0)806 1124 y fh(a)828 1130 y fg(2)p fw(;)p fg(1)873 1124 y fh(b)891 1130 y fg(1)506 1050 y fh(a)528 1056 y fg(0)p fw(;)p fg(1)573 1050 y fh(b)591 1056 y fg(1)206 1293 y fh(a)228 1299 y fg(1)p fw(;)p fg(1)273 1293 y fh(b)291 1299 y fg(1)619 1143 y fh(a)641 1149 y fg(1)p fw(;)p fg(2)686 1143 y fh(b)704 1149 y fg(2)335 1672 y fn(fig.)15 b fq(4.3)f(-)31 b fj(comp)n(actage)16 b(des)g(champs)h fk(a)f fj(et)g fk(b)g fj(dans)g(un)g(c)n(arr)o(\023) -22 b(e)16 b(de)h fe(z)1571 1656 y fg(2)146 1780 y fq(p)o(our)j(mettre) g(en)g(\023)-21 b(evidence)21 b(le)h(placemen)o(t)f(des)g(v)m(aleurs)h (sur)f(une)g(partie)g(seulemen)o(t)g(du)h(do-)75 1837

y(maine)d(de)g(r)o(\023)-21 b(ef)o(\023)g(erence,)17 b(nous)g(\023)-21 b(ecriv)o(ons)18 b(un)h(programme)e(plus)i(abstrait,) f fk(mavect3')p fq(,)f(don)o(t)h(les)h(rou-)75 1893 y(tages)d(sur)h (les)g(c)o(hamps)g fk(a)p fq(,)f fk(b)h fq(prennen)o(t)g(en)g(compte)g (le)g(placemen)o(t)h(des)f(v)m(aleurs)g(restrein)o(t)g(au)g(plan)75 1949 y fm(i)12 b fq(=)h(\()p fm(j)f ff(\000)f fm(k)q fq(\))k fm(mod)f(n)p fq(.)i(ce)f(programme)f(est)h(le)h(suiv)m(an)o(t)8 b(:)75 2063 y fk(matvect3')23 b(:)g(\(m,v\))15 b ff(7!)g fk(p)75 2078 y fx(8)75 2115 y(>)75 2128 y(>)75 2140 y(>)75 2153 y(>)75 2165 y(>)75 2178 y(<)75 2252 y(>)75 2265 y(>)75 2277 y(>)75 2290 y(>)75 2302 y(>)75 2315 y(:)137 2113 y fk(m)g fq(=)g fk(aligna)7 b(::)g(a0)137 2170 y(v)15 b fq(=)g fk(alignb)7 b(::)g(b0)137 2226 y(a')15 b fq(=)g fk(a0)g fa(c)h fk(spreada)532 2210 y fx(0)137 2283 y fk(b')f fq(=)g fk(b0)g fa(c)h fk(spreadb)532 2266 y fx(0)137 2339 y fk(p)f fq(=)g fk(prod)g fa(b)g fk(\(a'/&/)8 b(b'\))100 2400 y(spreada)268 2383 y fx(0)294 2400 y fq(=)15 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n)6 b(&)i(i)p fq(=)p fk(\(j-k\))23 b(mod)h(n\).\(i,j,0\))100 2456 y(spreadb)268 2440 y fx(0)294 2456 y fq(=)15 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j,k)p fm(<)p fk(n)6 b(&)i(i)p fq(=)p fk(\(j-k\))23 b(mod)h(n\).\(j,j,0\))146 2542 y fq(le)18 b(ra\016nemen)o(t)g(du)g(programme)f(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t)16 b(est)i(obten)o(u)g(en)g(r)o(\023)-21 b(e)o(\023)g(ecriv)m(an)o(t)17 b(les)g(\023)-21 b(equations)18 b(com-)p eop %%page: 60 62 60 61 bop 75 269 a fq(60)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)75 397 y fq(p)q(ortan)o(t)f(les)i(routages)7 b(:)281 499 y fk(a')15 b fq(=)g fk(a0)g fa(c)h fk(spreada)676 482 y fx(0)170 542 y fg(\(5)p fw(a)p fg(\))178 570 y ff(!)58 b fk(a')15 b fq(=)g fk(\(a0)24 b fa(c)g fk(spreada)o(\))15 b fa(c)g fk(diag)901 554 y fx(0)281 627 y fq(car)g fk(spreada)9 b ff(\016)h fk(diag)663 610 y fx(0)689 627 y fq(=)15 b fk(spreada)907 610 y fx(0)281 707 y fk(b')g fq(=)g fk(b0)g fa(c)h fk(spreadb)676 690 y fx(0)170 750 y fg(\(5)p fw(a)p fg(\))178 778 y ff(!)58 b fk(b')15 b fq(=)g fk(\(b0)24 b fa(c)g fk(spreadb)o(\))15 b fa(c)g fk(diag)901 762 y fx(0)281 835 y fq(car)g fk(spreadb)9 b ff(\016)h fk(diag)663 818 y fx(0)689 835 y fq(=)15 b fk(spreadb)907 818 y fx(0)146 914 y fq(en)f(renomman)o(t)f(les)i(expressions)g fk(a0)g fa(c)g fk(spreada)e fq(et)h fk(b0)h fa(c)h fk(spreadb)o fq(,)d(resp)q(ectiv)o(emen)o(t)i fk(a)f fq(et)g fk(b)p fq(,)75 970 y(nous)h(obtenons)g(p)q(our)h(l')o(\023)-21 b(equation)15 b(du)g(c)o(hamp)h fk(p)7 b fq(:)277 1072 y fk(p)15 b fq(=)h fk(prod)f fa(b)g fk(\(a)24 b fa(c)g fk(diag)755 1055 y fx(0)790 1072 y fk(/&/)7 b(b)24 b fa(c)g fk(diag)1072 1055 y fx(0)1083 1072 y fk(\))170 1115 y fg(\(8)p fw(b)p fg(\))176 1143 y ff(!)56 b fk(p)15 b fq(=)h fk(prod)f fa(b)g fk(\(\(a)23 b(/&/)8 b(b\))24 b fa(c)g fk(diag)930 1127 y fx(0)941 1143 y fk(\))277 1200 y fq(car)15 b fm(dom)p fq(\()p fk(diag)553 1183 y fx(0)564 1200 y fq(\))d ff(\022)h fm(dom)p fq(\()p fm(\033)772 1207 y fr(a)789 1200 y fq(\))c ff(\\)i fm(dom)p fq(\()p fm(\033)987 1207 y fr(b)1004 1200 y fq(\))e ff(^)i fm(img)r fq(\()p fk(diag)1266 1183 y fx(0)1275 1200 y fq(\))i ff(\022)g fm(dom)p fq(\()p fm(v)1480 1207 y fr(a)1497 1200 y fq(\))c ff(\\)i fm(dom)p fq(\()p fm(v)1691 1207 y fr(b)1708 1200 y fq(\))277 1256 y(\(de)16 b(la)f(derni)o(\022)-21

b(ere)15 b(\023)-21 b(equation)14 b(de)i fk(matvect3)p fq(\))75 1335 y(nous)f(obtenons)g(ainsi)i(le)f(programme)e fk(matvect3)g fq(par)h(ra\016nemen)o(t.)146 1450 y(nous)i(p)q(ouv)o (ons)h(main)o(tenan)o(t)g(expliciter)i(le)e(placemen)o(t)h(des)f(v)m (aleurs)h(sur)e(le)i(premier)g(plan)f(du)75 1506 y(cub)q(e.)e(nous)g (constatons)e(que)i(les)h(domaines)f(de)g(v)m(aleurs)g(de)g fk(a0)f fq(et)h fk(a')p fq(,)e(de)i(m)o(^)-21 b(eme)15 b(que)h(ceux)g(de)g fk(b0)75 1563 y fq(et)i fk(b')p fq(,)f(son)o(t)g (di\013)o(\023)-21 b(eren)o(ts)17 b(bien)i(que)f(leurs)h(cardinaux)g (soien)o(t)e(\023)-21 b(egaux.)16 b(nous)i(allons)h(mon)o(trer)e(que)h (le)75 1619 y(programme)c fk(matvect3")g fq(qui)i(suit,)f(est)g (ra\016n)o(\023)-21 b(e)14 b(par)h fk(matvect3')p fq(.)75 1734 y fk(matvect3")23 b(:)g(\(m,v\))15 b ff(7!)g fk(p)75 1748 y fx(8)75 1786 y(>)75 1798 y(>)75 1811 y(>)75 1823 y(>)75 1836 y(>)75 1848 y(<)75 1923 y(>)75 1935 y(>)75 1948 y(>)75 1960 y(>)75 1973 y(>)75 1985 y(:)137 1784 y fk(m)g fq(=)g fk(aligna)7 b(::)g(a0)132 b fq(\()p fm(i)p fq(\))137 1840 y fk(v)15 b fq(=)g fk(alignb)7 b(::)g(b0)132 b fq(\()p fm(ii)p fq(\))137 1897 y fk(a)15 b fq(=)g fk(a0)g fa(c)h fk(front)152 b fq(\()p fm(iii)p fq(\))137 1953 y fk(b)15 b fq(=)g fk(b0)g fa(c)h fk(spreadb)508 1937 y fx(00)612 1953 y fq(\()p fm(iv)r fq(\))137 2010 y fk(p)f fq(=)g fk(prod)g fa(b)g fk(\(a/&/)8 b(b\))100 2070 y(front)118 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(k)p fq(=)p fk(0\).\(i,j,k\))100 2127 y(spreadb)268 2110 y fx(00)338 2127 y fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(k)p fq(=)p fk(0\).\(j,j,0\))146 2213 y fq(p)o(our)19 b(cela,)h(nous)g(transformons)e(l'op)o(\023)-21 b(eration)19 b(de)h(routage)e(sur)i(le)h(c)o(hamp)e fk(b)p fq(,)g(en)i(utilisan)o(t)f(la)75 2270 y(propri)o(\023)-21 b(et)o(\023)g(e)14 b(2)g(qui)i(p)q(ermet)g(remplacer)g(toutes)e(les)i (o)q(ccurrences)g(de)g fk(b0)f fq(par)f fk(h)8 b(::)f(b0)p fq(,)15 b(o)q(\022)-24 b(u)15 b(la)g(bijection)75 2326 y fk(h)p fq(,)f(\023)-21 b(egale)14 b(\022)-23 b(a)15 b(son)g(in)o(v)o(erse,)g(est)8 b(:)364 2429 y fk(h)15 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(k)p fq(=)p fk(0\).\(i,j,\(j-i\)) 22 b(mod)i(n\))h fq(#)488 2485 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(\(j-k\))d(mod)h(n)8 b(&)g(k)p ff(6)p fq(=)p fk(0\).\(i,j,0\))488 2542 y fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p ff(6)p fq(=)p fk(\(j-k\))22 b(mod)h(n)8 b(&)g(k)p ff(6)p fq(=)p fk(0\).\(i,j,k\))p eop %%page: 61 63 61 62 bop 75 269 a fl(4.4.)29 b(repr)310 257 y(\023)304 269 y(esent)l(a)l(tion)17 b(du)e(pr)o(obl)867 257 y(\022)861 269 y(eme)h(en)f(deux)g(dimensions)298 b fq(61)75 397 y(nous)15 b(a)o(v)o(ons)7 b(:)340 497 y fk(v)15 b fq(=)g fk(alignb)7 b(::)g(b0)170 552 y fg(\()p fw(pr)q(op:)p fg(2)q(\))207 580 y ff(!)88 b fk(v)15 b fq(=)g fk(alignb)7 b(::)g(\(h)h(::)f(b0\))340 637 y fq(car)14 b fm(dom)p fq(\()p fm(\033)545 644 y fr(b0)578 637 y fq(\))e ff(\022)h fm(img)r fq(\()p fk(h)p fq(\))g(de)j(\()p fm(iv)r fq(\))170 691 y fg(\(7)p fw(b)p fg(\))176 720 y ff(!)119 b fk(v)15 b fq(=)g fk(\(alignb)9 b ff(\016)h fk(h\))d(::)h(b0)170 788 y ff(!)125 b fk(v)15 b fq(=)g fk(alignb)7 b(::)g(b0)340 845 y fq(car)14 b fk(alignb)c ff(\016)g fk(h)23 b(=)h(alignb)75 922 y fq(et)210 966 y fk(b)15 b fq(=)h fk(\(h)7 b(::)h(b0\))15 b fa(c)g fk(spreadb)716 949 y fx(00)100 1021 y fg(\(3)p fw(a)p fg(\))107 1049 y ff(!)58 b fk(b)15 b fq(=)h fk(h)7

b(::)g(\(b0)24 b fa(c)g fk(spreab)685 1033 y fx(0)696 1049 y fk(\))210 1106 y fq(car)15 b fk(spreadb)454 1089 y fx(0)480 1106 y fq(=)g fk(h)554 1089 y fq(-1)602 1106 y ff(\016)10 b fk(spreadb)803 1089 y fx(00)833 1106 y ff(\016)g fk(h)210 1162 y fq(et)15 b fm(dom)p fq(\()p fk(spreadb)535 1146 y fx(00)555 1162 y fq(\))10 b ff([)g fm(img)r fq(\()p fk(spreadb)889 1146 y fx(00)908 1162 y fq(\))i ff(\022)h fm(img)r fq(\()p fk(h)p fq(\))8 b ff(^)j fm(dom)p fq(\()p fm(v)1301 1169 y fr(b0)1333 1162 y fq(\))h ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))210 1219 y(car)i fm(dom)p fq(\()p fm(v)412 1226 y fr(b0)444 1219 y fq(\))e(=)g fm(dom)p fq(\()p fk(alignb)n fq(\))i(des)g(deuxi)o (\022)-21 b(emes)16 b(et)f(quatri)o(\022)-21 b(emes)14 b(\023)-21 b(equations)14 b(de)i fk(matvect3')146 1296 y fq(p)o(our)e(le)i(c)o(hamp)e fk(a)p fq(,)h(nous)g(v)o(oulons)g(in)o (tro)q(duire)h(l'op)o(\023)-21 b(eration)14 b(g)o(\023)-21 b(eom)o(\023)g(etrique)13 b fk(spreada)1633 1280 y fx(0)1659 1296 y fq(appliqu)o(\023)-21 b(ee)75 1353 y(sur)16 b fk(a0)p fq(.)g(il)h(s'agit)f(l\022)-23 b(a)16 b(d')o(\023)-21 b(equiv)m(alence)18 b(faible,)g(et)e(nous)g(rempla\030)-20 b(cons)17 b(l'expression)h fk(a)e fq(=)g fk(a0)f fa(c)h fk(front)o fq(,)75 1409 y(\022)-23 b(a)15 b(l'aide)h(de)g(la)f(propri)o (\023)-21 b(et)o(\023)g(e)14 b(4.\(1\),)f(par)i(la)h(nouv)o(elle)g (expression)9 b(:)614 1498 y fk(h)e(::)h(\(\(a0)23 b fa(c)h fk(front)o(\))g fa(c)g fk(spreada)1273 1481 y fx(0)1284 1498 y fk(\))75 1587 y fq(puisque)17 b(nous)e(v)o(\023)-21 b(eri\014ons)15 b(les)h(conditions)9 b(:)143 1669 y({)23 b fk(h)15 b fq(est)g(une)g(in)o(v)o(erse)h(gauc)o(he)f(de)h fk(spreada)914 1653 y fx(0)925 1647 y fq(-1)143 1759 y fq({)23 b fm(dom)p fq(\()p fk(front)n fq(\))13 b(=)g fm(img)r fq(\()p fk(spreada)756 1742 y fx(0)765 1759 y fq(\))189 1815 y(car)i fm(img)r fq(\()p fk(spreada)531 1799 y fx(0)540 1815 y fq(\))f(=)g fm(img)r fq(\()p fk(front)n fq(\))f ff(\022)h fm(dom)p fq(\()p fm(v)1043 1822 y fr(a0)1076 1815 y fq(\))f ff(\022)j fm(dom)p fq(\()p fk(aligna)o fq(\))d(=)h fm(img)r fq(\()p fk(front)n fq(\))h(de)i(\()p fm(i)p fq(\))189 1872 y(et)e(\()p fm(iii)p fq(\).)143 1961 y({)23 b fk(h)p fq(\()p fm(img)r fq(\()p fk(spreada)497 1945 y fx(0)505 1961 y fq(\)\))12 b ff(\022)h fm(dom)p fq(\()p fm(\033)731 1968 y fr(a0)764 1961 y fq(\))f ff(\022)h fm(img)r fq(\()p fk(h)p fq(\))189 2017 y(car)19 b fk(h)p fq(\()p fm(img)r fq(\()p fk(spreada)577 2001 y fx(0)586 2017 y fq(\)\))g(=)i fm(dom)p fq(\()p fk(spreada)969 2001 y fx(0)979 2017 y fq(\))f ff(\022)g fm(dom)p fq(\()p fk(front)o fq(\))f ff(\022)i fm(dom)p fq(\()p fm(\033)1518 2024 y fr(a0)1550 2017 y fq(\))f ff(\022)g fm(img)r fq(\()p fk(h)p fq(\))e(de)189 2074 y(\()p fm(iii)p fq(\).)146 2157 y(l')o(\023)-21 b(equation)14 b(d)o(\023)-21 b(e\014nissan)o(t)15 b(le)h(c)o(hamp)f fk(a)g fq(devien)o(t)h(ensuite)9 b(:)277 2257 y fk(a)15 b fq(=)h fk(h)7 b(::)h(\(\(a0)23 b fa(c)h fk(front)o(\))g fa(c)g fk(spreada)1026 2240 y fx(0)1037 2257 y fk(\))170 2311 y fg(\(5)p fw(b)p fg(\))176 2340 y ff(!)56 b fk(a)15 b fq(=)h fk(h)7 b(::)h(\(a0)23 b fa(c)h fk(front)10 b ff(\016)g fk(spreada)939 2324 y fx(0)949 2340 y fk(\))277 2396 y fq(car)15 b fm(img)r fq(\()p fk(spreada)619 2380 y fx(0)629 2396 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(front)n fq(\))g ff(\022)g fm(dom)p fq(\()p fm(\033)1138 2403 y fr(a0)1170 2396 y fq(\))170 2465 y ff(!)62 b fk(a)15 b fq(=)h fk(h)7 b(::)h(\(a0)23 b fa(c)h fk(spreada)776 2448 y fx(0)787

2465 y fk(\))277 2521 y fq(car)15 b fk(front)10 b ff(\016)g fk(spreada)684 2505 y fx(0)709 2521 y fq(=)16 b fk(spreada)928 2505 y fx(0)p eop %%page: 62 64 62 63 bop 75 269 a fq(62)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)146 397 y fq(l')o(\023)-21 b(equation)16 b(d)o(\023)-21 b(e\014nissan)o(t)16 b(le)i(c)o(hamp)e fk(p)p fq(,)h(en)g(utilisan)o(t) h(les)f(nouv)o(elles)h(d)o(\023)-21 b(e\014nitions)18 b(de)f fk(a)f fq(et)h fk(b)p fq(,)f(de-)75 453 y(vien)o(t)8 b(:)340 560 y fk(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(h)8 b(::)g(\(a0)23 b fa(c)h fk(spreada)1024 543 y fx(0)1035 560 y fk(\))g(/&/)7 b(h)h(::)g(\(b0)23 b fa(c)h fk(spreadb)1572 543 y fx(0)1583 560 y fk(\)\))170 615 y fg(\(11)p fw(a)p fg(\))186 643 y ff(!)109 b fk(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(h)8 b(::)g(\(a0)23 b fa(c)h fk(spreada)1024 627 y fx(0)1059 643 y fk(/&/)7 b(b0)24 b fa(c)g fk(spreadb)1437 627 y fx(0)1448 643 y fk(\)\))170 698 y fg(\(1)p fw(a)p fg(\))178 726 y ff(!)117 b fk(p)15 b fq(=)g fk(h)8 b(::)f(\(prod)23 b fa(b)h fk(\(a0)g fa(c)g fk(spreada)1041 710 y fx(0)1075 726 y fk(/&/)8 b(b0)24 b fa(c)g fk(spreadb)1454 710 y fx(0)1464 726 y fk(\)\))170 781 y fg(\()p fw(pr)q(op:)p fg(2)q(\))207 810 y ff(!)88 b fk(h)7 b(::)g(p)15 b fq(=)h fk(h)7 b(::)h(\(prod)23 b fa(b)h fk(\(a0)f fa(c)i fk(spreada)1128 793 y fx(0)1162 810 y fk(/&/)8 b(b0)23 b fa(c)i fk(spreadb)1541 793 y fx(0)1551 810 y fk(\)\))340 878 y fq(car)14 b fm(dom)p fq(\()p fm(\033)545 885 y fr(p)562 878 y fq(\))f ff(\022)f fm(img)r fq(\()p fk(h)p fq(\))170 933 y fg(\()p fw(i)r(i)r fg(\))178 961 y ff(!)117 b fk(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(a0)24 b fa(c)g fk(spreada)913 945 y fx(0)947 961 y fk(/&/)8 b(b0)24 b fa(c)g fk(spreadb)1326 945 y fx(0)1336 961 y fk(\))340 1018 y fq(car)14 b fm(dom)p fq(\()p fm(v)541 1025 y fr(p)558 1018 y fq(\))f ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\))146 1158 y(il)e(su\016t)f(main)o(tenan)o (t)f(de)i(nommer)f fk(a)g fq(l'expression)h fk(a0)k fa(c)g fk(spreada)1310 1142 y fx(0)1331 1158 y fq(et)10 b fk(b)g fq(l'expression)h fk(b0)k fa(c)h fk(spreadb)1934 1142 y fx(0)1944 1158 y fq(,)75 1214 y(p)q(our)f(obtenir)h(le)g(programme)e fk(matvect3')p fq(.)75 1373 y fw(4.4.2)55 b(suppression)19 b(d'une)f(dimension)146 1471 y fq(nous)13 b(p)q(ouv)o(ons)g(main)o (tenan)o(t)g(mon)o(trer)f(que)i(le)g(programme)e fk(matvect3")g fq(ra\016ne)h(le)h(programme)75 1527 y(suiv)m(an)o(t,)h(don)o(t)g(les)h (c)o(hamps)f(son)o(t)g(exprim)o(\023)-21 b(es)15 b(dans)g fe(z)1006 1511 y fg(2)1030 1527 y fq(:)75 1647 y fk(matvect4)23 b(:)g(\(m,v\))15 b ff(7!)h fk(p)75 1665 y fx(8)75 1702 y(>)75 1714 y(>)75 1727 y(>)75 1739 y(>)75 1752 y(>)75 1764 y(>)75 1777 y(>)75 1789 y(<)75 1864 y(>)75 1876 y(>)75 1889 y(>)75 1901 y(>)75 1914 y(>)75 1926 y(>)75 1939 y(>)75 1951 y(:)137 1697 y fk(a0)f fq(=)g fk(matrix)7 b(::)g(a0)137 1753 y(b0)15 b fq(=)g fk(matrix)7 b(::)g(b0)137 1810 y(m)15 b fq(=)g fk(a0)137 1866 y(v)g fq(=)g fk(align)7 b(::)h(b0)137 1922 y(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread)137 1979 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(a0/&/)8 b(b\))100 2051 y(matrix)49 b fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\))100 2108 y(align)73 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n)6 b(&)i(i)p fq(=)p fk(j\).\(i\))100 2164 y(spread)49 b fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(j,j\))146 2260 y fq(p)o(our)18 b(le)i(mon)o(trer,)d(il)j(su\016t)f(de)h

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col-1 s gr

col-1 s gr

gs 1 -1 scale (p = a*b) col-1 show gr /times-roman findfont 6.00 scalefont setfont 322 437 m gs 1 -1 scale (c = b0+c1) col-1 show gr /times-roman findfont 6.00 scalefont setfont 394 319 m gs 1 -1 scale (b0) col-1 show gr /times-roman findfont 6.00 scalefont setfont 649 419 m gs 1 -1 scale (c = b0+c0) col-1 show gr $f2psend %%enddocument @endspecial 225 w fy(\(a\))715 b(\(b\))363 2531 y fn(fig.)15 b fq(4.5)f(-)31 b fj(\(a\))15 b(a)o(lgorithme)i(initial)64 b(\(b\))15 b(a)o(lgorithme)h(am)o(\023)-22 b(elior)o(\023)g(e)p eop %%page: 70 72 70 71 bop 75 269 a fq(70)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)75 397 y fk(gaxpx1)23 b(:)h(\(a,x\))14 b ff(7!)i fk(c)75 411 y fx(8)75 449 y(>)75 461 y(>)75 473 y(>)75 486 y(>)75 498 y(>)75 511 y(>)75 523 y(>)75 536 y(>)75 548 y(>)75 561 y(>)75 573 y(<)75 648 y(>)75 660 y(>)75 673 y(>)75 685 y(>)75 698 y(>)75 710 y(>)75 723 y(>)75 735 y(>)75 747 y(>)75 760 y(>)75 772 y(:)137 452 y fk(a)f fq(=)g fk(matrix)7 b(::)g(a)137 509 y(x)15 b fq(=)g fk(align)7 b(::)h(b0)137 565 y(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread)137 622 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 678 y(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce)23 b fq(;)8 b fa(b)24 b fk(p\))137 735 y(c1)15 b fq(=)g fk(c0)g fa(c)h fk(col2row)137 791 y(c)f fq(=)g fk(add)g fa(b)h fk(\(b0)23 b(/&/)8 b(c1\))100 865 y(matrix)73 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))100 922 y(align)97 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p fq(=)p fk(0)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(j\))100 978 y(spread)73 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(0,j\))100 1035 y(reduce)73 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,n-1\))100 1091 y(col2row)f fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p fq(=)p fk(0)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(j,n-1\))100 1147 y(add)145 b fq(=)50 b fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+b\))100 1204 y(prod)121 b fq(=)50 b fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a*b\))100 1260 y(sum)145 b fq(=)50 b fk(id)10 b fq(#)g fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+sum)22 b(b\))75 1383 y fw(4.6.2)55 b(algorithme)16 b(a)n(v)n(ec)j(placemen)n(t)e(am)o(\023)-27 b(elior)o(\023)g(e)146 1469 y fq(si)17 b(on)g(s'autorise)f(la)i (di\013usion)g(des)f(v)m(aleurs)h(diagonales,)f(nous)g(p)q(ouv)o(ons)g (alors)g(concev)o(oir)g(une)75 1525 y(v)o(ersion)e(plus)g(e\016cace,)g (comme)f(esquiss)o(\023)-21 b(e)14 b(sur)h(la)g(\014gure)f(4.5\(b\).)f (en)h(e\013et,)g(les)h(alignemen)o(ts)h(de)e fm(x)h fq(et)75 1582 y fm(ax)i fq(sur)g(la)h(diagonale)f(fon)o(t)g(que)g(la)h(phase)f (de)h(transp)q(osition)f(devien)o(t)h(in)o(utile.)h(le)e(gain)h (engendr)o(\023)-21 b(e)75 1638 y(est)15 b(imp)q(ortan)o(t)g(d)o(\022) -21 b(es)14 b(lors)i(que)f(l'op)o(\023)-21 b(eration)15 b(est)f(r)o(\023)-21 b(ep)o(\023)g(et)o(\023)g(ee)13 b(de)j(nom)o(breuses)f(fois.)75 1758 y fo(r)o(\023)-25 b(e-alignmen)o(ts)19 b(du)e(v)o(ecteur)146 1843 y fq(l'id)o(\023)-21

b(ee)18 b(de)g(la)h(transformation)d(rep)q(ose)j(sur)f(le)h(r)o(\023) -21 b(e-alignemen)o(t)18 b(de)g fk(b0)g fq(et)g fk(c0)p fq(,)f(que)i(nous)f(allons)75 1900 y(placer)f(sur)g(la)g(diagonale,)g (comme)f(le)i(sugg)o(\022)-21 b(ere)15 b(la)i(\014gure)g(4.5.)e(cette)h (mo)q(di\014cation)i(du)f(placemen)o(t)75 1956 y(du)f(v)o(ecteur)f(v)m (a)g(ensuite)i(guider)f(la)f(reste)g(de)h(la)f(transformation.)146 2069 y(nous)d(c)o(hangeons)g(d'ab)q(ord)g(la)h(d)o(\023)-21 b(e\014nition)13 b(de)g fk(b0)f fq(fonction)g(de)h fk(x)f fq(en)h(appliquan)o(t)g(un)g(c)o(hangemen)o(t)75 2126 y(de)f(base)f(\()o(\023)-21 b(egal)10 b(\022)-23 b(a)11 b(son)h(in)o(v)o(erse\),)f(qui)g(\023)-21 b(ec)o(hange)10 b(les)i(v)m(aleurs)h(plac)o(\023)-21 b(ees)11 b(sur)g(la)h(diagonale)g (et)f(celles)j(plac)o(\023)-21 b(ees)75 2182 y(sur)15 b(la)h(premi)o(\022)-21 b(ere)15 b(ligne)9 b(:)401 2279 y fk(row2diag)j fq(=)49 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p fq(=)p fk(0)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(j,j\))25 b fq(#)689 2336 y fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p ff(6)p fq(=)p fk(0)6 b(&)i(i)p fq(=)p fk(j)g(&)g(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(0,j\))24 b fq(#)689 2392 y fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p ff(6)p fq(=)p fk(0)6 b(&)i(i)p ff(6)p fq(=)p fk(j)g(&)g(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))146 2486 y fq(des)15 b(d)o(\023)-21 b(e\014nitions)16 b(10)e(et)i(9,)e(nous)i(obtenons)f(un) h(programme)e(ra\016n)o(\023)-21 b(e)14 b(en)i(rempla\030)-20 b(can)o(t)15 b(toutes)g(les)75 2542 y(o)q(ccurrences)h(de)g fk(b0)f fq(par)g fk(\(row2diag)6 b(::)i(b0\))p fq(:)p eop %%page: 71 73 71 72 bop 75 269 a fl(4.6.)29 b(stra)l(t)336 257 y(\023)330 269 y(egie)16 b(4)7 b(:)15 b(modifica)l(tion)h(des)f(alignements)462 b fq(71)75 471 y fk(gaxpx1')23 b(:)h(\(a,x\))14 b ff(7!)i fk(c)75 485 y fx(8)75 522 y(>)75 535 y(>)75 547 y(>)75 560 y(>)75 572 y(>)75 585 y(>)75 597 y(>)75 610 y(>)75 622 y(>)75 635 y(>)75 647 y(<)75 722 y(>)75 734 y(>)75 747 y(>)75 759 y(>)75 772 y(>)75 784 y(>)75 796 y(>)75 809 y(>)75 821 y(>)75 834 y(>)75 846 y(:)137 526 y fk(a)f fq(=)g fk(matrix)7 b(::)g(a)137 583 y(x)15 b fq(=)g fk(align)7 b(::)h(\(row2diag)e(::)i(b0\))171 b fq(\()p fm(i)p fq(\))137 639 y fk(b)15 b fq(=)g fk(\(row2diag)7 b(::)h(b0)o(\))15 b fa(c)h fk(spread)129 b fq(\()p fm(ii)p fq(\))137 696 y fk(p)15 b fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 752 y(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce)23 b fq(;)8 b fa(b)24 b fk(p\))137 809 y(c1)15 b fq(=)g fk(c0)g fa(c)h fk(col2row)137 865 y(c)f fq(=)g fk(add)g fa(b)h fk(\(row2diag)6 b(::)i(b0)24 b(/&/)7 b(c1\))146 973 y fq(le)15 b(nouv)o(el)h(alignemen)o(t)g(de)g fk(b0)f fq(par)g(rapp)q(ort)f(\022)-23 b(a)15 b fk(x)g fq(s'obtien)o(t)g(en)h (appliquan)o(t)g(les)g(r)o(\022)-21 b(egles)7 b(:)100 1106 y(\()p fm(i)p fq(\))174 1077 y fg(\(7)p fw(b)p fg(\))166 1106 y ff(\000)-7 b(!)50 b fk(x)15 b fq(=)h fk(align)9 b ff(\016)h fk(row2diag)c(::)i(b0)166 1174 y ff(\000)-7 b(!)50 b fk(x)15 b fq(=)h fk(align2)6 b(::)i(b0)289 1231 y fq(a)o(v)o(ec)15 b fk(align2)d fq(=)h fk(align)c ff(\016)h fk(row2diag)h fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i=j)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(j\))75 1349 y fq(in)o(tuitiv)o(emen)o(t,)13 b(on)f(constate)g(que)g(le)h(c)o(hamp)g fk(b)f fq(a)g(les)h(m)o(^)-21 b(emes)11 b(v)m(aleurs)i(dans)g(les)g (deux)f(\023)-21 b(enonc)o(\023)g(es,)10 b(mais)75 1406 y(que)18 b(ces)h(v)m(aleurs)g(on)o(t)e(des)h(origines)h(di\013)o(\023) -21 b(eren)o(tes)7 b(;)17 b(ceci)j(est)d(un)i(cas)f(t)o(ypique)g(d')o

(\023)-21 b(equiv)m(alence)20 b(faible.)75 1462 y(de)d(ce)g(fait,)f (nous)h(rempla\030)-20 b(cons)18 b(le)f(mem)o(bre)g(droit)g(de)g(l')o (\023)-21 b(equation)16 b(\()p fm(ii)p fq(\))g(par)g(un)i(c)o(hamp)f (faiblemen)o(t)74 1519 y(\023)-21 b(equiv)m(alen)o(t)16 b(selon)g(les)g(d)o(\023)-21 b(e\014nitions)16 b(9)f(et)g(10)7 b(:)327 1673 y fk(b)13 b fq(=)g fk(\(row2diag)6 b(::)i(\(\(row2diag)f (::)g(b0\))24 b fa(c)g fk(spread)o(\)\))15 b fa(c)h fk(row2diag)75 1827 y fq(nous)c(simpli\014ons)j(main)o(tenan)o(t)d(cette)g(expression) h(p)q(our)g(obtenir)g(une)g(nouv)o(elle)h(expression)f(du)g(rou-)75 1884 y(tage)h(appliqu)o(\023)-21 b(e)17 b(sur)e fk(b0)p fq(.)f(dans)h(ce)h(qui)g(suit,)f fk(i)g fq(d)o(\023)-21 b(enote)14 b(l'iden)o(tit)o(\023)-21 b(e)16 b(dans)f fe(z)1374 1867 y fg(2)1389 1884 y fq(:)223 2001 y fk(b)g fq(=)g fk(\(row2diag)7 b(::)h(\(\(row2diag)e(::)i(b0\))24 b fa(c)g fk(spread)o(\)\))15 b fa(c)g fk(row2diag)106 2056 y fg(\(3)p fw(a)p fg(\))100 2085 y ff(\000)-7 b(!)50 b fk(b)15 b fq(=)g fk(\(row2diag)7 b(::)h(\(row2diag)f(::)g(\(b0)24 b fa(c)g fk(spread2)o(\)\)\))15 b fa(c)g fk(row2diag)223 2153 y fq(car)g fm(dom)p fq(\()p fk(spread)n fq(\))10 b ff([)g fm(img)r fq(\()p fk(spread)n fq(\))j ff(\022)f fm(img)r fq(\()p fk(row2diag)n fq(\))223 2209 y(et)j fm(dom)p fq(\()p fm(v)402 2216 y fr(b0)434 2209 y fq(\))e ff(\022)g fm(dom)p fq(\()p fk(row2diag)n fq(\))i(de)g(\()p fm(i)p fq(\),)223 2266 y(et)g(a)o(v)o(ec)g fk(spread2)f fq(=)h fk(row2diag)9 b ff(\016)h fk(spread)g ff(\016)f fk(row2diag)14 b fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(j,j\))108 2320 y fg(\(7)p fw(b)p fg(\))100 2349 y ff(\000)-7 b(!)50 b fk(b)15 b fq(=)g fk(\(i)p ff(j)373 2356 y fw(dom)p fg(\()p fr(row2di)q(ag)p fg(\))595 2349 y fk(::)8 b(\(b0)24 b fa(c)g fk(spread2)o(\)\))15 b fa(c)g fk(row2diag)100 2417 y ff(\000)-7 b(!)50 b fk(b)15 b fq(=)g fk(\(b0)24 b fa(c)g fk(spread2)o(\))15 b fa(c)h fk(row2diag)223 2474 y fq(simpli\014cation,)h(car)e fm(dom)p fq(\()p fk(spread2)n fq(\))e ff(\022)g fm(dom)p fq(\()p fk(row2diag)n fq(\))p eop %%page: 72 74 72 73 bop 75 269 a fq(72)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)108 377 y fg(\(5)p fw(b)p fg(\))100 406 y ff(\000)-7 b(!)50 b fk(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread2)9 b ff(\016)h fk(row2diag)223 462 y fq(car)15 b fm(img)r fq(\()p fk(row2diag)m fq(\))e ff(\022)g fm(dom)p fq(\()p fk(spread2)n fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\033)1143 469 y fr(b0)1175 462 y fq(\))100 531 y ff(\000)-7 b(!)50 b fk(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread2)223 587 y fq(d)o(\023)-21 b(e\014nition)16 b(de)g fk(spread2)p fq(,)d fk(spread2)f fq(=)h fk(spread2)c ff(\016)h fk(row2diag)75 682 y fq(les)16 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(ts)14 b(calculs)i(de)g(ra\016nemen)o(t)f(nous)g(am)o(\022)-21 b(enen)o(t)14 b(\022)-23 b(a)7 b(:)75 801 y fk(gaxpx2)23 b(:)h(\(a,x\))14 b ff(7!)i fk(c)75 815 y fx(8)75 852 y(>)75 865 y(>)75 877 y(>)75 890 y(>)75 902 y(>)75 915 y(>)75 927 y(>)75 939 y(>)75 952 y(>)75 964 y(>)75 977 y(<)75 1052 y(>)75 1064 y(>)75 1076 y(>)75 1089 y(>)75 1101 y(>)75 1114 y(>)75 1126 y(>)75 1139 y(>)75 1151 y(>)75 1164 y(>)75 1176 y(:)137 856 y fk(a)f fq(=)g fk(matrix)7 b(::)g(a)137 913 y(x)15 b fq(=)g fk(align2)7 b(::)g(b0)137 969 y(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread2)137 1025 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137

1082 y(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce)23 b fq(;)8 b fa(b)24 b fk(p\))137 1138 y(c1)15 b fq(=)g fk(c0)g fa(c)h fk(col2row)137 1195 y(c)f fq(=)g fk(add)g fa(b)h fk(\(row2diag)6 b(::)i(b0)24 b(/&/)7 b(c1\))75 1292 y fq(de)16 b(la)g(m)o(^)-21 b(eme)15 b(mani)o(\022)-21 b(ere,)16 b(nous)g(mo)q(di\014ons)h(l'alignemen)o(t)g(de)f fm(ax)g fq(\()p fk(c0)p fq(\))f(en)i(utilisan)o(t)g(le)g(c)o(hangemen)o (t)75 1348 y(de)f(base)7 b(:)354 1473 y fk(col2diag)k fq(=)49 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(j)p fq(=)p fk(n-1)6 b(&)i(0)p ff(\024)p fk(i)p fm(<)p fk(n\).\(i,i\))25 b fq(#)641 1529 y fm(\025)p fk(\(i,j\))p ff(j)p fk(\(j)p ff(6)p fq(=)p fk(n-1)6 b(&)i(i)p fq(=)p fk(j)g(&)g(0)p ff(\024)p fk(i)p fm(<)p fk(n\).\(i,n-1\))24 b fq(#)641 1586 y fm(\025)p fk(\(i,j\))p ff(j)p fk(\(j)p ff(6)p fq(=)p fk(n-1)6 b(&)i(i)p ff(6)p fq(=)p fk(j)g(&)g(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,j\))75 1707 y fq(t)l(oujours)15 b(en)g(utilisan)o(t)i(les)f(d)o(\023)-21 b(e\014nitions)16 b(9)e(et)h(10,)g(nous)g(p)q(ouv)o(ons)g(ra\016ner)g fk(gaxpx2)f fq(en)8 b(:)75 1826 y fk(gaxpx2')23 b(:)h(\(a,x\))14 b ff(7!)i fk(c)75 1840 y fx(8)75 1877 y(>)75 1890 y(>)75 1902 y(>)75 1915 y(>)75 1927 y(>)75 1939 y(>)75 1952 y(>)75 1964 y(>)75 1977 y(>)75 1989 y(>)75 2002 y(<)75 2076 y(>)75 2089 y(>)75 2101 y(>)75 2114 y(>)75 2126 y(>)75 2139 y(>)75 2151 y(>)75 2164 y(>)75 2176 y(>)75 2189 y(>)75 2201 y(:)137 1881 y fk(a)f fq(=)g fk(matrix)7 b(::)g(a)137 1937 y(x)15 b fq(=)g fk(align2)7 b(::)g(b0)137 1994 y(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread2)137 2050 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 2107 y(col2diag)e(::)i(c0)14 b fq(=)i fk(sum)f fa(b)g fk(\(reduce)23 b fq(;)8 b fa(b)24 b fk(p)o(\))50 b fq(\()p fm(iii)p fq(\))137 2163 y fk(c1)15 b fq(=)g fk(\(col2diag)7 b(::)h(c0)o(\))15 b fa(c)h fk(col2row)183 b fq(\()p fm(iv)r fq(\))137 2220 y fk(c)15 b fq(=)g fk(add)g fa(b)h fk(\(row2diag)6 b(::)i(b0)24 b(/&/)7 b(c1\))75 2316 y fq(comme)14 b(le)i(mon)o(tre)e (la)i(\014gure)f(4.5\(b\),)e(les)j(v)m(aleurs)g(de)f fk(p)g fq(son)o(t)f(rassem)o(bl)o(\023)-21 b(ees)14 b(sur)h(la)h (diagonale)f(p)q(our)75 2373 y(former)22 b fk(c0)p fq(.)f(les)i(c)o (hamps)g fk(p)f fq(et)g fk(\(col2diag)7 b(::)h(p)o(\))15 b fa(c)h fk(col2diag)21 b fq(son)o(t)h(faiblemen)o(t)g(\023)-21 b(equiv)m(alen)o(ts.)75 2429 y(nous)14 b(obtenons)g(une)h(nouv)o(elle)h (r)o(\023)-21 b(eduction)14 b(en)h(appliquan)o(t)g(les)f(\023)-21 b(etap)q(es)14 b(suiv)m(an)o(tes)h(de)f(ra\016nemen)o(t)g(\022)-23 b(a)75 2486 y(cette)15 b(expression)9 b(:)p eop %%page: 73 75 73 74 bop 75 269 a fl(4.6.)29 b(stra)l(t)336 257 y(\023)330 269 y(egie)16 b(4)7 b(:)15 b(modifica)l(tion)h(des)f(alignements)462 b fq(73)223 391 y fk(col2diag)6 b(::)i(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce)23 b fq(;)8 b fa(b)24 b fk(\(\(col2diag)6 b(::)i(p\))24 b fa(c)g fk(col2diag)o(\)\))106 446 y fg(\(3)p fw(a)p fg(\))100 474 y ff(\000)-7 b(!)50 b fk(col2diag)6 b(::)i(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce)23 b fq(;)8 b fa(b)24 b fk(\(col2diag)6 b(::)i(\(p)24 b fa(c)g fk(col2diag)o(\)\)\))223 531 y fq(car)15 b fm(dom)p fq(\()p fm(v)425 538 y fr(p)442 531 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(cold2diag)n fq(\))108 585 y fg(\(3)p fw(c)p fg(\))100 614 y ff(\000)-7 b(!)50 b fk(col2diag)6 b(::)i(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(col2diag)7 b(::)h(\(reduce2)23 b fq(;)8 b fa(b)23 b fk(\(p)h fa(c)g fk(col2diag)o(\)\))o(\))223 670 y fq(renommage)7 b(:)15 b fk(reduce2)35 b fq(=)13 b fk(col2diag)c ff(\016)g fk(reduce)h

ff(\016)g fk(col2diag)689 727 y fq(=)j fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p fm(<)p fk(n)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(i,i\))106 781 y fg(\(1)p fw(a)p fg(\))100 810 y ff(\000)-7 b(!)50 b fk(col2diag)6 b(::)i(c0)15 b fq(=)g fk(col2diag)7 b(::)g(\(sum)23 b fa(b)h fk(\(reduce2)f fq(;)8 b fa(b)24 b fk(\(p)f fa(c)h fk(col2diag)o(\)\)\))107 865 y fg(\()p fw(i)r(i)r fg(\))100 893 y ff(\000)-7 b(!)50 b fk(c0)15 b fq(=)g fk(sum)g fa(b)h fk(\(reduce2)22 b fq(;)8 b fa(b)24 b fk(\(p)f fa(c)i fk(col2diag)n(\)\))223 950 y fq(car)15 b fm(dom)p fq(\()p fm(v)425 957 y fr(c0)457 950 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(col2diag)o fq(\))h(de)i(l')o(\023)-21 b(equation)15 b(\()p fm(iv)r fq(\))75 1098 y(les)e(r)o(\023)-21 b(e)o(\023)g(ecritures)11 b(suiv)m(an)o(tes)i(nous)g(p)q(ermetten)o(t)f (de)h(supprimer)g(le)g(routage)f fk(col2diag)f fq(de)i(la)f(derni)o (\022)-21 b(ere)75 1154 y(expression)8 b(:)162 1313 y ff(\000)-7 b(!)51 b fk(c0)14 b fq(=)i fk(sum)f fa(b)g fk(\(reduce2)9 b ff(\016)h fk(col2diag)23 b fq(;)8 b fa(b)23 b fk(\(p)h fa(c)g fk(col2diag)o(\)\))286 1369 y fq(car)14 b fk(reduce2)23 b(=)h(reduce2)9 b ff(\016)h fk(col2diag)100 1412 y fg(\(6)p fw(;pr)q(op)p fg(1\))131 1441 y ff(\000)-7 b(!)82 b fk(c0)14 b fq(=)i fk(sum)f fa(b)g fk(\(reduce2)23 b fq(;)8 b fa(b)23 b fk(\(\(p)h fa(c)g fk(col2diag)o(\))g fa(c)g fk(col2diag)o(\)\))286 1497 y fq(car)14 b fk(col2diag)g fq(est)g(\023)-21 b(egale)15 b(\022)-23 b(a)15 b(son)g(in)o(v)o(erse)171 1540 y fg(\(5)p fw(b)p fg(\))162 1568 y ff(\000)-7 b(!)51 b fk(c0)14 b fq(=)i fk(sum)f fa(b)g fk(\(reduce2)23 b fq(;)8 b fa(b)23 b fk(\(p)h fa(c)g fk(i)p ff(j)998 1575 y fw(dom)p fg(\()p fr(col2dia)q(g)p fg(\))1212 1568 y fk(\)\))286 1625 y fq(car)14 b fm(dom)p fq(\()p fk(col2diag)o fq(\))e ff(\022)h fm(dom)p fq(\()p fm(\033)864 1632 y fr(p)881 1625 y fq(\))162 1693 y ff(\000)-7 b(!)51 b fk(c0)14 b fq(=)i fk(sum)f fa(b)g fk(\(reduce2)23 b fq(;)8 b fa(b)23 b fk(p\))286 1761 y fq(simpli\014cation,)17 b(car)40 b fm(img)r fq(\()p fk(col2diag)m fq(\))12 b ff(\022)h fm(dom)p fq(\()p fm(v)1171 1768 y fr(p)1188 1761 y fq(\))680 1818 y fm(dom)p fq(\()p fk(col2diag)n fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\033)1182 1825 y fr(p)1199 1818 y fq(\))75 1954 y(nous)i(a)o(v)o(ons)f(donc)8 b(:)75 2125 y fk(gaxpx3)23 b(:)h(\(a,x\))14 b ff(7!)i fk(c)75 2140 y fx(8)75 2177 y(>)75 2189 y(>)75 2202 y(>)75 2214 y(>)75 2227 y(>)75 2239 y(>)75 2252 y(>)75 2264 y(>)75 2277 y(>)75 2289 y(>)75 2301 y(<)75 2376 y(>)75 2389 y(>)75 2401 y(>)75 2414 y(>)75 2426 y(>)75 2438 y(>)75 2451 y(>)75 2463 y(>)75 2476 y(>)75 2488 y(>)75 2501 y(:)137 2181 y fk(a)f fq(=)g fk(matrix)7 b(::)g(a)137 2237 y(x)15 b fq(=)g fk(align2)7 b(::)g(b0)137 2294 y(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread2)137 2350 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 2407 y(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce2)23 b fq(;)8 b fa(b)24 b fk(p)o(\))137 2463 y(c1)15 b fq(=)g fk(\(col2diag)7 b(::)h(c0)o(\))15 b fa(c)h fk(col2row)57 b fq(\()p fm(v)r(i)p fq(\))137 2520 y fk(c)15 b fq(=)g fk(add)g fa(b)h fk(\(row2diag)6 b(::)i(b0/&/)f(c1\))p eop %%page: 74 76 74 75 bop 75 269 a fq(74)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)75 397 y fo(suppression)i(de)g(la)i(transp)q(osition)146 527 y fq(in)o(tuitiv)o(emen)o(t,)g(nous)h(observ)o(ons)f(que)g fk(c0)g fq(et)h fk(b0)f fq(son)o(t)f(tous)h(les)h(deux)g(align)o(\023)

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y fq(l')o(\023)-21 b(enonc)o(\023)g(e)19 b(\014nal)i(est)g(obten)o(u)f (en)h(appliquan)o(t)g(les)g(r)o(\022)-21 b(egles)20 b(\(11a\),)f (\(1a\))g(sur)h(la)h(derni)o(\022)-21 b(ere)20 b(\023)-21 b(equation,)75 2373 y(a\014n)19 b(de)g(factoriser)g(le)g(c)o(hangemen)o (t)g(de)g(base)g fk(row2diag)f fq(dans)h(la)g(partie)g(droite.)g(on)g (p)q(eut)h(alors)75 2429 y(simpli\014er)h(l')o(\023)-21 b(equation)18 b(en)i(rempla\030)-20 b(can)o(t)18 b(l'unique)j(o)q (ccurrence)f(de)f fk(c)g fq(par)f fk(row2diag)7 b(::)g(c)p fq(,)18 b(puis)i(en)75 2486 y(appliquan)o(t)c(la)g(r)o(\022)-21 b(egle)14 b(\(i)q(i\).)p eop %%page: 75 77 75 76 bop 75 269 a fl(4.6.)29 b(stra)l(t)336 257 y(\023)330 269 y(egie)16 b(4)7 b(:)15 b(modifica)l(tion)h(des)f(alignements)462 b fq(75)177 377 y fg(\(11)p fw(a)p fg(\))179 406 y ff(\000)-7 b(!)52 b fk(c)15 b fq(=)g fk(add)g fa(b)h fk(\(row2diag)6 b(::)i(\(b0)23 b(/&/)8 b(c0\)\))187 449 y fg(\(1)p fw(a)p fg(\))181 477 y ff(\000)-8 b(!)51 b fk(c)15 b fq(=)g fk(row2diag)7 b(::)g(\(add)23 b fa(b)i fk(\(b0)e(/&/)8 b(c0\))o(\))170 517 y fw(pr)q(op)p fg(2)176 544 y ff(\000)-8 b(!)56 b fk(row2diag)6 b(::)i(c)15 b fq(=)g fk(row2diag)7 b(::)g(\(add)23 b fa(b)h fk(\(b0)g(/&/)7 b(c0\)\))304 600 y fq(car)15 b fm(dom)p fq(\()p fm(\033)510 607 y fr(c)526 600 y fq(\))e ff(\022)g fm(img)r fq(\()p fk(row2diag)m fq(\))187 655 y fg(\()p fw(i)r(i)r fg(\))181 684 y ff(\000)-8 b(!)51 b fk(c)15 b fq(=)g fk(add)g fa(b)h fk(\(b0)23 b(/&/)8 b(c0\))304 740 y fq(car)15 b fm(dom)p fq(\()p fm(\033)510 747 y fr(c)526 740 y fq(\))e ff(\022)g fm(dom)p fq(\()p fk(row2diag)n fq(\))75 830 y(nous)18 b(obtenons)h(\014nalemen)o (t)g(un)g(programme)e(ra\016n)o(\023)-21 b(e,)17 b(don)o(t)h(le)h(v)o (ecteur)g(est)f(initialemen)o(t)i(align)o(\023)-21 b(e)75 886 y(sur)15 b(la)h(diagonale)8 b(:)75 999 y fk(gaxpx4)23 b(:)h(\(a,x\))14 b ff(7!)i fk(c)75 1021 y fx(8)75 1058 y(>)75 1070 y(>)75 1083 y(>)75 1095 y(>)75 1108 y(>)75 1120 y(>)75 1133 y(>)75 1145 y(<)75 1220 y(>)75 1232 y(>)75 1245 y(>)75 1257 y(>)75 1270 y(>)75 1282 y(>)75 1295 y(>)75 1307 y(:)137 1053 y fk(a)f fq(=)g fk(matrix)7 b(::)g(a)137 1109 y(x)15 b fq(=)g fk(align2)7 b(::)g(b0)137 1166 y(b)15 b fq(=)g fk(b0)g fa(c)h fk(spread2)137 1222 y(p)f fq(=)g fk(prod)g fa(b)g fk(\(a)24 b(/&/)8 b(b\))137 1278 y(c0)15 b fq(=)g fk(sum)g fa(b)g fk(\(reduce2)23 b fq(;)8 b fa(b)24 b fk(p)o(\))137 1335 y(c)15 b fq(=)g fk(add)g fa(b)h fk(\(b0/&/)7 b(c0\))100 1407 y(align2)73 b fq(=)50 b fm(\025)p fk(\(i,j\))p ff(j)p fk(\(i)p fq(=)p fk(j)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n\).\(j\))100 1464 y(spread2)49 b fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(j,j\))100 1520 y(reduce2)f fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(0)p ff(\024)p fk(i,j)p fm(<)p fk(n\).\(i,i\))p eop %%page: 76 78 76 77 bop 75 269 a fq(76)521 b fl(chapitre)16 b(4.)45 b(stra)l(t)1144 257 y(\023)1138 269 y(egies)16 b(de)f(transf)o(orma)l (tions)p eop %%page: 77 79 77 78 bop 1801 269 a fq(77)75 718 y fi(chapitre)32 b(5)75 935 y fh(pei)38 b fp(et)h(les)g(langages)g(data-parall)m(\022)-56 b(el)q(es)146 1168 y fq(nous)18 b(a)o(v)o(ons)f(mon)o(tr)o(\023)-21 b(e)16 b(dans)j([21)o(])e(que)i(le)g(formalisme)f fn(pei)g fq(et)g(le)h(mo)q(d)o(\022)-21 b(ele)18 b(de)h(programmation)75 1224 y(data-parall)o(\022)-21 b(ele)19 b(pr)o(\023)-21 b(esen)o(ten)o(t)17 b(de)i(nom)o(breuses)g(similitudes.)i(les)e(ob)s

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fm(v)145 1780 y fw(j)173 1773 y fq(=)p ff(f)p fq(2,3,4)p ff(g)p fq(.)e(puis)i(l'ensem)o(ble)i(des)f(pro)q(cesseurs)f(virtuels)h (v)m(alides)h(est)d(\023)-21 b(ev)m(alu)o(\023)g(e)7 b(:)j ff(f)p fq(\(2,2\),\(2,3\))o(,\()o(2,4\))o(,)75 1829 y(\(3,2\),\(3,3\))o(,\(3,)o(4\))p ff(g)e fq(et)13 b(en\014n)g(l'ensem)o(ble)g(des)g(pro)q(cesseurs)g(virtuels)g(actifs)8 b(:)j ff(f)p fq(\(2,3\),\(2,4\),)o(\(3,)o(4\))p ff(g)o fq(.)e(en-)75 1886 y(\014n)i(les)f(v)m(aleurs)i fk(a\(2,3\))p fq(,)p fk(a\(2,4\))p fq(,)p fk(a\(3,4\))6 b fq(son)o(t)k(resp)q(ectiv)o (emen)o(t)h(a\013ect)o(\023)-21 b(ees)9 b(\022)-23 b(a)9 b fk(a\(1,3\))p fq(,)p fk(a\(1,4\))p fq(,)p fk(a\(2,4\))p fq(.)75 2061 y fo(le)18 b(blo)q(c)h fk(forall)146 2147 y fq(il)c(faut)e(distinguer)i(l')p fj(instruction)f fk(forall)f fq(du)i(blo)q(c)g fk(forall)p fq(.)d(la)i(premi)o(\022)-21 b(ere)14 b(construction)h(n'au-)75 2204 y(torise)f(qu'une)g (a\013ectation)f(li)o(\023)-21 b(ee)14 b(au)g fk(forall)p fq(.)f(dans)g(le)i(deuxi)o(\022)-21 b(eme)14 b(cas,)f(plusieurs)j (instructions)f(p)q(eu-)75 2260 y(v)o(en)o(t)c(former)g(l'in)o(t)o (\023)-21 b(erieur)12 b(du)f fk(forall)p fq(,)g(qui)h(est)f(termin)o (\023)-21 b(e)11 b(par)g fk(endforall)p fq(.)f(le)i(blo)q(c)h(p)q(eut)f (notammen)o(t)75 2316 y(con)o(tenir)i(une)g(im)o(brication)h(de)f fk(forall)p fq(.)e(dans)h(ce)h(cas,)f(l'ensem)o(ble)i(des)f(pro)q (cesseurs)g(actifs)g(est)f(une)75 2373 y(restriction)j(de)f(l'ensem)o (ble)i(des)f(pro)q(cesseurs)f(actifs)g(d)o(\023)-21 b(e\014ni)16 b(par)f(les)h fk(forall)e fq(engloban)o(ts.)146 2486 y(cette)g(construction)h(nous)h(sera)e(utile)j(par)d(la)i(suite)f(car)g (hpf)g(imp)q(ose)h(une)f(restriction)h(imp)q(or-)75 2542 y(tan)o(te)7 b(:)19 b(un)h(indice)h(apparaissan)o(t)e(dans)g(un)h fm(tr)q(ipl)q(et)10 b ff(\000)h fm(spec)19 b fq(ne)h(p)q(eut)g(d)o (\023)-21 b(ep)q(endre)20 b(d'autres)f(indices)p eop %%page: 80 82 80 81 bop 75 269 a fq(80)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fq(apparaissan)o(t)e(dans)i(un)f (autre)g fm(tr)q(ipl)q(et)c ff(\000)g fm(spec)15 b fq(\([32)n(],)f (page)g(171\).)e(il)j(est)f(donc)h(n)o(\023)-21 b(ecessaire)14 b(d'utiliser)75 453 y(une)g(im)o(brication)h(de)f fk(forall)e fq(p)q(our)i(d)o(\023)-21 b(ecrire)14 b(des)g(domaines)g(qui)g(ne)g(d)o (\023)-21 b(ep)q(enden)o(t)14 b(pas)f(que)h(de)g(b)q(ornes)75 510 y(constan)o(tes.)g(p)o(ar)g(exemple,)j(un)e(domaine)h(triangulaire) g(sera)f(d)o(\023)-21 b(ecrit)15 b(par)7 b(:)147 610 y fk(forall)23 b(\(i=1:n\))242 667 y(forall)g(\(j=1:i\))337 723 y(a\(i,j\))h(=)f(....)242 780 y(endforall)147 836 y(endforall)75 964 y fo(la)18 b(b)q(oucle)h(ind)o(\023)-25 b(ep)q(endan)o(te)146 1052 y fq(notons)16 b(que)i(l'on)g(p)q(eut)f (\023)-21 b(egalemen)o(t)16 b(sp)o(\023)-21 b(eci\014er)18 b(un)g(traitemen)o(t)f(parall)o(\022)-21 b(ele)18 b(en)g(ins)o(\023)-21 b(eran)o(t)17 b(un)h(blo)q(c)75 1109 y(d'instructions)23 b(\022)-23 b(a)22 b(l'in)o(t)o(\023)-21 b(erieur)23 b(d'une)g(b)q (oucle)h fk(do)f(...)h(enddo)e fq(pr)o(\023)-21 b(ec)o(\023)g(ed)o (\023)g(ee)20 b(de)j(la)g(directiv)o(e)g fk($hpf!)75 1165 y(independent)p fq(.)12 b(le)j(programmeur)f(garan)o(tit)f(alors)h (au)g(compilateur)h(que)g(les)g(it)o(\023)-21 b(erations)14 b(son)o(t)f(ind)o(\023)-21 b(e-)75 1222 y(p)q(endan)o(tes)16 b(les)f(unes)h(des)f(autres,)f(p)q(ermettan)o(t)h(ainsi)h(au)f (compilateur)h(de)f(distribuer)h(c)o(hacune)g(des)75 1278 y(it)o(\023)-21 b(erations)17 b(sur)i(des)f(pro)q(cesseurs)h (di\013)o(\023)-21 b(eren)o(ts)17 b(p)q(our)i(une)g(ex)o(\023)-21 b(ecution)18 b(parall)o(\022)-21 b(ele.)19 b(nous)f(n'utiliserons)75 1335 y(pas)c(ici)i(ce)f(t)o(yp)q(e)f(de)h(b)q(oucle,)g(qui)g(se)g

(rappro)q(c)o(he)f(du)h(mo)q(d)o(\022)-21 b(ele)14 b(d'ex)o(\023)-21 b(ecution)15 b(mimd,)e(mais)i(nous)f(p)q(ou-)75 1391 y(v)o(ons)f(remarquer)g(que)h(l'instruction)h fk(forall)e fq(p)q(eut)g(^)-21 b(etre)12 b(remplac)o(\023)-21 b(ee)14 b(par)f(une)h(im)o(brication)h(parfaite)75 1447 y(de)g(b)q(oucles)h fk(do)e fq(ind)o(\023)-21 b(ep)q(endan)o(tes.)16 b(p)o(ar)d(exemple,)j (les)f(s)o(\023)-21 b(eman)o(tiques)14 b(du)h fk(forall)f fq(et)g(de)h(l'im)o(brication)75 1504 y(de)h(b)q(oucles)g(ind)o(\023) -21 b(ep)q(endan)o(tes)16 b(suiv)m(an)o(tes)g(son)o(t)e(\023)-21 b(equiv)m(alen)o(tes.)75 1604 y fk(forall)23 b(\(i=1:n,j=1:m\))237 b($hpf!)23 b(independent\(i,j\))290 1661 y(a\(i,j\)=b\(j,i\))332 b(do)24 b(i=1,n)1006 1717 y(do)f(j=1,m)1077 1774 y(a\(i,j\)=b\(j,i\)) 1006 1830 y(enddo)934 1887 y(enddo)75 2016 y fw(5.1.2)55 b(in)n(trinsics)146 2104 y fq(hpf)20 b(enric)o(hit)h(le)h(jeu)e(des)h (primitiv)o(es)h(de)e(f)l(ortran)g(90)g(propres)g(au)g(mo)q(d)o(\022) -21 b(ele)21 b(data-parall)o(\022)-21 b(ele,)75 2161 y(baptis)o(\023)g(ees)10 b fj(intrinsics)p fq(.)f(ces)i(primitiv)o(es)h (implan)o(ten)o(t)f(notammen)o(t)f(les)h(r)o(\023)-21 b(eductions)11 b(de)g(tableaux.)f(elles)75 2217 y(p)q(ermetten)o(t)20 b(ainsi)i(d')o(\023)-21 b(ecrire)21 b(des)g(programmes)e(plus)j(p)q (ortables)f(tout)f(en)h(gagnan)o(t)e(en)i(e\016cacit)o(\023)-21 b(e,)75 2274 y(ces)21 b(primitiv)o(es)g(p)q(ouv)m(an)o(t)f(^)-21 b(etre)19 b(optimis)o(\023)-21 b(ees)20 b(par)g(le)h(compilateur.)g (nous)f(ne)h(citerons)g(ici)g(que)g(les)75 2330 y(primitiv)o(es)f(de)f (la)h(famille)g fk(xxx)p 636 2330 15 2 v 17 w(scatter)e fq(que)h(nous)g(utiliserons)h(par)f(la)g(suite)h(p)q(our)f(traduire)g (les)75 2387 y(r)o(\023)-21 b(eductions)15 b(g)o(\023)-21 b(eom)o(\023)g(etriques)13 b(de)j fn(pei)p fq(.)f(p)o(armi)g(les)g(op)o (\023)-21 b(erateurs)14 b(de)i(r)o(\023)-21 b(eduction)15 b(disp)q(onibles)j(\(remplace)75 2443 y fk(xxx)p fq(\))c(on)i(trouv)o (e)e fk(sum)p fq(,)g fk(product)p fq(,)g fk(maxval)p fq(,)g fj(etc)p fq(.)h(la)g(primitiv)o(e)i(prend)f(comme)f(param)o (\022)-21 b(etres)6 b(:)143 2542 y({)23 b(un)15 b(tableau)h fm(a)f fq(de)h(dimension)h fm(m)e fq(qui)h(con)o(tien)o(t)f(les)h(donn) o(\023)-21 b(ees)15 b(\022)-23 b(a)15 b(r)o(\023)-21 b(eduire,)p eop %%page: 81 83 81 82 bop 75 269 a fl(5.1.)29 b(pr)246 257 y(\023)240 269 y(esent)l(a)l(tion)17 b(d'hpf)1103 b fq(81)143 397 y({)23 b(un)15 b(tableau)h fm(b)f fq(de)g(dimension)i fm(n)f fq(qui)g(con)o(tien)o(t)f(les)h(v)m(aleurs)g(initiales)h(des)f (r)o(\023)-21 b(eductions,)143 505 y({)23 b(un)15 b(ensem)o(ble)i(de)e (tableaux)h(d'indirection)h fm(x)985 512 y fg(1)1019 505 y fq(\022)-23 b(a)15 b fm(x)1083 512 y fw(n)1121 505 y fq(de)g(m)o(^)-21 b(eme)14 b(dimension)j(que)f fm(a)p fq(.)75 614 y(soit)k ff(a)g fq(l'ensem)o(ble)h(des)g(indices)g (de)g fm(a)p fq(,)e(et)h fm(z)i fq(un)e fm(m)p fq(-uplet)h ff(2)g fe(n)1243 597 y fw(m)1273 614 y fq(.)e(un)g(\023)-21 b(el)o(\023)g(emen)o(t)19 b(du)h(tableau)g(r)o(\023)-21 b(e-)75 670 y(sultat)15 b(calcul)o(\023)-21 b(e)17 b(par)e(la)h (primitiv)o(e)h(de)f(la)g(famille)h fk(xxx)p 1016 670 15 2 v 17 w(scatter)d fq(utilisan)o(t)j ff(\012)f fq(comme)f(op)o(\023) -21 b(erateur)14 b(de)75 727 y(r)o(\023)-21 b(eduction)15 b(est)8 b(:)474 794 y fm(r)q fq(\()p fm(i)530 801 y fg(1)548 794 y fm(;)g(:)g(:)g(:)d(;)j(i)666 801 y fw(n)687 794 y fq(\))13 b(=)889 762 y fx(n)958 794 y fm(a)p fq(\()p fm(z)r fq(\))786 850 y fw(z)q fx(2aj8)p fw(k)q fx(2)p fg([1)p fw(::n)p fg(])p fw(;x)1028 854 y fd(k)1044 850 y fg(\()p fw(z)q fg(\)=)p fw(i)1124 854 y fd(k)1173 794 y ff(\012)e fm(b)p fq(\()p fm(i)1273 801 y fg(1)1290

794 y fm(;)d(:)g(:)g(:)d(;)j(i)1408 801 y fw(n)1430 794 y fq(\))75 949 y(appuy)o(ons)16 b(nous)g(sur)g(un)h(exemple)g(de)f([27) o(],)f(page)h(90)f(p)q(our)i(expliquer)g(de)g(mani)o(\022)-21 b(ere)15 b(plus)i(in)o(tuitiv)o(e)75 1005 y(le)f(fonctionnemen)o(t)g (de)f(la)h(primitiv)o(e.)g(p)o(our)f(les)h(tableaux)8 b(:)146 1168 y fk(a)15 b fq(=)235 1071 y fx(2)235 1144 y(6)235 1171 y(4)284 1112 y fq(1)41 b(2)g(3)284 1168 y(4)g(5)g(6)284 1224 y(7)g(8)g(9)455 1071 y fx(3)455 1144 y(7)455 1171 y(5)498 1168 y fq(,)15 b fk(b)g fq(=)616 1071 y fx(2)616 1144 y(6)616 1171 y(4)664 1112 y ff(\000)p fq(1)42 b ff(\000)p fq(2)f ff(\000)p fq(3)664 1168 y ff(\000)p fq(4)h ff(\000)p fq(5)f ff(\000)p fq(6)664 1224 y ff(\000)p fq(7)h ff(\000)p fq(8)f ff(\000)p fq(9)942 1071 y fx(3)942 1144 y(7)942 1171 y(5)985 1168 y fq(,)15 b fk(x1)g fq(=)1126 1071 y fx(2)1126 1144 y(6)1126 1171 y(4)1174 1112 y fq(1)42 b(1)f(1)1174 1168 y(2)h(1)f(1)1174 1224 y(3)h(2)f(1)1346 1071 y fx(3)1346 1144 y(7)1346 1171 y(5)1389 1168 y fq(et)15 b fk(x2)g fq(=)1556 1071 y fx(2)1556 1144 y(6)1556 1171 y(4)1604 1112 y fq(1)41 b(2)h(3)1604 1168 y(1)f(1)h(2)1604 1224 y(1)f(1)h(1)1776 1071 y fx(3)1776 1144 y(7)1776 1171 y(5)75 1354 y fq(le)16 b(r)o(\023)-21 b(esultat)14 b(de)i fk(r)23 b(=)h(sum)p 521 1354 v 17 w(scatter\(a,b,x1,x2\))13 b fq(est)i fk(r)g fq(=)1140 1257 y fx(2)1140 1331 y(6)1140 1357 y(4)1188 1298 y fq(14)77 b(6)f(0)1211 1354 y(8)41 b ff(\000)p fq(5)h ff(\000)p fq(6)1211 1411 y(0)f ff(\000)p fq(8)h ff(\000)p fq(9)1454 1257 y fx(3)1454 1331 y(7)1454 1357 y(5)75 1468 y fq(l')o(\023)-21 b(el)o(\023)g(emen)o(t)16 b(de)h(tableau)h(r)o(\023)-21 b(esultat)16 b fk(r)p fq(\(1,2\))f(est)i (calcul)o(\023)-21 b(e)18 b(comme)f(suit)8 b(:)16 b(on)h(c)o(herc)o(he) h(tous)f(les)g(couples)75 1525 y fk(x1\()p fm(i;)8 b(j)s fk(\))p fq(,)p fk(x2\()p fm(i;)f(j)r fk(\))13 b fq(tels)k(que)g fk(x1\()p fm(i;)8 b(j)s fk(\))14 b fq(=)i(1)h(et)f fk(x2\()p fm(i;)8 b(j)s fk(\))14 b fq(=)i(2.)g(a)g(c)o(haque)h(fois)g(que)f(la)h (condition)h(est)75 1581 y(v)o(\023)-21 b(eri\014)o(\023)g(ee,)15 b(on)i(somme)f(la)g(v)m(aleur)i fk(a\()p fm(i;)8 b(j)s fk(\))p fq(.)14 b(nous)i(a)o(v)o(ons)f(ici)j(deux)f(couples)h(v)o(\023) -21 b(eri\014an)o(t)16 b(les)h(conditions,)75 1638 y(s)o(\023)-21 b(electionnan)o(t)15 b(deux)g(\023)-21 b(el)o(\023)g(emen)o(ts)14 b(dans)h fk(a)8 b fq(:)p 510 1700 902 2 v 509 1757 2 57 v 535 1740 a fk(x1)p fq(\(1,2\))13 b(=)j(1)49 b(et)h fk(x2)p fq(\(1,2\))13 b(=)j(2)p 1155 1757 v 49 w fk(a)p fq(\(1,2\))e(=)h(2)p 1411 1757 v 509 1813 v 535 1796 a fk(x1)p fq(\(2,3\))e(=)j(1)49 b(et)h fk(x2)p fq(\(2,3\))13 b(=)j(2)p 1155 1813 v 49 w fk(a)p fq(\(2,3\))e(=)h(6)p 1411 1813 v 510 1815 902 2 v 75 1909 a(soit)g fk(r)p fq(\(1,2\))f(=)h(2)g(+)g(6)g(+)h fk(b)p fq(\(1,2\))d(=)j(2)f(+)g(6)g (-2)g(=)h(6.)146 2026 y(des)c(primitiv)o(es)i(de)g(r)o(\023)-21 b(eduction)12 b(moins)i(g)o(\023)-21 b(en)o(\023)g(erales)11 b(implan)o(ten)o(t)j(les)f(op)o(\023)-21 b(erations)12 b(de)h(t)o(yp)q(e)g fj(p)n(ar)n(al)r(lel-)75 2082 y(pr)n(e\014x)k fq(du)h(mo)q(d)o(\022)-21 b(ele)17 b(data-parall)o(\022)-21 b(ele.)17 b(la)h(r)o(\023)-21 b(eduction)17 b(s'e\013ectue)g(selon)h (un)g(des)g(axes)f(canoniques)h(du)75 2139 y(tableau.)d(ces)g(primitiv) o(es)h(son)o(t)e(baptis)o(\023)-21 b(ees)14 b fk(xxx)p 913 2139 15 2 v 17 w(suffix)g fq(et)h fk(xxx)p 1213 2139 v 16 w(prefix)p fq(.)f(l'exemple)i(classique)g(de)75 2195 y(la)21 b(sommation)g(de)g fm(n)g fq(nom)o(bres)g(\(v)o(oir)g (page)f(15\))h(s')o(\023)-21 b(ecrit)20 b(ais)o(\023)-21 b(emen)o(t)20 b(a)o(v)o(ec)g(cette)h(primitiv)o(e.)h(p)o(our)75

2252 y(sommer)15 b(le)i(tableau)g fk(a)f fq(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t)15 b(selon)h(les)h(lignes,)g(nous)f(\023)-21 b(ecriv)o(ons)7 b(:)15 b fk(sum)p 1403 2252 v 17 w(prefix\(a,)23 b(dim)g(=)h(2\))75 2308 y fq(don)o(t)15 b(le)h(r)o(\023)-21 b(esultat)14 b(est)h(le)h(tableau)8 b(:)802 2370 y fx(2)802 2443 y(6)802 2469 y(4)854 2410 y fq(1)72 b(3)h(6)854 2466 y(4)f(9)50 b(15)854 2523 y(7)g(15)f(24)1092 2370 y fx(3)1092 2443 y(7)1092 2469 y(5)p eop %%page: 82 84 82 83 bop 75 269 a fq(82)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fw(5.1.3)55 b(alignemen)n(t)17 b(des)h(donn)o(\023)-27 b(ees)146 485 y fq(hpf)15 b(p)q(ermet)g(au)h (programmeur)e(d'ins)o(\023)-21 b(erer)15 b(dans)h(les)g(programmes,)e (des)i(directiv)o(es)g(\022)-23 b(a)16 b(l'in)o(ten-)75 542 y(tion)11 b(du)g(compilateur,)g(concernan)o(t)g(le)g(placemen)o(t)h (des)f(donn)o(\023)-21 b(ees)10 b(sur)h(les)g(pro)q(cesseurs.)g(le)g (placemen)o(t)75 598 y(in\015ue)16 b(de)e(mani)o(\022)-21 b(ere)14 b(consid)o(\023)-21 b(erable)15 b(sur)f(le)h(v)o(olume)f(de)h (comm)o(unications)f(in)o(ter-pro)q(cesseurs)h(et)f(donc)75 655 y(sur)k(les)h(p)q(erformances)f(d'un)h(programme.)e(le)h (compilateur)i(se)e(d)o(\023)-21 b(ec)o(harge)17 b(de)i(cette)f(t^)-23 b(ac)o(he)18 b(sur)g(le)75 711 y(programmeur)d(car)h(l'optimisation)h (automatique)f(du)h(placemen)o(t)g(des)f(donn)o(\023)-21 b(ees)16 b(est)g(un)h(probl)o(\022)-21 b(eme)75 768 y(di\016cile)18 b([17)o(,)c(43].)146 825 y(le)20 b(placemen)o(t)g(se)g(fait)g(en)g (plusieurs)g(\023)-21 b(etap)q(es)7 b(:)19 b(les)i(v)m(ariables)g (parall)o(\022)-21 b(eles)20 b(son)o(t)f(align)o(\023)-21 b(ees)20 b(en)o(tre)75 882 y(elles,)d(\023)-21 b(ev)o(en)o(tuellemen)o (t)17 b(\022)-23 b(a)16 b(tra)o(v)o(ers)f(des)i(tableaux)g(\014ctifs)g (app)q(el)o(\023)-21 b(es)17 b fj(templates)p fq(,)g(\022)-23 b(a)16 b(l'aide)h(de)g(la)g(direc-)75 938 y(tiv)o(e)d fk(align)p fq(.)f(les)i(v)m(ariables)g(parall)o(\022)-21 b(eles)15 b(son)o(t)e(ensuite)j(distribu)o(\023)-21 b(ees,)14 b(a)o(v)o(ec)g(la)g(directiv)o(e)h fk(distribute)p fq(,)75 995 y(sur)k(une)g(mac)o(hine)h(abstraite)f(constitu)o(\023)-21 b(ee)18 b(d'un)h(tableau)h(de)f(pro)q(cesseurs)g(virtuels)h(\(d)o(\023) -21 b(eclar)o(\023)g(ee)18 b(par)75 1051 y(la)g(directiv)o(e)h fk(processors)p fq(\).)d(en\014n,)i(le)h(tableau)f(de)h(pro)q(cesseurs) f(virtuels)h(est)e(plac)o(\023)-21 b(e)18 b(sur)g(les)h(pro-)75 1108 y(cesseurs)f(ph)o(ysiques)h(\(le)f(nom)o(bre)g(de)g(pro)q (cesseurs)g(ph)o(ysiques)h(est)e(supp)q(os)o(\023)-21 b(e)18 b(inf)o(\023)-21 b(erieur)18 b(ou)f(\023)-21 b(egal)17 b(au)75 1164 y(nom)o(bre)10 b(de)h(pro)q(cesseurs)f(virtuels\).)h(le)g (programmeur)e(ne)i(p)q(eut)g(in\015uencer)1401 1148 y fg(1)1431 1164 y fq(que)g(les)g(deux)g(premiers)75 1220 y(t)o(yp)q(es)i(de)h(placemen)o(ts.)g(notons)f(qu'\022)-23 b(a)12 b(ces)i(directiv)o(es)h(\(statiques\))d(corresp)q(onden)o(t)i (des)g(instructions)75 1277 y(\(dynamiques\))h(\023)-21 b(equiv)m(alen)o(tes)16 b(qui)g(son)o(t)e fk(realign)g fq(et)h fk(redistribute)p fq(.)75 1404 y fo(alignemen)o(t)146 1493 y fq(l'alignemen)o(t,)j(\022)-23 b(a)17 b(tra)o(v)o(ers)g(la)h (directiv)o(e)h fk(align)p fq(,)e(pr)o(\023)-21 b(ecise)18 b(les)g(relations)g(de)h(placemen)o(t)f(des)g(v)m(a-)75 1549 y(riables)e(parall)o(\022)-21 b(eles,)16 b(les)f(unes)h(par)f (rapp)q(ort)f(aux)h(autres,)g(ou)g(par)f(rapp)q(ort)h(\022)-23 b(a)15 b(un)g(template.)g(dans)g(le)75 1606 y(cas)e(o)q(\022)-24 b(u)14 b(une)g(v)m(ariable)h(parall)o(\022)-21 b(ele)14 b(est)f(align)o(\023)-21 b(ee)14 b(sur)f(une)h(autre,)f(le)h (compilateur)h(aligne)f(implicitemen)o(t)75 1662 y(la)e(v)m(ariable)i

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g(et)f(non)g(les)h(o)q(ccurrences)g(initiales)i(des)d(v)m(ariables.)146 510 y(hpf,)c(en)i(rev)m(anc)o(he,)f(n'imp)q(ose)h(pas)f(cette)h(con)o (train)o(te.)e(les)i(indices)h(des)f(v)m(ariables)g(parall)o(\022)-21 b(eles)13 b(ne)75 566 y(r)o(\023)-21 b(ef)o(\023)g(erencen)o(t)14 b(pas)h(les)i(pro)q(cesseurs)f(virtuels)g(mais)g(son)o(t)f(r)o(\023)-21 b(eellemen)o(t)16 b(les)g(indices)i(des)e(tableaux.)f(p)o(ar)75 623 y(cons)o(\023)-21 b(equen)o(t,)16 b(le)i(placemen)o(t)g(ne)g(mo)q (di\014e)g(pas)f(la)g(s)o(\023)-21 b(eman)o(tique)17 b(d'un)h(programme.)d(p)o(our)i(un)g(m)o(^)-21 b(eme)75 679 y(texte)17 b(de)h(programme,)d(mais)j(don)o(t)f(les)h(v)m(ariables) g(ne)g(pas)f(plac)o(\023)-21 b(ees)17 b(de)h(la)g(m)o(^)-21 b(eme)16 b(mani)o(\022)-21 b(ere,)17 b(seul)h(le)75 736 y(temps)13 b(d'ex)o(\023)-21 b(ecution)14 b(di\013)o(\022)-21 b(ere.)12 b(ainsi,)j(la)e(\014gure)h(5.1)e(illustre)j(cette)e(di\013)o (\023)-21 b(erence)14 b(sur)f(une)h(transp)q(osition)75 792 y(de)19 b(matrice,)f(\023)-21 b(ecrite)19 b(en)g(c)545 776 y fx(\003)583 792 y fq(\(\022)-23 b(a)19 b(gauc)o(he\))g(et)g(le)g (programme)e(\023)-21 b(equiv)m(alen)o(t)20 b(en)g(hpf)e(\(\022)-23 b(a)19 b(droite\).)f(en)75 849 y(c)108 832 y fx(\003)127 849 y fq(,)c(les)g(indices)i fk(i)e fq(et)g fk(j)g fq(repr)o(\023)-21 b(esen)o(ten)o(t)13 b(des)h(n)o(um)o(\023)-21 b(eros)13 b(de)i(pro)q(cesseurs)f(virtuels,)h(c'est-\022)-23 b(a-dire)14 b(que)h(les)75 905 y(v)m(aleurs)g(de)g fk(b)g fq(ne)g(son)o(t)e(pas)i (transp)q(os)o(\023)-21 b(ees)6 b(;)14 b(en)h(hpf)g(au)f(con)o(traire,) g(les)h(indices)i(son)o(t)c(les)j(indices)g(des)75 962 y(matrices.)i(le)h(programme)f(hpf)g(sans)g(les)h(directiv)o(es)h(de)f (placemen)o(t)g(\(lignes)g(1)g(\022)-23 b(a)18 b(3\))g(pro)q(duirait)75 1018 y(les)k(m)o(^)-21 b(emes)21 b(r)o(\023)-21 b(esultats)20 b(que)i(le)g(programme)f(list)o(\023)-21 b(e,)21 b(mais)h(a)o(v)o(ec)f (une)h(p)q(erformance)g(moindre)g(si)g(le)75 1074 y(compilateur)16 b(ne)g(place)g(pas)f(automatiquemen)o(t)g(les)h(matrices)f(comme)g (indiqu)o(\023)-21 b(e.)75 1213 y fk(shape)23 b(mat[100][100];)332 b(real)24 b(a\(100,100\),)e(b\(100,100\))75 1269 y(float)h(:)h(mat)f (i,j,a,b;)309 b(!hpf$)24 b(processors)e(m\(100,100\))862 1326 y(!hpf$)i(align)f(a\(i,j\))g(with)g(b\(j,i\))123 1382 y(i=pc_coord\(0\);)403 b(!hpf$)24 b(distribute)e(a\(block\))h (onto)g(m)123 1438 y(j=pc_coord\(1\);)123 1495 y([j][i])g(b)g(=)h ([i][j])f(a;)286 b(forall\(i=1:100,j=1:00\))21 b(b\(j,i\))i(=)h (a\(i,j\))425 1643 y fn(fig.)15 b fq(5.1)f(-)30 b fj(t)m(r)n(ansp)n (osition)14 b(de)j(matric)n(e)f(en)g(c)1253 1626 y fx(\003)1288 1643 y fj(et)g(en)g(hpf)75 1833 y ft(5.3)66 b(sc)n(h)o(\023)-32 b(ema)22 b(d'ensem)n(ble)g(du)h(m)o(\023)-32 b(ecanisme)21 b(de)i(traduction)146 1934 y fq(un)9 b(\023)-21 b(enonc)o(\023)g(e)10 b fn(pei)g fq(quelconque)j(n'est,)c(de)j(mani)o(\022)-21 b(ere)10 b(g)o(\023)-21 b(en)o(\023)g(erale,)9 b(pas)h(directemen)o(t)i (traduisible)g(en)f(un)75 1991 y(co)q(de)16 b(data-parall)o(\022)-21 b(ele.)15 b(les)h(caract)o(\023)-21 b(eristiques)15 b(de)h fn(pei)f fq(emp)o(^)-21 b(ec)o(han)o(t)15 b(cette)g(traduction)g(son)o (t)g(de)h(deux)75 2047 y(sortes)7 b(:)13 b(l'asp)q(ect)h(\023)-21 b(equationnel)14 b(des)g(\023)-21 b(enonc)o(\023)g(es)12 b(et)i(l'absence)h(d'ordonnancemen)o(t)f(op)o(\023)-21 b(erationnel)14 b(p)q(our)74 2104 y(\023)-21 b(ev)m(aluer)12 b(la)h(solution)f(d'une)g(\023)-21 b(equation.)11 b(p)o(ar)g(cons)o (\023)-21 b(equen)o(t,)11 b(certaines)i(transformations)e fj(s)o(\023)-22 b(emantiques)p fq(,)75 2160 y(utilisan)o(t)14 b(les)g(r)o(\022)-21 b(egles)11 b(\023)-21 b(enonc)o(\023)g(ees)12 b(dans)i(les)f(c)o(hapitres)h(3)f(et)g(4,)f(doiv)o(en)o(t)g(^)-21 b(etre)12 b(pr)o(\023)-21 b(ealablemen)o(t)13 b(e\013ectu)o(\023)-21 b(ees)75 2217 y(p)q(our)15 b(obtenir)h(un)f(\023)-21 b(enonc)o(\023)g(e)14 b(ra\016n)o(\023)-21 b(e)14 b(sous)h(une)h(forme)

e fj(tr)n(aduisible)p fq(.)189 2316 y fo(equations)8 b fq(:)18 b(nous)g(ne)g(p)q(ouv)o(ons)g(traduire)h(que)f(des)f(\023)-21 b(equations)18 b(assimilables)i(\022)-23 b(a)18 b(des)g(a\013ec-)189 2373 y(tations,)g(c'est-\022)-23 b(a-dire)19 b(celles)h(don)o(t)f(au)f (moins)h(un)g(des)g(mem)o(bres)g(n'est)f(comp)q(os)o(\023)-21 b(e)18 b(que)h(d'un)189 2429 y(iden)o(ti\014cateur)d(de)g(c)o(hamp.)f (p)o(ar)g(exemple,)h(nous)g(ne)g(sa)o(v)o(ons)e(pas)h(traduire)h(l')o (\023)-21 b(equation)15 b fk(f)7 b(::)h(x)189 2486 y fq(=)15 b fk(g)8 b(::)f(y)p fq(.)15 b(il)h(est)f(donc)h(n)o(\023)-21 b(ecessaire)16 b(de)f(faire)h(appara)-5 b(^)-18 b(\020tre)15 b(uniquemen)o(t)i fk(x)e fq(ou)g fk(y)h fq(dans)f(l'un)h(des)189 2542 y(mem)o(bres,)g(\022)-23 b(a)16 b(l'aide)h(des)g(r)o(\022)-21 b(egles)16 b(exp)q(os)o(\023)-21 b(ees)16 b(en)h(section)g(3.3.)e(en)i (l'o)q(ccurrence,)g(on)f(p)q(eut)h(faire)p eop %%page: 85 87 85 86 bop 75 269 a fl(5.3.)29 b(sch)274 257 y(\023)268 269 y(ema)15 b(d'ensemble)g(du)g(m)839 257 y(\023)833 269 y(ecanisme)g(de)g(traduction)306 b fq(85)189 397 y(appara)-5 b(^)-18 b(\020tre)14 b fk(x)h fq(uniquemen)o(t)h(en)f(mem)o (bre)g(gauc)o(he)g(si)h fk(f)e fq(est)h(in)o(v)o(ersible,)h(et)f(en)g (appliquan)o(t)h fk(f)1809 380 y fq(-1)189 453 y fq(sur)h(les)h(deux)g (mem)o(bres)f(\(r)o(\022)-21 b(egles)16 b(\(i\))h(et)g(\(7b\))g(\022) -23 b(a)16 b(condition)j(que)f(les)f(domaines)h(de)g(r)o(\023)-21 b(ef)o(\023)g(erence)189 510 y(de)14 b fk(x)f fq(et)h(de)g fk(y)f fq(soien)o(t)h(inclus)i(dans)d fm(dom)p fq(\()p fk(h)p fq(\)\),)f(puis)j(en)f(calculan)o(t)h(le)f(nouv)o(eau)g(c)o (hangemen)o(t)f(de)189 566 y(base)j fk(f)315 550 y fq(-1)362 566 y ff(\016)10 b fk(g)15 b fq(appliqu)o(\023)-21 b(e)17 b(\022)-23 b(a)16 b fk(y)p fq(,)f(ce)h(qui)h(donne)f(l')o(\023)-21 b(equation)16 b fk(x)g fq(=)g fk(\(f)1328 550 y fq(-1)1375 566 y ff(\016)10 b fk(g\))d(::)g(y)p fq(.)16 b(dans)f(d'autres)189 623 y(cas,)f(la)i(traduction)g(est)f(imp)q(ossible)10 b(:)15 b(par)g(exemple,)h(aucune)g(r)o(\022)-21 b(egle)15 b(de)h(ra\016nemen)o(t)f(n'existe)189 679 y(actuellemen)o(t)j(p)q(our)g (r)o(\023)-21 b(e)o(\023)g(ecrire)17 b(l')o(\023)-21 b(equation)17 b fk(x)e fa(c)g fk(f)j fq(=)g fk(y)d fa(c)g fk(g)j fq(de)g(mani)o(\022)-21 b(ere)17 b(\022)-23 b(a)17 b(faire)h(appara)-5 b(^)-18 b(\020tre)189 736 y fk(x)21 b fq(ou)g fk(y)g fq(seul)h(sur)f(l'un)h(des)g(mem)o(bres,)f(dans)g(le)h (cas)f(o)q(\022)-24 b(u)21 b fk(f)h fq(et)f fk(g)g fq(son)o(t)f(des)i (fonctions)f(non)189 792 y(in)o(v)o(ersibles.)h(p)o(ar)e(cons)o(\023) -21 b(equen)o(t,)20 b(l')o(\023)-21 b(enonc)o(\023)g(e)20 b(con)o(tenan)o(t)g(une)i(telle)f(\023)-21 b(equation)21 b(ne)g(p)q(ourra)f(^)-21 b(etre)189 849 y(traduit.)189 953 y fo(cadencemen)o(t)16 b(des)e(\023)-25 b(equations)17 b(r)o(\023)-25 b(ecurren)o(tes)6 b fq(:)13 b(nous)i(ne)f(traduisons)g (que)h(les)e(\023)-21 b(equations)189 1009 y(r)o(\023)g(ecurren)o(tes) 20 b(p)q(our)i(lesquelles)i(nous)e(p)q(ouv)o(ons)g(d)o(\023)-21 b(eterminer)22 b(un)g(cadencemen)o(t)g(des)g(calculs)189 1066 y(dans)17 b(le)g(temps.)f(ce)h(probl)o(\022)-21 b(eme)17 b(est)f(largemen)o(t)h(ab)q(ord)o(\023)-21 b(e)16 b(dans)h(la)g(litt)o(\023)-21 b(erature)16 b(concernan)o(t)g(la)189 1122 y(parall)o(\023)-21 b(elisation)22 b(automatique,)e(et)h(nous)h (sa)o(v)o(ons)e(qu'il)i(consiste)g(\022)-23 b(a)21 b(trouv)o(er)f(des)h (fonctions)189 1179 y(de)f(cadencemen)o(t)g(et)g(de)g(placemen)o(t.)g (en)g fn(pei)p fq(,)f(cette)h(transformation)e(se)i(traduit)g(par)f(un) 189 1235 y(c)o(hangemen)o(t)d(de)h(base)g(appliqu)o(\023)-21 b(e)17 b(\022)-23 b(a)17 b(l')o(\023)-21 b(enonc)o(\023)g(e.)15 b(un)i(exemple)h(simple)g(est)f(celui)h(du)f(pro)q(cessus)189 1292 y(de)e(di\013usion)i(discr)o(\022)-21 b(ete)15 b(de)g(la)h

(\014gure)f(5.2.)p 179 1407 1564 2 v 179 1744 2 338 v 193 1449 a fk(do)g fm(i)d fq(=)h(1)p fm(;)8 b(n)311 1506 y fk(do)15 b fm(j)g fq(=)e(1)p fm(;)8 b(n)429 1562 y(y)r fq(\()p fm(i;)g(j)s fq(\))i(=)j fm(f)5 b fq(\()p fm(y)r fq(\()p fm(i)10 b ff(\000)g fq(1)p fm(;)e(j)s fq(\))p fm(;)g(y)r fq(\()p fm(i)o(;)g(j)h ff(\000)h fq(1\)\))311 1619 y fk(enddo)193 1675 y(enddo)p 1741 1744 v 179 1746 1564 2 v 547 1823 a fn(fig.)15 b fq(5.2)f(-)30 b fj(pr)n(o)n(c)n(essus) 15 b(de)h(di\013usion)g(discr)o(\022)-22 b(ete)189 1984 y fq(qui)16 b(s')o(\023)-21 b(ecrit)14 b(en)h fn(pei)p fq(,)g(si)h fk(x)f fq(est)g(le)h(c)o(hamp)f(con)o(tenan)o(t)g(les)g(v)m (aleurs)i(initiales)9 b(:)510 2090 y fk(y)15 b fq(=)g fk(f)g fa(b)h fk(\(x)23 b(/&/)8 b(\(y)24 b fa(c)g fk(f1\))f(/&/)8 b(\(y)23 b fa(c)i fk(f2)o(\)\))95 b fq(\(1\))510 2147 y fk(f1)15 b fq(=)g fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)7 b(&)h(1)p ff(\024)p fk(j)p ff(\024)p fk(n\).\(i-1,j\))510 2203 y(f2)15 b fq(=)g fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)7 b(&)h(1)p ff(\024)p fk(j)p ff(\024)p fk(n\).\(i,j-1\))189 2309 y fq(nous)22 b(sa)o(v)o(ons)g(qu'il)i(existe) f(dans)g(ce)g(cas,)g(une)g(transformation)f(a\016ne)g(temps)h ff(\002)g fq(espace)189 2366 y(p)q(ermettan)o(t)14 b(de)i(parall)o (\023)-21 b(eliser)16 b(la)g(r)o(\023)-21 b(ecurrence.)14 b(en)i(l'o)q(ccurrence,)g(cette)f(transformation)f(p)q(eut)188 2422 y(^)-21 b(etre)6 b(:)757 2436 y fx( )811 2480 y fm(t)811 2536 y(p)854 2436 y fx(!)900 2508 y fq(=)948 2436 y fx( )1001 2480 y fq(1)42 b(1)1001 2536 y(1)g(0)1109 2436 y fx(!)8 b( )1203 2480 y fm(i)1203 2536 y(j)1245 2436 y fx(!)p eop %%page: 86 88 86 87 bop 75 269 a fq(86)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)189 397 y fq(o)q(\022)-24 b(u)11 b(l'indice)j fm(t)e fq(repr)o(\023)-21 b(esen)o(te)11 b(le)i(temps)e(et)h fm(p)f fq(les)h(pro)q(cesseurs.)g(en)g fn(pei)p fq(,)f(cette)h(transformation)e(est)189 453 y(r)o(\023)-21 b(ealis)o(\023)g(ee)12 b(par)g(l'application)i(de)f(c)o (hangemen)o(t)g(de)g(base)g fk(h)f fq(=)h fm(\025)p fk (\(t,p\).\(t+p,t\))d fq(sur)j(l')o(\023)-21 b(equation)189 510 y(\(1\).)13 b(apr)o(\022)-21 b(es)15 b(application)i(des)e(r)o (\022)-21 b(egles,)15 b(nous)g(obtenons)7 b(:)462 598 y fk(y')15 b fq(=)h fk(f)f fa(b)g fk(\(x')23 b(/&/)8 b(\(y')23 b fa(c)i fk(g1)o(\))f(/&/)8 b(\(y')23 b fa(c)h fk(g2\)\))95 b fq(\(2\))462 654 y fk(g1)15 b fq(=)h fm(\025)p fk(\(t,p\))p ff(j)p fk(\(1)p ff(\024)p fk(p)p ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(t-p)p ff(\024)p fk(n\).\(t-1,p-1\))462 710 y(g2)15 b fq(=)h fm(\025)p fk(\(t,p\))p ff(j)p fk(\(1)p ff(\024)p fk(p)p ff(\024)p fk(n)6 b(&)i(1)p ff(\024)p fk(t-p)p ff(\024)p fk(n\).\(t-1,p\))189 801 y fq(o)q(\022)-24 b(u)18 b fk(g1)g fq(=)h fk(h)12 b ff(\016)g fk(f1)g ff(\016)g fk(h)565 784 y fq(-1)621 801 y fq(et)18 b fk(g2)g fq(=)h fk(h)12 b ff(\016)g fk(f2)g ff(\016)g fk(h)987 784 y fq(-1)1025 801 y fq(.)18 b(une)h(dimension)h(d)o(\023)-21 b(e\014nissan)o(t)18 b(les)h(fonctions)189 857 y(g)o(\023)-21 b(eom)o(\023)g(etriques)9 b fk(g1)i fq(et)g fk(g2)g fq(comme)f(des)i (translations,)f(nous)g(v)o(errons)f(que)i(cette)f(transformation)189 914 y(nous)h(p)q(ermet)h(de)g(traduire)g(la)g(r)o(\023)-21 b(ecurrence)13 b(par)f(une)h(b)q(oucle)i(s)o(\023)-21 b(equen)o(tielle)14 b(sur)e fm(t)h fq(et)g(une)g(b)q(oucle)189 970 y(parall)o(\022)-21 b(ele)16 b(sur)f fm(p)p fq(.)189

1043 y(notons)e(que)i(dans)g(cet)f(exemple,)i(une)f(seule)g(direction)h (repr)o(\023)-21 b(esen)o(te)14 b(le)h(temps,)f(sous)g(la)h(forme)189 1100 y(d'une)f(translation)h(pr)o(\023)-21 b(esen)o(te)13 b(dans)i(toutes)e(fonctions)i(g)o(\023)-21 b(eom)o(\023)g(etriques.)12 b(notre)i(traducteur)g(ac-)189 1156 y(tuel)h(ne)g(reconna)-5 b(^)-18 b(\020t)15 b(que)g(cette)g(forme)f(de)h(cadencemen)o(t)g(lin)o (\023)-21 b(eaire,)16 b(mais)e(d'autres)g(tec)o(hniques)189 1213 y(de)i(cadencemen)o(t)h(p)q(ourraien)o(t)e(^)-21 b(etre)15 b(incluses,)j(par)e(exemple)h(la)g(tec)o(hnique)g(de)g (cadencemen)o(t)189 1269 y(m)o(ulti-dimensionnel)23 b(exp)q(os)o(\023) -21 b(ee)19 b(dans)h([16)o(].)e(de)i(nouv)o(elles)h(formes)e(d')o(\023) -21 b(equations)18 b(p)q(ourraien)o(t)189 1326 y(alors)j(^)-21 b(etre)21 b(traduites)7 b(:)22 b(en)o(tre)g(autres,)g(l')o(\023)-21 b(equation)22 b(\(1\))f(p)q(ourrait)g(^)-21 b(etre)22 b(traduite)g(par)g(deux)189 1382 y(b)q(oucles)16 b(s)o(\023)-21 b(equen)o(tielles,)16 b(donnan)o(t)g(le)f(programme)g(f)l(ortran)f(de)h (la)h(\014gure)f(5.2.)75 1473 y(apr)o(\022)-21 b(es)14 b(a)o(v)o(oir)g(transform)o(\023)-21 b(e)13 b(l')o(\023)-21 b(enonc)o(\023)g(e)14 b(de)i(mani)o(\022)-21 b(ere)14 b(\022)-23 b(a)15 b(resp)q(ecter)h(ces)f(princip)q(es,)i(nous)e(p)q (ouv)o(ons)g(en)o(ta-)75 1530 y(mer)e(un)g(pro)q(cessus)h(de)f (transformation)f(de)i(nature)e fj(syntaxique)p fq(,)h(que)g(nous)h(d)o (\023)-21 b(ecriv)o(ons)12 b(main)o(tenan)o(t.)75 1671 y ft(5.4)66 b(le)23 b(traducteur)h fc(pei2hpf)146 1772 y fq(la)18 b(traduction)g(n)o(\023)-21 b(ecessite)18 b(que)g(les)g(\023)-21 b(enonc)o(\023)g(es)16 b fn(pei)i fq(accept)o(\023)-21 b(es)17 b(en)h(en)o(tr)o(\023)-21 b(ee)17 b(aien)o(t)h(une)h(forme)e(bien)75 1829 y(pr)o(\023)-21 b(ecise,)13 b(que)g(nous)g(app)q(elons)h fj(forme)g(tr)n(aduisible)p fq(.)e(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t,)11 b(les)h(\023)-21 b(enonc)o(\023)g(es)12 b(transform)o(\023)-21 b(es)10 b(par)j(le)75 1885 y(calcul)i(de)e(ra\016nemen)o(t)g fn(pei)g fq(p)q(our)h(obtenir)f(des)g(\023)-21 b(equations)12 b(assimilables)k(\022)-23 b(a)13 b(des)g(a\013ectations,)f(et)h(des)74 1942 y(\023)-21 b(equations)14 b(r)o(\023)-21 b(ecurren)o(tes)14 b(ordonnanc)o(\023)-21 b(ees,)15 b(ne)g(son)o(t)f(pas)h(sous)g(cette)g (forme)g(traduisible.)h(cep)q(endan)o(t,)75 1998 y(si)d(elle)h(existe,) f(la)g(mise)g(sous)g(forme)f(traduisible)j(d'un)e(tel)f(\023)-21 b(enonc)o(\023)g(e)11 b(est)h(un)i(pro)q(cessus)f(automatisable)75 2055 y(de)19 b(transformation)e(syn)o(taxique)8 b(:)18 b(il)i(est)f(r)o(\023)-21 b(ealis)o(\023)g(e)17 b(dans)i(notre)f (protot)o(yp)q(e)g(par)h(un)g(pr)o(\023)-21 b(e-pro)q(cesseur)75 2111 y(baptis)o(\023)g(e)17 b fk(normhpf)p fq(.)f(dans)h(la)h(suite,)f (nous)h(donnons)f(d'ab)q(ord)h(la)f(forme)g(traduisible)i(c)o(hoisie,)f (ainsi)75 2167 y(que)i(les)g(raisons)f(qui)h(nous)g(on)o(t)e(amen)o (\023)-21 b(e)19 b(\022)-23 b(a)19 b(ce)h(c)o(hoix,)f(puis)h(nous)g (mon)o(trons)e(les)i(transformations)75 2224 y(syn)o(taxiques)15 b(op)o(\023)-21 b(er)o(\023)g(ees)14 b(par)h fk(normhpf)p fq(.)75 2344 y fw(5.4.1)55 b(f)-5 b(orme)18 b(traduisible)146 2429 y fq(le)c(traducteur)f(accepte)h(en)h(en)o(tr)o(\023)-21 b(ee)12 b(un)i(sous-ensem)o(bles)h(des)f(programmes)f fn(pei)p fq(,)g(constitu)o(\023)-21 b(e)13 b(des)74 2486 y(\023)-21 b(equations)15 b(a)o(y)o(an)o(t)e(la)j(forme)e(suiv)m(an)o (te)9 b(:)p eop %%page: 87 89 87 88 bop 75 269 a fl(5.4.)29 b(le)16 b(traducteur)f fk(pei2hpf)1034 b fq(87)p 75 353 1822 2 v 74 409 2 57 v 100 392 a fj(f)m(orme)16 b(tr)n(aduisible)p 1896 409 v 75 411 1822 2 v 74 1021 2 610 v 155 492 a fq(1.)23 b(op)o(\023)-21 b(eration)14 b(g)o(\023)-21 b(eom)o(\023)g(etrique)6

b(:)15 b fk(y)g fq(=)g fk(x)g fa(c)h fk(g)155 586 y fq(2.)23 b(op)o(\023)-21 b(eration)14 b(fonctionnelle)c(:)15 b fk(y)g fq(=)g fk(f)g fa(b)h fk(\(x)942 593 y fg(1)984 586 y fk(/&/)8 b fm(:)g(:)g(:)22 b fk(/&/)7 b(x)1244 593 y fw(n)1267 586 y fk(\))155 680 y fq(3.)23 b(op)o(\023)-21 b(eration)14 b(de)i(c)o(hangemen)o(t)f(de)h(base)7 b(:)15 b fk(y)g fq(=)h fk(h)7 b(::)g(x)155 773 y fq(4.)23 b(propri)o(\023)-21 b(et)o(\023)g(e)13 b(de)j(c)o(hamp)f(de)g(donn)o(\023)-21 b(ees)8 b(:)15 b fk(y)g fq(=)g fk(h)7 b(::)h(y)155 867 y fq(5.)23 b(r)o(\023)-21 b(eduction)15 b(g)o(\023)-21 b(eom)o(\023)g(etrique)6 b(:)15 b fk(y)g fq(=)h fk(fr)e fa(b)i fk(\(gr)23 b fq(;)8 b fa(b)24 b fk(x\))155 961 y fq(6.)f(d)o(\023)-21 b(e\014nition)15 b(r)o(\023)-21 b(ecursiv)o(e)7 b(:)15 b fk(y)g fq(=)h fk(fc)f fa(b)g fk(\(x)24 b(/&/)7 b(\(y)24 b fa(c)g fk(g)1144 968 y fg(1)1162 961 y fk(\))g(/&/)8 b fm(:)g(:)g(:)22 b fk(/&/)7 b(\(y)24 b fa(c)g fk(g)1601 968 y fw(n)1623 961 y fk(\)\))p 1896 1021 v 75 1023 1822 2 v 75 1033 v 74 1089 2 57 v 100 1072 a fj(contr)n(aintes)p 1896 1089 v 75 1091 1822 2 v 74 1494 2 404 v 168 1172 a fq({)f(dans)k(5.)h(la)g(d)o(\023)-21 b(e\014nition)29 b(de)g fk(fr)f fq(doit)f(^)-21 b(etre)27 b(de)i(la)f(forme)g(r)o(\023)-21 b(ecursiv)o(e)28 b(pr)o(\023)-21 b(e-d)o(\023)g(e\014nie)28 b fk(fr)g fq(=)214 1229 y fk(id)9 b fq(#)h fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a)23 b ff(\012)i fk(fr\(b\)\))o fq(,)16 b(o)q(\022)-24 b(u)17 b ff(\012)h fq(est)e(un)i(op)o(\023)-21 b(erateur)16 b(asso)q(ciatif,)h(comm)o(utatif)f(a)o(v)o(ec)f(\023)-21 b(el)o(\023)g(e-)214 1285 y(men)o(t)14 b(neutre.)i(dans)e(le)i(protot)o (yp)q(e)f(actuel,)g(seuls)h fk(+)f fq(et)g fk(*)g fq(son)o(t)g(accept)o (\023)-21 b(es.)168 1379 y({)23 b(dans)16 b(6.)g fk(fc)h fq(doit)f(^)-21 b(etre)15 b(de)j(la)f(forme)f fk(fc)h fq(=)g fk(id)34 b fq(#)10 b fm(\025)p fk(\(a)p fq(;)n fk(b)p fq(;)p fm(:)e(:)g(:)22 b fq(;)o fk(z\).\(a)i ff(\014)1517 1386 y fg(1)1560 1379 y fk(b)f ff(\014)1642 1386 y fg(2)1669 1379 y fm(:)8 b(:)g(:)d ff(\014)1764 1386 y fw(n)1811 1379 y fk(z\))p fq(,)214 1435 y(o)q(\022)-24 b(u)15 b(les)h ff(\014)378 1442 y fw(i)407 1435 y fq(son)o(t)e(des)i(op)o(\023)-21 b(erateurs)14 b(quelconques)i(sur)g(des)f(scalaires.)p 1896 1494 v 75 1496 1822 2 v 146 1556 a(nous)j(dirons)i(qu'un)e(\023) -21 b(enonc)o(\023)g(e)17 b fn(pei)i fq(est)f fj(tr)n(aduisible)h fq(si)g(toutes)f(ses)g(\023)-21 b(equations)18 b(son)o(t)g(sous)h (l'une)75 1613 y(des)e(formes)f(d)o(\023)-21 b(ecrites)17 b(dans)g(le)g(tableau)h fj(f)m(orme)f(tr)n(aduisible)f fq(et)h(si)g(elles)i(resp)q(ecten)o(t)e(les)g(conditions)74 1669 y(\023)-21 b(enonc)o(\023)g(ees)14 b(dans)h(le)h(tableau)g fj(contr)n(aintes)p fq(.)146 1782 y(plusieurs)23 b(raisons)f(nous)g(on) o(t)g(amen)o(\023)-21 b(e)21 b(\022)-23 b(a)22 b(ne)g(consid)o(\023)-21 b(erer)23 b(que)f(cet)g(ensem)o(ble)h(restrein)o(t)f(d'ex-)75 1838 y(pressions.)f(elles)i(nous)e(p)q(ermetten)o(t)g(d'asso)q(cier)h (\022)-23 b(a)20 b(c)o(hacune)i(d'en)o(tre)f(elles,)i(un)e(ensem)o(ble) i(d'ins-)75 1895 y(tructions)16 b(hpf,)g(sans)g(a)o(v)o(oir)g(b)q (esoin)h(d'informations)f(suppl)o(\023)-21 b(emen)o(taires)17 b(con)o(textuelles)h(\(tir)o(\023)-21 b(ees)15 b(du)75 1951 y(programme\).)j(p)o(ar)h(ailleurs,)i(la)f(traduction)g (d'expressions)g(de)h(c)o(hamps)e(de)h(donn)o(\023)-21 b(ees)20 b(plus)h(com-)75 2008 y(pliqu)o(\023)-21 b(ees)17 b(n'est)e(pas)g(toujours)g(p)q(ossible.)i(examinons)e(les)h(probl)o (\022)-21 b(emes)16 b(li)o(\023)-21 b(es)16 b(\022)-23 b(a)15 b(certaines)h(expressions)75 2064 y fn(pei)7 b fq(:)189 2147 y fo(op)o(\023)-25 b(eration)20 b(g)o(\023)-25

b(eom)o(\023)g(etrique)8 b fq(:)16 b(on)i(se)f(limite)i(\022)-23 b(a)17 b(la)g(traduction)g(d')o(\023)-21 b(equations)16 b(de)i(la)f(forme)g fk(y)189 2204 y fq(=)g fk(x)e fa(c)g fk(g)p fq(.)h(consid)o(\023)-21 b(erons)16 b(en)i(e\013et,)d(un)j(c)o (hamp)e(de)h(donn)o(\023)-21 b(ees)17 b(d)o(\023)-21 b(e\014ni)17 b(par)f fk(y)h fq(=)g fk(\(x)24 b fa(c)g fk(g\))15 b fa(c)g fk(f)p fq(.)189 2260 y(la)g(traduction)h(de)h(cette) e(expression)i(supp)q(oserait)f(de)g(la)h(ra\016ner)e(en)h(une)g (expression)h(de)f(la)189 2316 y(forme)f fk(y)g fq(=)h fk(x)f fa(c)g fk(\(g)10 b ff(\016)g fk(f\))p fq(,)15 b(en)h(v)o(\023)-21 b(eri\014an)o(t)15 b(les)h(conditions)h(de)f(la)g (r)o(\022)-21 b(egle)15 b(\(5b\))g(dans)g(l'en)o(vironne-)189 2373 y(men)o(t)i fn(pei)p fq(,)f(et)i(de)f(calculer)i(la)f(comp)q (osition)g(de)g(fonctions)f fk(g)10 b ff(\016)g fk(f)17 b fq(car)g(hpf)g(ne)h(p)q(ermet)g(pas)189 2429 y(d'exprimer)g(la)f (comp)q(osition)i(de)f(fonctions)f(agissan)o(t)g(sur)g(des)h(indices)i (de)d(tableau.)h(on)g(est)189 2486 y(donc)d(conduit)h(\022)-23 b(a)15 b(exprimer)h(ces)f(deux)h(op)o(\023)-21 b(erations)14 b(g)o(\023)-21 b(eom)o(\023)g(etriques)13 b(successiv)o(es)j(en)g(d)o (\023)-21 b(ecomp)q(o-)189 2542 y(san)o(t)16 b(l'expression)i(initiale) h(en)e(deux)f(\023)-21 b(equations)17 b fk(y)f fq(=)h fk(x)1182 2526 y fx(0)1209 2542 y fa(c)f fk(f)g fq(et)h fk(x)1379 2526 y fx(0)1407 2542 y fq(=)g fk(x)e fa(c)h fk(g)p fq(.)g(c'est)g(ce)h(que)p eop %%page: 88 90 88 89 bop 75 269 a fq(88)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)189 397 y fq(fera)f(la)i(phase)f(de)h (normalisation)g(pr)o(\023)-21 b(esen)o(t)o(\023)g(ee)13 b(plus)k(loin.)189 550 y fo(op)o(\023)-25 b(eration)17 b(fonctionnelle)10 b fq(:)k(nous)g(imp)q(osons)h(que)g(tous)e(les)i(c)o (hamps)g(corresp)q(ondan)o(t)f(aux)189 606 y(argumen)o(ts)i(de)i(la)g (fonction)g(utilis)o(\023)-21 b(ee)19 b(par)e(l'op)o(\023)-21 b(eration)17 b(fonctionnelle)i(soien)o(t)f(pr)o(\023)-21 b(esen)o(ts)16 b(dans)189 663 y(l')o(\023)-21 b(equation.)11 b(consid)o(\023)-21 b(erons)12 b(par)g(exemple)h(la)g(fonction)f fk(f)g fq(=)h fm(\025)p fk(\(a)p fq(;)n fk(b\).\(a-\(b*a\)\))e fq(et)h(l')o(\023)-21 b(equation)189 719 y fk(y)12 b fq(=)g fk(f)j fa(b)h fk(\(x1/&/)7 b(x2\))p fq(.)k(sous)i(cette)f (forme,)f(il)j(est)e(tr)o(\022)-21 b(es)11 b(simple)j(de)e(substituer)h (aux)f(argumen)o(ts)189 776 y fk(a)k fq(et)g fk(b)h fq(les)g(c)o(hamps) f fk(x1)h fq(et)f fk(x2)g fq(resp)q(ectiv)o(emen)o(t.)h(dans)f(le)i (cas)e(con)o(traire,)g(par)g(exemple)i fk(y)e fq(=)189 832 y fk(f)f fa(b)g fk(x)p fq(,)g(il)h(est)f(n)o(\023)-21 b(ecessaire)15 b(de)h(consulter)g(dans)f(le)h(reste)f(du)h(programme)e (la)h(d)o(\023)-21 b(e\014nition)16 b(de)g fk(x)p fq(.)189 985 y fo(op)o(\023)-25 b(eration)22 b(in)o(v)o(erse)d(g)o(\023)-25 b(eom)o(\023)g(etrique)7 b fq(:)18 b(l'op)o(\023)-21 b(eration)17 b(de)h(r)o(\023)-21 b(eduction)18 b(g)o(\023)-21 b(eom)o(\023)g(etrique)16 b(de)j fn(pei)189 1041 y fq(p)q(ose)c(le)i (probl)o(\022)-21 b(eme)15 b(du)h(non-d)o(\023)-21 b(eterminisme)17 b(des)e(solutions)i(obten)o(ues.)e(un)g(\023)-21 b(enonc)o(\023)g(e)14 b(constitu)o(\023)-21 b(e)189 1098 y(de)16 b(la)g(seule)g(\023)-21 b(equation)15 b fk(y)h fq(=)g fk(f)f fq(;)8 b fa(b)15 b fk(x)h fq(n'est)f(pas)h(un)g(programme)f fn(pei)p fq(,)g(si)i(la)f (fonction)g fk(f)g fq(n'est)189 1154 y(pas)f(injectiv)o(e)j(car)d(le)i (c)o(hamp)f(solution)h fk(y)f fq(n'est)f(pas)h(unique)i(\(les)e(s)o (\023)-21 b(equences)16 b(de)g(v)m(aleurs)h(on)o(t)189 1211 y(un)g(ordre)g(quelconque\).)h(or,)f(le)h(langage)f(hpf)f(ne)i(p)q (ermet)f(ce)g(t)o(yp)q(e)g(d'op)o(\023)-21 b(eration)16

b(que)i(si)f(la)189 1267 y(comm)o(unication)d(est)g(accompagn)o(\023) -21 b(ee)13 b(d'un)i(op)o(\023)-21 b(erateur)12 b(de)j(r)o(\023)-21 b(eduction,)13 b(\022)-23 b(a)14 b(l'aide)h(des)f fj(intrinsics)189 1324 y fq(de)j(la)g(famille)h fk(xxx)p 527 1324 15 2 v 17 w(scatter)p fq(.)d(c'est)h(p)q(ourquoi)h(nous)g(a)o(v)o(ons)f(c)o (hoisi)i(de)f(ne)g(traduire)g(que)g(les)189 1380 y(r)o(\023)-21 b(eductions)15 b fn(pei)g fq(de)g(la)h(forme)e(5.)75 1562 y fw(5.4.2)55 b(normalisation)17 b(a)n(v)m(an)n(t)i(traduction)146 1649 y fq(comme)f(nous)i(l'a)o(v)o(ons)f(dit)h(en)g(in)o(tro)q (duction,)g(certains)e(\023)-21 b(enonc)o(\023)g(es)19 b(p)q(euv)o(en)o(t)g(^)-21 b(etre)18 b(transform)o(\023)-21 b(es)75 1706 y(syn)o(taxiquemen)o(t)17 b(en)h(des)e(\023)-21 b(enonc)o(\023)g(es)16 b(traduisibles)j(\022)-23 b(a)17 b(l'aide)i(d'un)e(pr)o(\023)-21 b(e-pro)q(cesseur)17 b(agissan)o(t)g(comme)75 1762 y(un)22 b(\014ltre)246 1746 y fg(2)285 1762 y fq(a)o(v)m(an)o(t)f(la)g(traduction)g(propremen) o(t)g(dite.)h(ce)f(pr)o(\023)-21 b(e-pro)q(cesseur)21 b(nomm)o(\023)-21 b(e)20 b fk(normhpf)g fq(est)75 1819 y(en)o(tre)e(autres)f(c)o(harg)o(\023)-21 b(e)17 b(d'in)o(tro)q(duire)h (les)h(d)o(\023)-21 b(e\014nitions)19 b(de)f(c)o(hamps)g(de)g(donn)o (\023)-21 b(ees)18 b(in)o(term)o(\023)-21 b(ediaires)18 b(n)o(\023)-21 b(e-)75 1875 y(cessaires,)19 b(et)f(plus)i(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t)17 b(de)i(transformer)e(l')o(\023)-21 b(enonc)o(\023)g(e)17 b(initial)k(en)d(\023)-21 b(enonc)o(\023)g(e)17 b(traduisible)k(par)75 1931 y(de)15 b(simples)h(a)s(justemen)o(ts)d (syn)o(taxiques.)i(p)o(ar)f(exemple,)h(une)g(\023)-21 b(equation)14 b(non-r)o(\023)-21 b(ecursiv)o(e)14 b(de)h(la)g(forme)75 1988 y fk(y)g fq(=)h fm(e)186 1995 y fg(1)205 1988 y fq(\()p fm(e)244 1995 y fg(2)262 1988 y fq(\()p fm(:)8 b(:)g(:)e(e)362 1995 y fw(n)385 1988 y fq(\()p fk(x)p fq(\))i fm(:)g(:)g(:)l fq(\)\),)15 b(o)q(\022)-24 b(u)15 b fm(e)652 1995 y fw(i)682 1988 y fq(est)g(une)h(simple)h(op)o(\023)-21 b(eration)15 b(g)o(\023)-21 b(eom)o(\023)g(etrique,)13 b(fonctionnelle,)18 b(ou)d(un)75 2044 y(c)o(hangemen)o(t)g(de)h(base,)e (est)h(remplac)o(\023)-21 b(ee)16 b(par)f(l'ensem)o(ble)h(d')o(\023)-21 b(equations)7 b(:)75 2058 y fx(8)75 2096 y(>)75 2108 y(>)75 2120 y(>)75 2133 y(<)75 2208 y(>)75 2220 y(>)75 2233 y(>)75 2245 y(:)133 2097 y fk(y)12 b fq(=)h fm(e)238 2104 y fg(1)257 2097 y fq(\()p fk(y)299 2104 y fr(1)316 2097 y fq(\))133 2153 y fk(y)157 2160 y fr(1)187 2153 y fq(=)g fm(e)256 2160 y fg(2)274 2153 y fq(\()p fk(y)316 2160 y fr(2)333 2153 y fq(\))133 2210 y fm(:)8 b(:)g(:)133 2266 y fk(y)157 2273 y fr(n)187 2266 y fq(=)13 b fm(e)256 2273 y fw(n)278 2266 y fq(\()p fk(x)p fq(\))75 2344 y(l')o(\023)-21 b(enonc)o(\023)g(e)18 b fn(pei)g fq(obten)o(u)h(apr)o(\022)-21 b(es)18 b(ces)i(transformations)d(est)h(\023)-21 b(equiv)m(alen)o(t)19 b(\022)-23 b(a)19 b(l')o(\023)-21 b(enonc)o(\023)g(e)18 b(initial,)j(comme)75 2401 y(nous)15 b(l'a)o(v)o(ons)f(vu)g(en)i (section)f(3.2.)e(la)i(\014gure)g(5.3)e(illustre)k(l'ensem)o(ble)f(des) f(transformations)e(que)i(le)75 2457 y(pr)o(\023)-21 b(e-pro)q(cesseur)15 b(p)q(eut)h(e\013ectuer)f(p)q(our)h(rendre)f (traduisible)i(un)e(\023)-21 b(enonc)o(\023)g(e)6 b(:)p 75 2500 709 2 v 124 2542 a fd(2)p fc(:)j fd(au)14 b(sens)f fi(unix)g fd(du)h(terme.)p eop %%page: 89 91 89 90 bop 75 269 a fl(5.4.)29 b(le)16 b(traducteur)f fk(pei2hpf)1034 b fq(89)p 75 353 1758 2 v 75 790 2 438 v 104 397 a fk(sample)23 b(:)h(\(a,b,c\))14 b ff(7!)i fk(z)104 412 y fx(8)104 449 y(>)104 462 y(>)104 474 y(>)104 487 y(<)104 562 y(>)104 574 y(>)104 586 y(>)104 599 y(:)166

451 y fk(w)f fq(=)h fk(x)f(/&/)7 b(y)166 507 y(z)15 b fq(=)h fk(fc1)e fa(b)i fk(w)166 564 y(x)f fq(=)h fk(fb1)7 b(::)g(\(\(a)24 b fa(c)g fk(fg1)o(\))g fa(c)g fk(fg2\))166 620 y(y)15 b fq(=)h fk(fc3)e fa(b)i fk(\(fc2)23 b fa(b)h fk(\(b)g(/&/)8 b(c\))23 b(/&/)8 b(y)23 b fa(c)i fk(fg3)o(\))1095 397 y(sample)e(:)g(\(a,b,c\))15 b ff(7!)g fk(z)1095 419 y fx(8)1095 456 y(>)1095 469 y(>)1095 481 y(>)1095 493 y(>)1095 506 y(>)1095 518 y(>)1095 531 y(>)1095 543 y(<)1095 618 y(>)1095 630 y(>)1095 643 y(>)1095 655 y(>)1095 668 y(>)1095 680 y(>)1095 693 y(>)1095 705 y(:)1156 451 y fk(z)g fq(=)h fk(fc1)f fa(b)g fk(\(x)24 b(/&/)7 b(y\))1156 507 y(x)15 b fq(=)h fk(fb1)7 b(::)g(x')1156 564 y(x')15 b fq(=)h fk(x'')f fa(c)g fk(fg2)1156 620 y(x'')g fq(=)h fk(a)f fa(c)g fk(fg1)1156 677 y(y1)g fq(=)h fk(fc2)f fa(b)g fk(\(b)24 b(/&/)7 b(c\))1156 733 y(y)15 b fq(=)h fk(fc3)f fa(b)g fk(\(y1)24 b(/&/)7 b(y)24 b fa(c)g fk(fg3\))p 1831 790 v 75 792 1758 2 v 344 869 a fn(fig.)15 b fq(5.3)f(-)31 b fj(t)m(r)n(ansformations)15 b(syntaxiques)g(r)o(\023)-22 b(e)n(alis)o(\023)g(e)n(es)15 b(p)n(ar)h fk(normhpf)75 1012 y fw(5.4.3)55 b(ordonnancemen)n(t)18 b(a)n(v)m(an)n(t)h (traduction)146 1101 y fq(p)o(our)d(pro)q(duire)j(un)f(co)q(de)g (data-parall)o(\022)-21 b(ele,)18 b(nous)f(dev)o(ons)g(\023)-21 b(egalemen)o(t)17 b(ordonner)g(les)g(\023)-21 b(equations)75 1157 y(du)22 b(programme)f fn(pei)g fq(sous)g(forme)g(traduisible.)i (nous)f(utilisons)h(p)q(our)f(cela)g(l'ordre)g(induit)h(par)75 1214 y(les)e(d)o(\023)-21 b(e\014nitions)20 b(de)g(c)o(hamps)g(de)g (donn)o(\023)-21 b(ees)8 b(:)19 b(nous)h(disons)g(qu'un)g(c)o(hamp)g fk(x)g fq(d)o(\023)-21 b(ep)q(end)20 b(causalemen)o(t)75 1270 y(d'un)c(c)o(hamp)g fk(y)g fq(si)g fk(x)g fq(la)g(d)o(\023)-21 b(e\014nition)17 b(de)f fk(x)g fq(con)o(tien)o(t)g(au)f(moins)i(une)f (o)q(ccurrence)h(de)f fk(y)p fq(.)g(a)g(partir)f(des)75 1327 y(c)o(hamps)k(en)h(en)o(tr)o(\023)-21 b(ee,)18 b(nous)h(p)q(ouv)o (ons)h(construire)f(un)h(graphe)f(don)o(t)g(les)h(no)q(euds)g(son)o(t)f (les)h(c)o(hamps)75 1383 y(de)d(donn)o(\023)-21 b(ees)16 b(du)i(programme,)d(et)h(les)i(arcs)e(repr)o(\023)-21 b(esen)o(ten)o(t)15 b(la)i(relation)g fj(\()-9 b(\()q fq(d)o(\023)-21 b(ep)q(end)17 b(causalemen)o(t)g(de)p fj(\))-9 b(\))q fq(.)75 1440 y(l')o(\023)-21 b(enonc)o(\023)g(e)13 b fn(pei)h fq(est)g(traduisible)i(si)e(ce)h(graphe)f(est)g(sans)g (circuit.)h(dans)e(ce)i(cas,)f(dans)g(le)h(programme)75 1496 y(imp)o(\023)-21 b(eratif,)17 b(tout)f(c)o(hamp)i fk(x)f fq(d)o(\023)-21 b(ep)q(endan)o(t)17 b(causalemen)o(t)h(d'un)g(c) o(hamp)f fk(y)g fq(devra)g(appara)-5 b(^)-18 b(\020tre)18 b(apr)o(\022)-21 b(es)16 b fk(y)75 1553 y fq(dans)k(l'ordre)f(des)h (instructions.)g(cette)f(relation)h(sur)g(l'ensem)o(ble)h(des)e(c)o (hamps)h(d'un)g(tel)g(graphe)74 1609 y(\023)-21 b(etan)o(t)15 b(d'ordre)h(partiel,)h(nous)f(p)q(ouv)o(ons)h(c)o(hoisir)g(p)q(our)f (deux)h(c)o(hamps)g(non)f(comparables)h(un)g(ordre)75 1666 y(d'apparition)h(quelconque.)g(de)g(plus,)g(p)q(our)f(p)q(ouv)o (oir)h(traduire)f(les)g(\023)-21 b(equations)17 b(en)h(a\013ectations,) e(le)75 1722 y(c)o(hamp)f(d)o(\023)-21 b(ep)q(endan)o(t)15 b(causalemen)o(t)h(d'une)g(expression)g(ne)f(p)q(eut)g(^)-21 b(etre)14 b(qu'un)i(iden)o(ti\014cateur)1692 1705 y fg(3)1712 1722 y fq(.)146 1780 y(ainsi,)21 b(le)g(programme)f(de)h(droite)f(de)h (la)g(\014gure)g(5.3)e(est)h(ordonnanc)o(\023)-21 b(e)20 b(comme)g(le)i(mon)o(tre)d(la)75 1837 y(\014gure)e(5.4.)f(l'ordre)h (d'apparition)h(des)e(\023)-21 b(equations)17 b(dans)g(le)h(syst)o (\022)-21 b(eme)16 b(sera)g(celui)j(des)f(instructions)75 1893 y(du)e(programme)e(imp)o(\023)-21 b(eratif.)146 2010 y(cette)16 b(m)o(\023)-21 b(etho)q(de)16 b(d'ordonnancemen)o(t)g

(est)h(simple)h(car)e(elle)i(tire)f(partie)g(de)g(l'ordonnancemen)o(t) 75 2066 y(propre)f(au)h(mo)q(d)o(\022)-21 b(ele)16 b(data-parall)o (\022)-21 b(ele)8 b(:)16 b(comme)g(nous)g(l'a)o(v)o(ons)g(d)o(\023)-21 b(ej\022)e(a)15 b(vu,)h(le)h(\014l)g(d'ex)o(\023)-21 b(ecution)17 b(des)f(op)o(\023)-21 b(e-)75 2123 y(rations)14 b(parall)o(\022)-21 b(eles)15 b(est)g(en)g(e\013et)f(s)o(\023)-21 b(equen)o(tiel.)14 b(ici,)i(les)f(c)o(hamps)f(de)h(donn)o(\023)-21 b(ees)15 b(ou)f(v)m(ariables)i(parall)o(\022)-21 b(eles)75 2179 y(son)o(t)14 b(consid)o(\023)-21 b(er)o(\023)g(ees)15 b(dans)g(leur)h(en)o(ti)o(\022)-21 b(eret)o(\023)g(e.)14 b(plus)h(pr)o(\023)-21 b(ecis)o(\023)g(emen)o(t,)14 b(cela)i (signi\014e)h(que)e(l'ordonnancemen)o(t)75 2236 y(e\013ectu)o(\023)-21 b(e)16 b(implique)k(qu'une)e(v)m(ariable)g(parall)o(\022)-21 b(ele)18 b(est)f(compl)o(\022)-21 b(etemen)o(t)16 b(d)o(\023)-21 b(e\014nie)18 b(par)f(une)h(instruction,)75 2292 y(c'est-\022)-23 b(a-dire)17 b(que)g(toutes)f(ses)g(v)m(aleurs)i(son)o(t)c(\023)-21 b(ev)m(alu)o(\023)g(ees)16 b(lors)h(de)g(cette)f(instruction.)h(p)o(ar) f(opp)q(osition,)75 2348 y(nous)c(v)o(errons)f(en)i(section)f(5.10.2)e (qu'un)i(ordonnancemen)o(t)g(issu)h(des)f(tec)o(hniques)h(de)f(parall)o (\023)-21 b(elisation)75 2405 y(p)q(ermet)14 b(d')o(\023)-21 b(ev)m(aluer)13 b(certains)h(sous-ensem)o(bles)h(de)f(v)m(aleurs)g (d'une)g(v)m(ariable)h(parall)o(\022)-21 b(ele)14 b(\022)-23 b(a)13 b(des)h(instan)o(ts)75 2461 y(distincts.)p 75 2500 709 2 v 124 2542 a fd(3)p fc(:)9 b fd(une)k(exception)i(est)e (faite)g(actuellemen)o(t)j(p)q(our)d(les)h(alignemen)o(ts)h(d)o(\023) -18 b(eduits)15 b(d'un)e(c)o(hangemen)o(t)i(de)e(base)p eop %%page: 90 92 90 91 bop 75 269 a fq(90)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 383 y fk(sample)23 b(:)h(\(a,b,c\))14 b ff(7!)i fk(z)75 405 y fx(8)75 442 y(>)75 454 y(>)75 467 y(>)75 479 y(>)75 492 y(>)75 504 y(>)75 517 y(>)75 529 y(<)75 604 y(>)75 616 y(>)75 629 y(>)75 641 y(>)75 654 y(>)75 666 y(>)75 679 y(>)75 691 y(:)137 437 y fk(x'')f fq(=)g fk(a)g fa(c)h fk(fg1)137 493 y(y1)f fq(=)g fk(fc2)g fa(b)g fk(\(b)24 b(/&/)8 b(c\))137 550 y(y)15 b fq(=)g fk(fc3)g fa(b)h fk(\(y1)23 b(/&/)8 b(y)23 b fa(c)i fk(fg3)o(\))137 606 y(x')15 b fq(=)g fk(x'')g fa(c)g fk(fg2)137 662 y(x)g fq(=)g fk(fb1)7 b(::)h(x')137 719 y(z)15 b fq(=)g fk(fc1)g fa(b)h fk(\(x)23 b(/&/)8 b(y\))293 862 y fn(fig.)15 b fq(5.4)f(-)31 b fj(t)m(ri)15 b(des)g(\023)-22 b(e)n(quations)16 b(selon)f(l'or)n(dr)n(e)h(des)g(d)o(\023)-22 b(ep)n(endanc)n(es)15 b(c)n(ausales)146 1004 y fq(pratiquemen)o(t)10 b(dans)g(le)h(protot)o (yp)q(e,)e(l'ordonnancemen)o(t)i(est)f(r)o(\023)-21 b(ealis)o(\023)g(e) 9 b(par)i(un)f(autre)g(\014ltre)h(baptis)o(\023)-21 b(e)75 1060 y fk(ord)p fq(,)14 b(appliqu)o(\023)-21 b(e)16 b(apr)o(\022)-21 b(es)15 b fk(normhpf)p fq(.)75 1195 y fw(5.4.4)55 b(synopsis)19 b(du)g(fonctionnemen)n(t)d(du)j(traducteur)146 1285 y fq(lorsqu'elles)f(ab)q(outissen)o(t)g(\022)-23 b(a)17 b(une)h(forme)f(traduisible,)i(des)f(transformations)e(pr)o(\023)-21 b(ec)o(\023)g(eden)o(tes)16 b(r)o(\023)-21 b(e-)75 1341 y(sulte)14 b(un)g(programme)e fn(pei)i fq(don)o(t)f(la)g(structure)h (est)f(sem)o(blable)i(\022)-23 b(a)13 b(celle)i(d'un)f(programme)e (parall)o(\022)-21 b(ele.)75 1398 y(le)22 b(tra)o(v)m(ail)g(restan)o(t) f(\022)-23 b(a)21 b(r)o(\023)-21 b(ealiser)21 b(se)h(r)o(\023)-21 b(eduit)22 b(main)o(tenan)o(t)f(\022)-23 b(a)21 b(la)h(traduction)g (des)f(\023)-21 b(equations)21 b(une)h(\022)-23 b(a)75 1454 y(une,)17 b(et)f(\022)-23 b(a)15 b(l'inf)o(\023)-21 b(erence)18 b(des)e(dimensions)i(des)f(v)m(ariables)g(parall)o(\022)-21

b(eles)17 b(utilis)o(\023)-21 b(ees.)17 b(le)g(m)o(\023)-21 b(ecanisme)16 b(de)g(la)75 1511 y(traduction)f(p)q(eut)g(^)-21 b(etre)14 b(r)o(\023)-21 b(esum)o(\023)g(e)14 b(de)h(la)h(mani)o(\022) -21 b(ere)15 b(suiv)m(an)o(te)8 b(:)131 1614 y(1.)22 b(analyse)g(lexicale)i(et)d(syn)o(taxique)h(du)g(programme)e fn(pei)i fq(et)f(construction)h(d'un)g(arbre)f(de)189 1670 y(syn)o(taxe)14 b(abstraite)h(\(ast\).)131 1773 y(2.)22 b(p)o(arcours)15 b(de)i(l'ast)g(p)q(our)f(sto)q(c)o(k)o(er)g (les)h(informations)g(pr)o(\023)-21 b(esen)o(tes)15 b(dans)i(l')o(\023) -21 b(equation)16 b(dans)g(un)189 1829 y(format)9 b(de)j(donn)o(\023) -21 b(ees)10 b(in)o(term)o(\023)-21 b(ediaire.)11 b(on)h(v)o(\023)-21 b(eri\014e)11 b(\022)-23 b(a)11 b(ce)g(momen)o(t)f(que)i(les)f (informations)g(retir)o(\023)-21 b(ees)189 1886 y(des)14 b(\023)-21 b(equations)15 b(p)q(ourron)o(t)e(^)-21 b(etre)14 b(exploit)o(\023)-21 b(ees)16 b(p)q(our)f(la)h(pro)q(duction)g(de)g(co) q(de.)131 1989 y(3.)22 b(p)o(asse)f(suppl)o(\023)-21 b(emen)o(taire)23 b(sur)g(l'ensem)o(ble)h(des)d(\023)-21 b(equations)22 b(p)q(our)h(inf)o(\023)-21 b(erer)22 b(des)h (informations)189 2045 y(globales)f(concernan)o(t)g(les)h(domaines)g (des)f(c)o(hamps)g(de)g(donn)o(\023)-21 b(ees.)22 b(on)g(compl)o(\022) -21 b(ete)22 b(alors)g(les)189 2102 y(informations)15 b(du)h(format)d(in)o(term)o(\023)-21 b(ediaire.)131 2204 y(4.)22 b(p)o(assage)16 b(des)i(informations)f(\022)-23 b(a)18 b(un)g(formateur)e(de)i(texte)f(qui)i(exploite)f(les)h (informations)e(du)189 2261 y(format)c(in)o(term)o(\023)-21 b(ediaire)16 b(p)q(our)g(pro)q(duire)g(le)g(co)q(de)g(hpf.)75 2396 y fw(5.4.5)55 b(t)-5 b(raduction)19 b(hpf)g(des)f(\023)-27 b(equations)17 b fv(pei)146 2486 y fq(nous)c(nous)g(in)o(t)o(\023)-21 b(eressons)13 b(main)o(tenan)o(t)g(aux)g(instructions)h(hpf)f(asso)q (ci)o(\023)-21 b(ees)13 b(\022)-23 b(a)14 b(c)o(hacune)g(des)e(\023)-21 b(equa-)75 2542 y(tions)15 b(traduisibles.)p eop %%page: 91 93 91 92 bop 75 269 a fl(5.4.)29 b(le)16 b(traducteur)f fk(pei2hpf)1034 b fq(91)75 397 y fo(op)o(\023)-25 b(eration)19 b(g)o(\023)-25 b(eom)o(\023)g(etrique)146 491 y fq(soit)15 b(l')o(\023)-21 b(equation)15 b(de)g(la)h(forme)f(1.:)862 561 y fk(y)d fq(=)h fk(x)i fa(c)h fk(g)75 660 y fq(rapp)q(elons)k(que)e fk(g)g fq(est)g(de)h(la)g(forme)e fm(\025)p fq(\()p fm(z)r fq(\))g ff(j)g fm(dom)p fq(\()p fk(g)p fq(\))p fm(:)p fk(g)p fq(\()n fm(z)r fq(\))h(et)g(que)h(le)g(c)o(hamp)f fk(y)g fq(est)g(\()p fm(v)1641 667 y fr(x)1668 660 y ff(\016)10 b fk(g)18 b fq(:)f fm(\033)1799 667 y fr(x)1816 660 y fq(\).)75 717 y(l'instruction)i fk(forall)e fq(d'hpf)g(autorise)h (une)g(traduction)g(directe)h(de)f(l'op)o(\023)-21 b(eration)17 b(g)o(\023)-21 b(eom)o(\023)g(etrique.)75 773 y(nous)15 b(traduisons)g(l')o(\023)-21 b(equation)15 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)14 b(par)8 b(:)563 890 y fk(forall)23 b(\()p fm(z)15 b ff(2)e fm(dom)p fq(\()p fk(g)p fq(\))p fk(\))23 b(y\()p fm(z)r fk(\))g(=)h(x\(g\()p fm(z)r fk(\)\))75 1001 y fq(si)19 b(le)f(domaine)h(de)f fk(g)g fq(n'est)f(pas)h(pr)o (\023)-21 b(ecis)o(\023)g(e)17 b(dans)h(la)g(d)o(\023)-21 b(e\014nition)19 b(de)g(la)f(fonction,)g(le)g(domaine)h(utilis)o(\023) -21 b(e)75 1058 y(est)17 b fm(dom)p fq(\()p fm(\033)278 1065 y fr(x)294 1058 y fq(\).)f(de)h(plus,)h(la)f(con)o(train)o(te)f fm(img)r fq(\()p fk(g)p fq(\))e ff(\022)h fm(dom)p fq(\()p fm(v)1122 1065 y fr(x)1139 1058 y fq(\))h(garan)o(tit)g(que)h(les)h (indices)h fk(g\()p fm(z)r fk(\))d fq(son)o(t)75 1114 y(v)m(alides)h(p)q(our)e(le)h(tableau)g fk(x)p fq(.)f(p)o(ar)f (exemple,)i(la)g(traduction)f(de)h(l')o(\023)-21 b(equation)690 1226 y fk(y)15 b fq(=)g fk(x)g fa(c)h fk(right)555 1282

y fq(a)o(v)o(ec)50 b fk(right)14 b fq(=)i fm(\025)p fk(\(i\))p ff(j)p fk(\(1)p fm(<)p fk(i)p ff(\024)p fk(100\).\(i-1\))75 1394 y fq(est)f fk(forall)23 b(\(i=2:100\))g(y\(i\))g(=)h(x\(i-1\))p fq(.)75 1540 y fo(op)o(\023)-25 b(eration)19 b(fonctionnelle)146 1634 y fq(soit)c(l')o(\023)-21 b(equation)15 b(de)g(la)h(forme)f(2.)7 b(:)676 1750 y fk(y)12 b fq(=)h fk(f)i fa(b)h fk(\(x)898 1757 y fg(1)940 1750 y fk(/&/)7 b fm(:)h(:)g(:)22 b fk(/&/)8 b(x)1200 1757 y fw(n)1222 1750 y fk(\))75 1865 y fq(cette)14 b(\023)-21 b(equation)14 b(est)h(traduite)h(par)f(l'instruction)h(hpf)7 b(:)284 1981 y fk(forall)23 b(\()p fm(z)15 b ff(2)e fm(dom)p fk(\()p fm(\033)690 1988 y fr(x)706 1981 y fk(\))24 b(,)g fm(z)15 b ff(2)e fm(dom)p fk(\(f\)\))23 b(y\()p fm(z)r fk(\))g(=)h(f\(x)1324 1988 y fg(1)1342 1981 y fq(\()p fm(z)r fq(\))p fk(,)p fm(:)8 b(:)g(:)d fk(,x)1533 1988 y fw(n)1555 1981 y fq(\()p fm(z)r fq(\))p fk(\))75 2093 y fq(notons)17 b(que)h(nous)g(supp)q(osons)g(ici)h fm(dom)p fq(\()p fm(\033)841 2100 y fr(x)858 2093 y fq(\))d(=)i fm(dom)p fq(\()p fm(\033)1075 2100 y fr(x)1090 2104 y fo(1)1104 2093 y fq(\))f(=)g fm(:)8 b(:)g(:)15 b fq(=)i fm(dom)p fq(\()p fm(\033)1443 2100 y fr(x)1458 2104 y fo(n)1473 2093 y fq(\).)g(cette)g(con)o(train)o(te)75 2150 y(actuelle)g(p)q(eut)d(^)-21 b(etre)15 b(lev)o(\023)-21 b(ee)15 b(dans)g(une)h(v)o(ersion)f(future)h(du)f(traducteur.)75 2206 y(la)j(restriction)g(de)g(l'ensem)o(ble)h(des)e(pro)q(cesseurs)h (virtuels)h(v)m(alides)g(\022)-23 b(a)18 b fm(dom)p fq(\()p fk(f)p fq(\))e(est)h(repr)o(\023)-21 b(esen)o(t)o(\023)g(e)16 b(par)75 2262 y(la)g(partie)f fj(masque)g fq(de)h(l'instruction)g fk(forall)p fq(.)e(nous)h(traduirons)h(par)f(exemple,)h(l')o(\023)-21 b(equation)7 b(:)646 2374 y fk(y)15 b fq(=)g fk(divide)g fa(b)g fk(\(x1/&/)7 b(x2\))646 2431 y(divide)14 b fq(=)i fm(\025)p fk(\(a)p fq(;)o fk(b\))p ff(j)p fk(\(b)p ff(6)p fq(=)p fk(0\).\(a/b\))75 2542 y fq(en)g fk(forall)23 b(\()p fm(z)14 b ff(2)f fm(dom)p fk(\()p fm(\033)541 2549 y fr(x2)574 2542 y fk(\),x2\()p fm(z)r fk(\).neq.0\))22 b(y\()p fm(z)r fk(\))h(=)h(x1\()p fm(z)r fk(\)/x2\()p fm(z)r fk(\))p eop %%page: 92 94 92 93 bop 75 269 a fq(92)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fo(propri)o(\023)-25 b(et)o(\023)g(e)17 b(de)h(c)o(hamp)f(de)g(donn)o(\023)-25 b(ees)146 491 y fq(l')o(\023)k(equation)14 b fk(x)h fq(=)g fk(h)8 b(::)f(x)15 b fq(exprime)h(que)g(l'application)h(de)e(la)g (fonction)h(iden)o(tit)o(\023)-21 b(e)15 b(sur)g(le)h(domaine)75 548 y(des)e(v)m(aleurs)g(et)f(r)o(\023)-21 b(ef)o(\023)g(erence)12 b(de)h fk(x)g fq(ne)h(mo)q(di\014e)g(pas)f fk(x)p fq(.)g(dans)g(ce)g(t) o(yp)q(e)g(d')o(\023)-21 b(equation,)13 b(la)g(fonction)h(utilis)o (\023)-21 b(ee)75 604 y(dans)14 b(le)h(c)o(hangemen)o(t)e(de)i(base)f (ne)g(p)q(eut)g(^)-21 b(etre)12 b(autre)i(que)g(l'iden)o(tit)o(\023)-21 b(e.)14 b(en)g(e\013et,)g(de)g(par)g(la)g(d)o(\023)-21 b(e\014nition)75 661 y(du)18 b(c)o(hangemen)o(t)g(de)g(base,)f(nous)h (dev)o(ons)g(v)o(\023)-21 b(eri\014er)18 b(les)f(\023)-21 b(egalit)o(\023)g(es)6 b(:)18 b fm(v)1267 668 y fr(x)1302 661 y fq(=)g fm(v)1377 668 y fr(x)1405 661 y ff(\016)10 b fk(h)1462 644 y fq(-1)1517 661 y fq(et)17 b fm(\033)1598 668 y fr(x)1633 661 y fq(=)h fm(\033)1712 668 y fr(x)1740 661 y ff(\016)10 b fk(h)1797 644 y fq(-1)1834 661 y fq(.)75 717 y(nous)22 b(p)q(ouv)o(ons)h(d)o(\023)-21 b(eduire)23 b(de)g(cette)e(\023)-21 b(equation)22 b(les)h(dimensions)h(de)f(la)g(v) m(ariable)h fk(x)e fq(utilis)o(\023)-21 b(ee)23 b(dans)75

773 y(le)e(programme)e(hpf,)g(car)h(les)h(v)m(aleurs)g(du)g(c)o(hamp)f fk(x)g fq(son)o(t)f(plac)o(\023)-21 b(ees)20 b(dans)h fm(dom)p fq(\()p fm(\033)1578 780 y fr(x)1594 773 y fq(\))g ff(\022)g fm(dom)p fq(\()p fk(h)p fq(\).)75 830 y(cette)11 b(forme)f(d')o(\023)-21 b(equation)11 b(n'est)f(donc)i(pas)f (directemen)o(t)h(traduite)f(sous)g(forme)f(d'instructions)i(hpf,)75 886 y(mais)18 b(utilis)o(\023)-21 b(ee)18 b(p)q(our)g(dimensionner)i (le)e(tableau)g(utilis)o(\023)-21 b(e)18 b(p)q(our)g(la)g(traduction)g (du)g(c)o(hamp)f fk(x)p fq(.)g(cette)75 943 y(information)12 b(est)f(utilis)o(\023)-21 b(ee)13 b(conjoin)o(temen)o(t)e(a)o(v)o(ec)g (les)i(c)o(hangemen)o(ts)e(de)h(base)g(lian)o(t)g(di\013)o(\023)-21 b(eren)o(ts)11 b(c)o(hamps)75 999 y(de)16 b(donn)o(\023)-21 b(ees.)75 1144 y fo(op)o(\023)c(eration)19 b(de)e(c)o(hangemen)o(t)g (de)h(base)146 1238 y fq(une)e(\023)-21 b(equation)15 b(de)i(la)g(forme)f fk(y)g fq(=)h fk(h)7 b(::)h(x)16 b fq(indique)j(que)e(toute)e(v)m(aleur)j(plac)o(\023)-21 b(ee)17 b(au)f(p)q(oin)o(t)h fm(z)h fq(dans)75 1295 y(le)g(c)o(hamp)g fk(y)f fq(est)g(aussi)h(la)g(v)m(aleur)h(plac)o(\023)-21 b(ee)17 b(au)h(p)q(oin)o(t)g fk(h\()p fm(z)r fk(\))f fq(dans)h(le)g(c)o(hamp)g fk(x)p fq(.)f(on)h(p)q(eut)g(imaginer)75 1351 y(deux)e(mani)o(\022)-21 b(eres)15 b(de)g(traduire)h(cette)e(\023) -21 b(equation.)146 1412 y(la)16 b(premi)o(\022)-21 b(ere)17 b(consiste)g(\022)-23 b(a)16 b(d)o(\023)-21 b(eterminer)17 b(quel)h(tableau)f(est)f(calcul)o(\023)-21 b(e)18 b(le)f(premier)h (lors)e(de)h(l'ex)o(\023)-21 b(ecu-)75 1468 y(tion)16 b(du)h(programme)d(\(supp)q(osons)j(que)f(ce)g(soit)g fk(x)p fq(\),)f(et)h(\022)-23 b(a)15 b(recopier)i(les)g(v)m(aleurs)g fk(x\()p fm(z)r fk(\))e fq(en)i fk(y\(h\()p fm(z)r fk(\)\))p fq(.)75 1525 y(cette)11 b(fa\030)-20 b(con)12 b(de)g(pro)q(c)o(\023)-21 b(eder)12 b(implique)i(des)e(comm)o(unications)g(lors)g(de)h(l'ex)o (\023)-21 b(ecution,)11 b(mais)h(est)g(l'unique)75 1581 y(p)q(ossibilit)o(\023)-21 b(e)14 b(de)g(traduction)f(dans)g(des)g (langages)f(data-parall)o(\022)-21 b(eles)13 b(n'o\013ran)o(t)e(pas)i (la)g(p)q(ossibilit)o(\023)-21 b(e)15 b(d'ali-)75 1638 y(gnemen)o(t.)146 1698 y(l'autre)20 b(c)o(hoix)h(de)g(traduction)g (consid)o(\022)-21 b(ere)21 b(que)g(l')o(\023)-21 b(equation)20 b(indique)j(commen)o(t)d(les)h(tableaux)75 1755 y(son)o(t)d(plac)o (\023)-21 b(es)18 b(les)i(uns)f(par)f(rapp)q(ort)g(aux)g(autres)g(sur)h (la)g(mac)o(hine)g(virtuelle.)h(nous)f(a)o(v)o(ons)e(vu)i(que)75 1811 y(hpf)i(o\013re)f(la)i(p)q(ossibilit)o(\023)-21 b(e)22 b(de)g(traduire)f(cette)g(notion)h(a)o(v)o(ec)e(la)i(directiv)o (e)g fk(align)p fq(.)e(ce)h(c)o(hoix)h(est)75 1868 y(pr)o(\023)-21 b(ef)o(\023)g(erable)16 b(dans)i(la)f(mesure)h(o)q(\022)-24 b(u)17 b(il)i(n'induit)g(pas)e(de)h(comm)o(unication)g(\022)-23 b(a)17 b(l'ex)o(\023)-21 b(ecution.)17 b(av)o(ec)h(cette)75 1924 y(m)o(\023)-21 b(etho)q(de)12 b(nous)g(c)o(hoisissons)h(d'aligner) g(les)g(tableaux)f fk(x)g fq(et)g fk(y)g fq(relativ)o(emen)o(t)h(\022) -23 b(a)12 b(un)g(template)h(comm)o(un)75 1981 y fk(t)p fq(.)i(sc)o(h)o(\023)-21 b(ematiquemen)o(t,)14 b(les)i(alignemen)o(ts)g (corresp)q(ondan)o(ts)f(\022)-23 b(a)15 b(l')o(\023)-21 b(equation)15 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)14 b(son)o(t)7 b(:)687 2092 y fk(align)23 b(x\()p fm(z)r fk(\))h(with)f(t\()p fm(z)r fk(\))687 2148 y(align)g(y\()p fm(z)r fk(\))h(with)f(t\(h\()p fm(z)r fk(\)\))75 2256 y fq(le)d(template)f(doit)g(p)q(ouv)o(oir)h(con)o(tenir)f(les)h(di\013) o(\023)-21 b(eren)o(ts)18 b(tableaux)i(du)f(programme.)f(p)o(our)g (cela,)i(ses)75 2312 y(b)q(ornes)13 b(son)o(t)f(calcul)o(\023)-21 b(ees)14 b(de)f(fa\030)-20 b(con)13 b(\022)-23 b(a)13 b(d)o(\023)-21 b(e\014nir)13 b(l'en)o(v)o(elopp)q(e)h(parall)o(\023)-21 b(el)o(\023)g(epip)o(\022)g(edique)15 b(de)e fm(dom)p

fq(\()p fk(h)p fq(\))5 b ff([)g fm(img)r fq(\()p fk(h)p fq(\).)75 2369 y(en)12 b(supp)q(osan)o(t)f(que)h(les)h(autres)d(\023) -21 b(equations)11 b(du)h(programme)e fn(pei)i fq(nous)f(app)q(orten)o (t)h(des)g(informations)75 2425 y(sur)23 b(les)g(b)q(ornes)g(des)g (tableaux,)g(et)g(le)g(t)o(yp)q(e)g fm(ty)r(pe)g fq(des)g(v)m(aleurs)h (sto)q(c)o(k)o(\023)-21 b(ees,)21 b(nous)i(p)q(ouv)o(ons)g(alors)75 2482 y(pro)q(duire)16 b(les)g(d)o(\023)-21 b(eclarations)15 b(des)h(tableaux)f(et)g(du)h(template.)146 2542 y(supp)q(osons)f(qu'on) g(inf)o(\022)-21 b(ere)14 b(du)i(programme)d(que)j(les)f(tableaux)g fk(x)g fq(et)g fk(y)f fq(son)o(t)g(de)h(taille)i([1)8 b fm(:)g(:)g(:)t(n)p fq(])h ff(\002)p eop %%page: 93 95 93 94 bop 75 269 a fl(5.4.)29 b(le)16 b(traducteur)f fk(pei2hpf)1034 b fq(93)75 397 y([1)8 b fm(:)g(:)g(:)d(n)p fq(])15 b(alors)g(les)g(d)o(\023)-21 b(eclarations)15 b(des)h(tableaux)g(et)f(du)g(template)h(son)o(t)6 b(:)637 504 y fm(ty)r(pe)24 b fk(x\(n\),)f fm(ty)r(pe)h fk(y\(h\()p fq(1)p fk(\):h\()p fm(n)p fk(\)\))637 560 y(template)e(t\()p fm(dom)p fq(\()p fk(h)p fq(\))9 b ff([)i fm(img)r fq(\()p fk(h)p fq(\))p fk(\))75 662 y fq(en\014n,)k(\022)-23 b(a)14 b(l'ex)o(\023)-21 b(ecution,)14 b(il)i(reste)e(\022)-23 b(a)14 b(recopier)h(les)g(v)m(aleurs)h(du)e(tableau)h(d)o(\023)-21 b(ej\022)e(a)14 b(calcul)o(\023)-21 b(e)15 b(dans)f(le)i(nouv)o(eau)75 718 y(a\014n)11 b(de)g(resp)q(ecter)h(l')o(\023)-21 b(egalit)o(\023)g (e)10 b(des)h(v)m(aleurs)h(sp)o(\023)-21 b(eci\014)o(\023)g(ee)10 b(par)h(l')o(\023)-21 b(equation)10 b fn(pei)h fq(\(con)o(trairemen)o (t)f(au)h(premier)75 775 y(mo)q(de)j(de)f(traduction,)g(cette)h(copie)g (n'engendre)g(pas)f(de)h(comm)o(unication)g(du)g(fait)f(des)h (alignemen)o(ts)75 831 y(r)o(\023)-21 b(ealis)o(\023)g(es\).)17 b(illustrons)j(simplemen)o(t)f(ce)g(pro)q(c)o(\023)-21 b(ed)o(\023)g(e)18 b(a)o(v)o(ec)g(l'exemple)i(d'un)f(d)o(\023)-21 b(ecalage)18 b(en)o(tre)h(tableaux.)75 888 y(consid)o(\023)-21 b(erons)15 b(les)g(\023)-21 b(equations)14 b(suiv)m(an)o(tes,)i(o)q (\022)-24 b(u)15 b fk(x)g fq(est)g(un)g(c)o(hamp)h(en)f(en)o(tr)o(\023) -21 b(ee)7 b(:)630 992 y fk(x)15 b fq(=)h fk(matrix)6 b(::)i(x)630 1048 y(y)15 b fq(=)h fk(transl)6 b(::)i(x)496 1105 y fq(a)o(v)o(ec)49 b fk(transl)14 b fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(i,j)p ff(\024)p fk(n\).\(i+1,j\))496 1161 y fq(et)96 b fk(matrix)14 b fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(i,j)p ff(\024)p fk(n\).\(i,j\))75 1263 y fq(la)24 b(premi)o(\022)-21 b(ere)22 b(\023)-21 b(equation,)23 b(comme)h(nous)g(l'a)o(v)o(ons)f(vu)h(pr)o(\023)-21 b(ec)o(\023)g (edemmen)o(t,)22 b(con)o(train)o(t)h(les)h(domaines)75 1319 y(de)c(r)o(\023)-21 b(ef)o(\023)g(erence)19 b(et)g(de)i(v)m (aleurs)f(de)h fk(x)e fq(au)h(domaine)g(de)h fk(matrix)1188 1302 y fq(-1)1225 1319 y fq(,)e(c'est-\022)-23 b(a-dire)21 b(que)f(ces)g(domaines)75 1376 y(son)o(t)i(inclus)i(dans)e ff(f)p fq(\()p fm(i;)8 b(j)s fq(\))22 b ff(j)j fq(1)p ff(\024)p fm(i;)8 b(j)s ff(\024)n fm(n)p ff(g)p fq(.)22 b(la)h(seconde)f(\023)-21 b(equation,)21 b(toujours)h(selon)h(la)g(d)o (\023)-21 b(e\014nition)75 1432 y(du)22 b(c)o(hangemen)o(t)f(de)h (base,)f(con)o(train)o(t)g fm(dom)p fq(\()p fm(v)917 1439 y fr(y)934 1432 y fq(\))g(et)g fm(dom)p fq(\()p fm(\033)1162 1439 y fr(y)1179 1432 y fq(\))7 b(:)21 b(par)g(exemple)i fm(v)1529 1439 y fr(y)1568 1432 y fq(=)f fm(v)1647 1439 y fr(x)1696 1432 y ff(\016)10 b fk(h)1753 1416 y fq(-1)1811 1432 y fq(=)75 1489 y fm(v)97 1496 y fr(x)125 1489 y ff(\016)f fm(\025)p fk(\(i,j\))p ff(j)p fk(\(2)p ff(\024)p fk(i)p ff(\024)p fk(n+1)e(&)h(1)p ff(\024)p fk(j)p ff(\024)p

fk(n\).\(i-1,j\))o fq(,)14 b(qui)j(implique)g fm(dom)p fq(\()p fm(v)1343 1496 y fr(y)1360 1489 y fq(\))12 b ff(\022)h(f)p fq(\()p fm(i;)8 b(j)s fq(\))i ff(j)j fq(2)p ff(\024)p fm(i)p ff(\024)p fm(n)d fq(+)h(1)e ff(^)75 1545 y fq(1)p ff(\024)p fm(j)s ff(\024)p fm(n)p ff(g)p fq(.)15 b(nous)g(allons)h(donc)f(traduire)h(ces)e(\023)-21 b(equation)15 b(par)7 b(:)507 1652 y fm(ty)r(pe)25 b fk(x\(1:n\),)e fm(ty)r(pe)h fk(y\(2:n+1\))507 1709 y(template)f (t\(n+1,n\))507 1765 y(align)h(x\(i,j\))f(with)g(t\(i,j\))507 1821 y(align)h(y\(i,j\))f(with)g(t\(i+1,j\))507 1878 y(forall)g(\(i=1:n,j=1:n\))g(y\(i+1,j\))f(=)i(x\(i,j\))75 1979 y fj(r)n(estrictions)6 b fq(:)75 2095 y(l'expressivit)o(\023)-21 b(e)17 b(de)g(la)g(directiv)o(e)h fk(align)e fq(n')o(\023)-21 b(etan)o(t)15 b(pas)h(aussi)h(grande)g(que)g(celles)i(des)e(fonctions)g (ap-)75 2151 y(paraissan)o(t)g(dans)g(le)i(c)o(hangemen)o(t)e(de)h (base,)f(nous)g(sommes)g(amen)o(\023)-21 b(es)17 b(\022)-23 b(a)17 b(consid)o(\023)-21 b(erer)17 b(plusieurs)j(res-)75 2208 y(trictions.)e(la)f(premi)o(\022)-21 b(ere)18 b(est)f(li)o(\023) -21 b(ee)18 b(\022)-23 b(a)18 b(la)g(forme)f(a\016ne)g(des)h(alignemen) o(ts)h(hpf.)e(p)o(ar)f(cons)o(\023)-21 b(equen)o(t)17 b(la)75 2264 y(fonction)f(de)f(c)o(hangemen)o(t)g(de)h(base)f(doit)f(^) -21 b(etre)14 b(de)i(la)g(forme)7 b(:)443 2375 y fk(h)13 b fq(=)g fm(\025)p fk(\()p fm(i)595 2382 y fg(1)612 2375 y fm(;)8 b(:)g(:)g(:)d(;)j(i)730 2382 y fw(m)761 2375 y fk(\).\()p fm(f)855 2382 y fg(1)873 2375 y fq(\()p fm(i)907 2382 y fg(1)925 2375 y fm(;)g(:)g(:)g(:)d(;)j(i)1043 2382 y fw(m)1073 2375 y fq(\))p fm(;)g(:)g(:)g(:)d(;)j(f)1215 2382 y fw(n)1237 2375 y fq(\()p fm(i)1271 2382 y fg(1)1289 2375 y fm(;)g(:)g(:)g(:)t(;)g(i)1406 2382 y fw(m)1437 2375 y fq(\))p fk(\))75 2486 y fq(o)q(\022)-24 b(u)15 b(les)h fm(f)226 2493 y fw(i)255 2486 y fq(son)o(t)f(des)g(fonctions)h (a\016nes.)p eop %%page: 94 96 94 95 bop 75 269 a fq(94)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)146 397 y fq(une)i(autre)f(con)o(train)o(te) g(survien)o(t)h(quand,)f(p)q(our)h fk(h)8 b fq(:)16 b fe(z)1108 380 y fw(m)1151 397 y ff(7!)g fe(z)1245 380 y fw(n)1265 397 y fq(,)g(nous)g(a)o(v)o(ons)g fm(m)f ff(6)p fq(=)g fm(n)p fq(.)h(dans)g(ce)75 453 y(cas,)c(la)h(syn)o(taxe)e (de)i(la)g(directiv)o(e)h(nous)e(oblige)i(\022)-23 b(a)12 b(aligner)h(le)g(tableau)g(a)o(y)o(an)o(t)e(le)i(moins)g(de)g (dimensions)75 510 y(sur)h(celui)h(a)o(y)o(an)o(t)e(le)h(plus)h(de)f (dimensions.)h(nous)f(distinguons)h(donc)f(deux)h(cas)e(p)q(our)h(la)g (traduction)8 b(:)143 584 y({)23 b(si)15 b fm(m)e(>)g(n)i fq(alors)40 b fk(align)23 b(x\()p fm(z)r fk(\))h(with)f(t\()p fm(z)r fk(\))511 641 y(align)g(y\()p fm(dom)p fq(\()p fk(h)830 624 y fq(-1)867 641 y fq(\))p fk(\))g(with)g(t\()p fm(img)r fq(\()p fk(h)1221 624 y fq(-1)1257 641 y fq(\))p fk(\))189 693 y fq(a)o(v)o(ec)14 b fm(z)h ff(2)e fe(z)400 677 y fw(m)429 693 y fq(.)143 779 y({)23 b(si)15 b fm(m)e(<)g(n)i fq(alors)40 b fk(align)23 b(x\()p fm(dom)p fq(\()p fk(h)p fq(\))p fk(\))g(with)g(t\()p fm(img)r fq(\()p fk(h)p fq(\))p fk(\))511 836 y(align)g(y\()p fm(dom)p fq(\()p fk(h)830 819 y fq(-1)867 836 y fq(\))p fk(\))g(with)g(t\()p fm(img)r fq(\()p fk(h)1221 819 y fq(-1)1257 836 y fq(\))p fk(\))75 907 y fq(ainsi,)c(l'alignemen)o(t)h(d'un)e(v)o(ecteur)h(sur)f (la)g(premi)o(\022)-21 b(ere)18 b(colonne)i(d'un)e(v)o(ecteur)g(est)g (souv)o(en)o(t)g(d)o(\023)-21 b(e\014ni)75 963 y(par)16 b(l')o(\023)-21 b(equation)16 b fk(y)g fq(=)h fk(h)7

b(::)h(x)o fq(.)16 b(le)h(c)o(hangemen)o(t)f(de)h(base)g fk(h)f fq(est)g(alors)g fm(\025)p fk(\(i,j\))p ff(j)p fk(\(j)p fq(=)p fk(0\).\(i\))e fq(si)j fk(y)f fq(est)75 1020 y(le)k(v)o(ecteur,)e(ou)h(bien)i(son)e(in)o(v)o(erse)g fm(\025)p fk(\(i\).\(i,0\))e fq(si)j fk(x)f fq(est)g(le)g(v)o(ecteur.)g (si)h fk(y)f fq(est)g(le)g(v)o(ecteur)g(nous)75 1076 y(pro)q(duisons)d(les)g(directiv)o(es)9 b(:)170 1176 y fk(align)24 b(x\(i,j\))f(with)g(t\(i,j\))170 1233 y(align)h(y\(i\))f (with)g(t\(i,0\))75 1317 y fq(dans)14 b(ce)i(cas,)e(la)h(seule)h(con)o (train)o(te)f(implicite)i(de)e(l')o(\023)-21 b(equation)15 b(est)f(que)i(les)f(v)m(aleurs)h(du)f(v)o(ecteur)g(son)o(t)75 1374 y(celles)i(de)f(la)f(premi)o(\022)-21 b(ere)15 b(colonne)h(de)g (la)f(matrice)h(\()p fm(v)972 1381 y fr(y)1004 1374 y fq(=)g fm(v)1077 1381 y fr(x)1104 1374 y ff(\016)10 b fm(\025)p fk(\(i\).\(i,0\))n fq(\).)146 1430 y(p)o(ar)i(con)o(tre,)g (si)i fk(x)e fq(est)h(le)h(v)o(ecteur,)f(le)g(tableau)h fk(y)f fq(ne)g(p)q(eut)h(a)o(v)o(oir)e(de)i(v)m(aleurs)g(que)f(sur)g (sa)g(premi)o(\022)-21 b(ere)75 1487 y(colonne)18 b(\()p fm(dom)p fq(\()p fm(v)385 1494 y fr(y)402 1487 y fq(\))f(=)g fm(dom)p fq(\()p fm(\033)619 1494 y fr(y)636 1487 y fq(\))g(et)g fm(dom)p fq(\()p fm(\033)856 1494 y fr(y)873 1487 y fq(\))g ff(\022)f(f)p fq(\()p fm(i;)8 b(j)s fq(\))14 b ff(j)i fm(j)s fq(=)o(0)p ff(g)p fq(\).)g(p)o(ar)h(cons)o(\023)-21 b(equen)o(t)16 b(les)i(directiv)o(es)75 1543 y(g)o(\023)-21 b(en)o(\023)g(er)o(\023)g(ees)13 b(son)o(t)7 b(:)170 1643 y fk(align)24 b(x\(i\))f(with)g(t\(i,0\))170 1700 y(align)h(y\(i,0\))f(with)g(t\(i,0\))146 1784 y fq(le)f(traducteur)h (a\016c)o(he)f(d'ailleurs)i(un)e(message)g(d'a)o(v)o(ertissemen)o(t)g (dans)g(ce)h(dernier)g(cas)f(de)75 1841 y(\014gure,)15 b(car)g(l'utilisation)i(de)f(la)f(premi)o(\022)-21 b(ere)15 b(colonne)h(seulemen)o(t)h(d'une)e(matrice)h(est)f(inhabituelle.)75 1957 y fo(r)o(\023)-25 b(eduction)19 b(g)o(\023)-25 b(eom)o(\023)g (etrique)17 b(com)o(bin)o(\023)-25 b(ee)17 b(a)o(v)o(ec)g(une)h(op)o (\023)-25 b(eration)18 b(fonctionnelle)146 2043 y fq(seules)f(les)f (fonctions)g fk(xxx)p 618 2043 15 2 v 17 w(scatter)f fq(de)h(r)o(\023)-21 b(eduction)15 b(de)i(tableaux)f(d'hpf)f(\(v)o(oir) g(section)i(5.1.2\))75 2100 y(p)q(ermetten)o(t)e(d'exprimer)h(les)g(r)o (\023)-21 b(eductions)15 b fn(pei)g fq(en)g(toute)g(g)o(\023)-21 b(en)o(\023)g(eralit)o(\023)g(e.)13 b(soit)i(l')o(\023)-21 b(equation)15 b fn(pei)7 b fq(:)684 2170 y fk(y)15 b fq(=)g fk(fr)g fa(b)h fk(\(gr)23 b fq(;)8 b fa(b)24 b fk(x\))586 2227 y fq(o)q(\022)-24 b(u)50 b fk(fr)15 b fq(=)g fk(id)10 b fq(#)g fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a)23 b ff(\012)h fk(fr\(b\)\))75 2297 y fq(et)f(soit)g fm(e)251 2304 y fx(\012)302 2297 y fq(l')o(\023)-21 b(el)o(\023)g(emen)o(t)22 b(neutre)i(de)f ff(\012)p fq(.)g(p)o(our)g(traduire)g(cette)f(\023)-21 b(equation)23 b(nous)g(a)o(v)o(ons)f(b)q(esoin)i(de)75 2354 y(construire)14 b(les)g(tableaux)g(d'indirections)h fm(x)841 2361 y fg(1)873 2354 y fq(\022)-23 b(a)13 b fm(x)935 2361 y fw(n)971 2354 y fq(utilis)o(\023)-21 b(es)14 b(par)g(hpf.)e(p)o(our)h(cela,)h(nous)f(d)o(\023)-21 b(e\014nissons)75 2410 y(la)16 b(fonction)f fm(index)g fq(qui)h(s)o(\023)-21 b(electionne)16 b(la)f fm(k)812 2394 y fw(i)q fg(\022)-18 b fw(eme)903 2410 y fq(dimension)16 b(d'un)g fm(m)p fq(-uplet)9 b(:)650 2485 y fm(index)k fq(:)41 b fe(n)865 2469 y fw(m)905 2485 y ff(\002)11 b fq([1)d fm(:)g(:)g(:)t(m)p fq(])41 b ff(7!)h fe(n)831 2542 y fq(\(\()p fm(i)883 2549 y fg(1)900 2542 y fm(;)8 b(::;)g(i)984 2549 y fw(m)1013 2542 y fq(\))p fm(;)g(k)q fq(\))53 b ff(!)42 b fm(i)1251 2549 y fw(k)p eop

%%page: 95 97 95 96 bop 75 269 a fl(5.4.)29 b(le)16 b(traducteur)f fk(pei2hpf)1034 b fq(95)75 397 y(les)16 b(tableaux)f fm(x)368 404 y fw(k)389 397 y fm(;)8 b ff(8)p fm(k)13 b ff(2)g fq([1)8 b fm(:)g(:)g(:)t(n)p fq(])15 b(son)o(t)g(construits)g (par)7 b(:)327 495 y ff(8)p fq(\()p fm(i)386 502 y fg(1)404 495 y fm(;)h(:)g(:)g(:)d(;)j(i)522 502 y fw(m)552 495 y fq(\))k ff(2)h fm(dom)p fq(\()p fm(\033)755 502 y fr(x)772 495 y fq(\))p fm(;)8 b(x)837 502 y fw(k)856 495 y fq(\()p fm(i)890 502 y fg(1)908 495 y fm(;)g(:)g(:)g(:)d(;)j(i)1026 502 y fw(m)1057 495 y fq(\))k(=)h fm(index)p fq(\()p fk(gr)o fq(\()p fm(i)1348 502 y fg(1)1366 495 y fm(;)8 b(:)g(:)g(:)d(;)j(i)1484 502 y fw(m)1514 495 y fq(\))p fm(;)g(k)q fq(\))146 592 y(ainsi,)14 b(nous)f(construisons)h(une)g (application)h(de)f(l'ensem)o(ble)h(des)e(\()p fm(i)1338 599 y fg(1)1356 592 y fm(;)8 b(:)g(:)g(:)d(;)j(i)1474 599 y fw(m)1505 592 y fq(\))k ff(2)h fm(dom)p fq(\()p fm(\033)1708 599 y fr(x)1725 592 y fq(\))f(dans)75 649 y(l'ensem)o(ble)j(des)d(\023)-21 b(el)o(\023)g(emen)o(ts)12 b(\()p fm(x)596 656 y fg(1)614 649 y fq(\()p fm(i)648 656 y fg(1)666 649 y fm(;)c(:)g(:)g(:)d(;)j(i)784 656 y fw(m)815 649 y fq(\))p fm(;)g(:)g(:)g(:)t(;)g(x)960 656 y fw(n)982 649 y fq(\()p fm(i)1016 656 y fg(1)1034 649 y fm(;)g(:)g(:)g(:)d(;)j(i)1152 656 y fw(m)1182 649 y fq(\)\),)k(comme)h(sp)o(\023)-21 b(eci\014)o(\023)g(e)13 b(dans)g([27)o(],)g(p.89.)146 705 y(notons)j(que)i(nous)f(sommes)f(ici) j(dans)e(le)h(cas)f(particulier)i(o)q(\022)-24 b(u)17 b fm(n)f fq(=)g fm(m)h fq(puisque)i(les)f(r)o(\023)-21 b(eductions)75 762 y(g)o(\023)g(eom)o(\023)g(etriques)13 b(de)j fn(pei)f fq(son)o(t)f(d)o(\023)-21 b(e\014nies)16 b(par)f(des)h(fonctions)f(de)h fe(z)1204 745 y fw(m)1245 762 y ff(7!)d fe(z)1337 745 y fw(m)1365 762 y fq(.)146 818 y(la)i(traduction)g(de)h(l')o(\023)-21 b(equation)15 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)14 b(est)h(alors)7 b(:)625 908 y fk(y)15 b fq(=)h ff(\012)p 753 908 15 2 v 17 w fk(scatter\(x,)p fm(e)1028 915 y fx(\012)1055 908 y fk(,)p fm(x)1105 915 y fg(1)1123 908 y fk(,)p fm(:)8 b(:)g(:)n fk(,)p fm(x)1250 915 y fw(n)1273 908 y fk(\))146 998 y fq(nous)15 b(traduisons)g(par)g(exemple)h(l')o(\023)-21 b(equation)690 1092 y fk(y)15 b fq(=)g fk(sum)g fa(b)g fk(\(red)24 b fq(;)8 b fa(b)23 b fk(x\))555 1149 y fq(a)o(v)o(ec)50 b fk(sum)14 b fq(=)i fk(id)10 b fq(#)g fm(\025)p fk(\(a)p fq(;)o fk(b\).\(a+sum\(b\)\))690 1205 y(red)k fq(=)i fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(i,j)p ff(\024)p fk(3\).\(i,i\))75 1287 y fq(par)404 1432 y fk(y)f fq(=)h fk(sum)p 569 1432 v 17 w(scatter\(x,0,)893 1335 y fx(2)893 1408 y(6)893 1435 y(4)945 1375 y fk(1)50 b(2)g(3)945 1432 y(1)g(2)g(3)945 1488 y(1)g(2)g(3)1142 1335 y fx(3)1142 1408 y(7)1142 1435 y(5)1169 1432 y fk(,)1217 1335 y fx(2)1217 1408 y(6)1217 1435 y(4)1270 1375 y fk(1)f(2)h(3)1270 1432 y(1)f(2)h(3)1270 1488 y(1)f(2)h(3)1466 1335 y fx(3)1466 1408 y(7)1466 1435 y(5)1493 1432 y fk(\))146 1577 y fq(une)11 b(am)o(\023)-21 b(elioration)10 b(consisterait)h(\022)-23 b(a)10 b(examiner)h(les)g(cas)g(particuliers)h(de)f(r)o(\023)-21 b(eductions)10 b(selon)i(les)f(axes)75 1634 y(canoniques)k(du)f (domaine)h(de)f(r)o(\023)-21 b(ef)o(\023)g(erence,)12 b(a\014n)i(d'utiliser)h(les)g(primitiv)o(es)g(plus)g(simples)g fk(xxx)p 1690 1634 v 16 w(suffix)75 1690 y fq(ou)g fk(xxx)p 213 1690 v 17 w(prefix)p fq(.)75 1809 y fo(d)o(\023)-25 b(e\014nition)19 b(r)o(\023)-25 b(ecursiv)o(e)75 1895

y fq(consid)o(\023)k(erons)15 b(l')o(\023)-21 b(equation)15 b(suiv)m(an)o(te)8 b(:)457 1993 y fk(y)13 b fq(=)g fk(fc)i fa(b)g fk(\(x)24 b(/&/)7 b(\(y)24 b fa(c)g fk(g)961 2000 y fg(1)980 1993 y fk(\)/&/)7 b fm(:)h(:)g(:)22 b fk(/&/)7 b(\(y)24 b fa(c)g fk(g)1394 2000 y fw(n)1417 1993 y fk(\)\))75 2091 y fq(t)l(rouv)o(er)16 b(la)g(solution)i fk(y)e fq(d'une)h(telle)f (\023)-21 b(equation)16 b(revien)o(t)h(donc)f(\022)-23 b(a)16 b(r)o(\023)-21 b(esoudre)16 b(une)g(\023)-21 b(equation)15 b(de)i(p)q(oin)o(t)75 2147 y(\014xe.)h(une)h(traduction)f(na)-5 b(\177)-18 b(\020v)o(e)19 b(consiste)f(\022)-23 b(a)18 b(calculer)i(une)f(suite)f(de)h(c)o(hamps)f fk(y)g fq(don)o(t)g(le)h (dernier)g(est)75 2204 y(solution)e(de)g(l')o(\023)-21 b(equation.)16 b(c'est)f(la)i(solution)g(reten)o(ue)g(dans)g([21)o(],)e (o)q(\022)-24 b(u)17 b(une)g(b)q(oucle)h(s)o(\023)-21 b(equen)o(tielle)18 b(sur)75 2260 y(un)e(indice)h(suppl)o(\023)-21 b(emen)o(taire)16 b(repr)o(\023)-21 b(esen)o(te)14 b(la)i(suite)g(des)f (c)o(hamps)g fk(y)g fq(calcul)o(\023)-21 b(es.)146 2373 y(une)21 b(optimisation)h(consiste)f(\022)-23 b(a)21 b(confondre)h(cette)f(b)q(oucle)h(a)o(v)o(ec)f(la)g(ou)g(les)h (directions)g(repr)o(\023)-21 b(e-)75 2429 y(sen)o(tan)o(t)18 b(le)h(temps,)f(selon)h(que)g(l'on)g(utilise)h(des)f(cadencemen)o(ts)g (mono)f(o)q(\022)-24 b(u)18 b(m)o(ulti-dimensionnel)q(s.)75 2486 y(rapp)q(elons)i(en)e(e\013et)g(qu'une)g(telle)g(\023)-21 b(equation)18 b(n'est)f(traduisible)j(que)f(si)f(elle)i(d)o(\023)-21 b(e\014nit)18 b(un)g(cadence-)75 2542 y(men)o(t)g(des)h(calculs)9 b(:)18 b(les)h(termes)f(successifs)i(de)f(la)f(r)o(\023)-21 b(ecurrence)19 b(doiv)o(en)o(t)f(alors)f(^)-21 b(etre)18 b(calcul)o(\023)-21 b(es)19 b(selon)p eop %%page: 96 98 96 97 bop 75 269 a fq(96)404 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fq(ce)k(cadencemen)o(t.)g(ceci)g (se)f(traduit)h(de)g(mani)o(\022)-21 b(ere)18 b(g)o(\023)-21 b(en)o(\023)g(erale,)16 b(par)j(un)f(blo)q(c)i(constitu)o(\023)-21 b(e,)17 b(par)i(autan)o(t)75 453 y(de)f(b)q(oucles)i(s)o(\023)-21 b(equen)o(tielles)19 b(im)o(briqu)o(\023)-21 b(ees)18 b(qu'il)h(y)f(a)f(de)i(dimensions)g(de)f(temps,)g(et)f(\022)-23 b(a)18 b(l'in)o(t)o(\023)-21 b(erieur)18 b(de)75 510 y(ces)d(b)q(oucles,)g(par)g(une)g(op)o(\023)-21 b(eration)13 b(parall)o(\022)-21 b(ele)15 b fk(forall)f fq(sur)g(les)i(dimensions)g (non)e(temp)q(orelles.)i(l'im-)75 566 y(brication)i(des)g(b)q(oucles)g (supp)q(ose)g(que)g(l'indice)i(d'une)d(b)q(oucle)i(soit)e(ind)o(\023) -21 b(ep)q(endan)o(t)18 b(des)g(indices)h(des)75 623 y(b)q(oucles)h(in)o(ternes.)f(si)h(ce)f(n'est)f(pas)h(le)g(cas,)f(on)h (r)o(\023)-21 b(esout)17 b(classiquemen)o(t)j(ce)f(probl)o(\022)-21 b(eme)19 b(par)f(l'algo-)75 679 y(rithme)e(d')o(\023)-21 b(elimination)16 b(de)f(f)l(ourier.)146 794 y(la)20 b(traduction)g(est) g(constitu)o(\023)-21 b(ee)20 b(de)h(deux)g(blo)q(cs)g(d'instructions.) g(le)f(premier)h(e\013ectue)g(l'ini-)75 850 y(tialisation)d(de)g fk(y)f fq(a)o(v)o(ec)f(les)i(v)m(aleurs)g(de)f fk(x)g fq(utilis)o(\023)-21 b(ees)18 b(au)f(premier)h(instan)o(t)f(de)h (calcul,)g(tandis)f(que)h(le)75 906 y(deuxi)o(\022)-21 b(eme)16 b(est)f(une)h(b)q(oucle)g(s)o(\023)-21 b(equen)o(tielle)16 b(\023)-21 b(en)o(um)o(\023)g(eran)o(t)13 b(les)j(di\013)o(\023)-21 b(eren)o(ts)14 b(instan)o(ts)h(de)h(calcul.)146 1021 y(dans)f(l')o(\023)-21 b(equation)15 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te,)14 b(les)i(fonctions)f fk(g)995 1028 y fw(k)1016 1021 y fm(;)8 b(k)13 b ff(2)g fq([1)8 b fm(:)g(:)g(:)d(n)p fq(])15 b(d)o(\023)-21 b(e\014nies)16 b(de)g fe(z)1532 1004 y fw(m)1576 1021 y fq(dans)f fe(z)1716

1004 y fw(m)1760 1021 y fq(p)q(eu-)75 1077 y(v)o(en)o(t)g(s')o(\023)-21 b(ecrire)577 1138 y fk(g)601 1145 y fw(k)634 1138 y fq(=)13 b(\()p fm(f)722 1145 y fw(k)q fg(1)759 1138 y fq(\()p fm(i)793 1145 y fg(1)811 1138 y fm(;)8 b(:)g(:)g(:)t(i)907 1145 y fw(m)939 1138 y fq(\))p fm(;)g(:)g(:)g(:)t(;)g(f)1080 1145 y fw(k)q(m)1129 1138 y fq(\()p fm(i)1163 1145 y fg(1)1181 1138 y fm(;)g(:)g(:)g(:)d(i)1278 1145 y fw(m)1309 1138 y fq(\)\))146 1228 y fn(pei)19 b fq(autorise)h(l'utilisation,)h (dans)f(une)g(m)o(^)-21 b(eme)19 b(d)o(\023)-21 b(e\014nition)20 b(r)o(\023)-21 b(ecursiv)o(e,)19 b(de)h(c)o(hamps)g(a)o(y)o(an)o(t)e (des)75 1284 y(domaines)e(de)g(v)m(aleurs)g(di\013)o(\023)-21 b(eren)o(ts.)14 b(la)h(cons)o(\023)-21 b(equence)16 b(est)f(qu'il)h (est)f(di\016cile)j(de)e(d)o(\023)-21 b(eterminer)15 b(quelles)75 1341 y(v)m(aleurs,)j(pro)o(v)o(enan)o(t)f(des)i(di\013)o (\023)-21 b(eren)o(ts)17 b(c)o(hamps)g(sup)q(erp)q(os)o(\023)-21 b(es,)18 b(son)o(t)f(utilis)o(\023)-21 b(ees)19 b(\022)-23 b(a)18 b(c)o(hacune)g(des)f(\023)-21 b(etap)q(es)75 1397 y(de)19 b(la)h(r)o(\023)-21 b(ecurrence.)18 b(le)h(probl)o(\022)-21 b(eme)19 b(se)g(p)q(ose)h(notammen)o(t)d(p)q(our)j(la)f(traduction)g (hpf)7 b(:)19 b(l'op)o(\023)-21 b(erateur)75 1454 y fk(fc)21 b fq(est)g(d)o(\023)-21 b(e\014ni)21 b(en)h fn(pei)f fq(\(comme)f(dans)i(tout)e(langage)h(fonctionnel\))h(sur)f(une)h(liste) g(de)g(longueur)75 1510 y(quelconque,)13 b(alors)e(qu'en)g(hpf,)g(nous) g(devrons)h(distinguer)g(les)g(cas)f(p)q(our)h(lesquels)h(l'op)o(\023) -21 b(erateur)10 b(a)h(un)75 1566 y(nom)o(bre)j(di\013)o(\023)-21 b(eren)o(t)14 b(d'argumen)o(ts.)e(le)j(probl)o(\022)-21 b(eme)14 b(se)h(r)o(\023)-21 b(esume)13 b(\022)-23 b(a)14 b(d)o(\023)-21 b(ecomp)q(oser)14 b(l'union)h(des)g(domaines)75 1623 y(de)21 b(v)m(aleurs)g(utilis)o(\023)-21 b(es,)21 b(en)g(sous-domaines)g(sur)f(lesquels)j(le)e(nom)o(bre)f(de)h(v)m (aleurs)g(sup)q(erp)q(os)o(\023)-21 b(ees)21 b(est)75 1679 y(constan)o(t.)9 b(un)i(tel)g(algorithme)f(de)h(d)o(\023)-21 b(ecomp)q(osition)11 b(a)f(par)g(exemple)h(\023)-21 b(et)o(\023)g(e)8 b(implan)o(t)o(\023)-21 b(e)10 b(p)q(our)h(un)g(probl)o(\022)-21 b(eme)75 1736 y(similaire)15 b(dans)d(une)h(v)o(ersion)g(r)o(\023)-21 b(ecen)o(te)12 b(de)h(la)g(librairie)h(p)o(olylib[63)q(].)e(nous)g(ne)h (nous)g(in)o(t)o(\023)-21 b(eressons)11 b(p)q(our)75 1792 y(l'instan)o(t)20 b(qu'\022)-23 b(a)20 b(deux)h(cas)g (particuliers)h(corresp)q(ondan)o(ts)e(\022)-23 b(a)20 b(la)g(grande)h(ma)s(jorit)o(\023)-21 b(e)18 b(des)j(probl)o(\022)-21 b(emes)75 1849 y(rencon)o(tr)o(\023)g(es)14 b(dans)h(la)h(pratique)8 b(:)143 1948 y({)23 b(nous)10 b(consid)o(\023)-21 b(erons)10 b(le)h(cas)f(o)q(\022)-24 b(u)10 b(le)h(domaine)g(des)f(v)m(aleurs)h (du)g(c)o(hamp)f(initial)i(est)e(disjoin)o(t)h(de)g(celui)189 2005 y(des)17 b(op)o(\023)-21 b(erations)15 b(g)o(\023)-21 b(eom)o(\023)g(etriques)15 b(appliqu)o(\023)-21 b(ees)18 b(r)o(\023)-21 b(ecursiv)o(emen)o(t.)15 b(c'est)h(le)i(cas)e(de)h(l')o (\023)-21 b(elimination)189 2061 y(de)12 b(gauss)g(par)g(exemple,)h(o)q (\022)-24 b(u)12 b(les)g(v)m(aleurs)i(du)e(c)o(hamp)g(initial)i(son)o (t)e(uniquemen)o(t)h(utilis)o(\023)-21 b(ees)13 b(p)q(our)189 2118 y(d)o(\023)-21 b(e\014nir)14 b(le)g(premier)g(pas)g(de)g(la)f(r)o (\023)-21 b(ecurrence.)13 b(dans)h(ce)f(cas)h(l'op)o(\023)-21 b(erateur)12 b(dans)i(la)g(r)o(\023)-21 b(ecurrence)13 b(ne)189 2174 y(p)q(orte)i(pas)g(sur)g(les)h(v)m(aleurs)g(du)g(c)o (hamp)f(initial.)143 2273 y({)23 b(le)13 b(deuxi)o(\022)-21 b(eme)13 b(cas)g(consid)o(\023)-21 b(er)o(\023)g(e)12 b(est)g(celui)j(o)q(\022)-24 b(u)13 b(les)g(v)m(aleurs)h(du)f(c)o(hamp) g(initial)i(son)o(t)d(utilis)o(\023)-21 b(ees)13 b(tout)189 2330 y(au)j(long)g(de)g(la)g(r)o(\023)-21 b(ecurrence.)16 b(un)g(exemple)h(t)o(ypique)g(est)f(la)g(somme)f(pr)o(\023)-21

b(e\014xe)16 b(d'un)g(v)o(ecteur)g(de)189 2386 y(nom)o(bres.)h(dans)h (ce)g(cas)g(l'op)o(\023)-21 b(erateur)17 b(utilise)j(\022)-23 b(a)17 b(toute)g(\023)-21 b(etap)q(e)17 b(de)i(la)f(r)o(\023)-21 b(ecurrence)18 b(une)h(v)m(aleur)189 2443 y(du)c(v)o(ecteur)h(initial.) 146 2542 y(dans)f(le)h(cas)f(particulier)i(d'un)f(cadencemen)o(t)g (a\016ne)f(mono-dimensionnel,)j(supp)q(osons)e(que)f(la)p eop %%page: 97 99 97 98 bop 75 269 a fl(5.4.)29 b(le)16 b(traducteur)f fk(pei2hpf)1034 b fq(97)75 397 y(dimension)17 b fm(i)306 404 y fg(1)340 397 y fq(soit)f(la)f(dimension)j(temp)q(orelle.)e(les)g (fonctions)g fm(f)1227 404 y fw(k)q fg(1)1280 397 y fq(repr)o(\023)-21 b(esen)o(ten)o(t)14 b(alors)i(les)g(d)o(\023)-21 b(ep)q(en-)75 453 y(dances)16 b(temp)q(orelles)g(p)q(our)g(tout)e fm(k)g ff(2)f fq([1)8 b fm(:)g(:)g(:)t(n)p fq(].)15 b(ces)g(fonctions)h(son)o (t)e(soit)h(de)h(la)f(forme)669 572 y fm(f)5 b fq(\()p fm(i)730 579 y fg(1)748 572 y fm(;)j(:)g(:)g(:)d(;)j(i)866 579 y fw(m)896 572 y fq(\))k(=)h fm(f)1001 553 y fx(0)1013 572 y fq(\()p fm(i)1047 579 y fg(1)1065 572 y fq(\))f(=)h fm(i)1159 579 y fg(1)1188 572 y ff(\000)d fm(c)75 691 y fq(o)q(\022)-24 b(u)15 b fm(c)g fq(est)g(une)h(constan)o(te)e(p)q (ositiv)o(e,)i(soit)f(de)h(la)f(forme)757 809 y fm(f)779 816 y fw(k)q fg(1)816 809 y fq(\()p fm(i)850 816 y fg(1)868 809 y fm(;)8 b(:)g(:)g(:)d(;)j(i)986 816 y fw(m)1016 809 y fq(\))13 b(=)g fm(exp)75 928 y fq(o)q(\022)-24 b(u)12 b fm(exp)f fq(a)h(une)g(v)m(aleur)h(ind)o(\023)-21 b(ep)q(endan)o(te)13 b(de)f fm(i)815 935 y fg(1)833 928 y fq(,)g(mais)f(est)h(inf)o(\023)-21 b(erieure)13 b(\022)-23 b(a)11 b fm(i)1276 935 y fg(1)1306 928 y fq(p)q(our)h(tout)f fm(i)1521 935 y fg(1)1551 928 y fq(dans)h fm(dom)p fq(\()p fm(f)1780 935 y fw(k)q fg(1)1816 928 y fq(\).)146 1046 y(p)o(our)j(nous)h(situer)h(dans)f(un)h(cas)f(simple)h(\()o(\023)-21 b(evitan)o(t)15 b(la)h(d)o(\023)-21 b(ecomp)q(osition)17 b(des)f(domaines)g(\023)-21 b(ev)o(o)q(qu)o(\023)g(ee)75 1103 y(pr)o(\023)g(ec)o(\023)g(edemmen)o(t\),)10 b(supp)q(osons)j(que)g (la)f(d)o(\023)-21 b(ep)q(endance)13 b(temp)q(orelle)h(est)e fm(f)1306 1086 y fx(0)1318 1103 y fq(\()p fm(i)1352 1110 y fg(1)1370 1103 y fq(\))g(=)h fm(i)1464 1110 y fg(1)1486 1103 y ff(\000)t fm(c)g fq(p)q(our)f(toutes)g(les)75 1159 y(fonctions)h fk(g)290 1166 y fw(k)310 1159 y fq(.)g(nous)f(c)o (herc)o(hons)h(les)h(b)q(ornes)f(inf)o(\023)-21 b(erieures)13 b(et)g(sup)o(\023)-21 b(erieures)13 b(de)g(la)g(b)q(oucle)i(s)o(\023) -21 b(equen)o(tielle,)75 1216 y(que)16 b(nous)f(app)q(elons)h fm(i)470 1223 y fg(1)487 1227 y fd(min)561 1216 y fq(et)f fm(i)630 1223 y fg(1)647 1227 y fd(max)723 1216 y fq(resp)q(ectiv)o (emen)o(t.)146 1278 y(l'initialisation)20 b(de)d(certaines)h(v)m (aleurs)h(de)e fk(y)h fq(a)o(v)o(ec)f(celles)i(du)e(c)o(hamp)h(initial) h fk(x)e fq(est)h(n)o(\023)-21 b(ecessaire)75 1334 y(p)q(our)15 b(amorcer)f(la)h(r)o(\023)-21 b(ecurrence.)14 b(les)g(\023)-21 b(el)o(\023)g(emen)o(ts)13 b(de)j(tableau)f(de)g fk(y)f fq(dev)m(an)o(t)g(^)-21 b(etre)14 b(initialis)o(\023)-21 b(es)16 b(son)o(t)e(ceux)75 1391 y(r)o(\023)-21 b(ef)o(\023)g(erenc)o (\023)g(es)19 b(\022)-23 b(a)20 b(l'instan)o(t)h fm(i)533 1398 y fg(1)574 1391 y fq(=)h fm(i)647 1398 y fg(1)664 1402 y fd(min)722 1391 y fq(.)f(soit)g fm(t)871 1398 y fg(0)912 1391 y fq(=)i fm(f)997 1374 y fx(0)1009 1391 y fq(\()p fm(i)1043 1398 y fg(1)1060 1402 y fd(min)1118 1391 y fq(\))d(l'instan)o(t)h(pr)o(\023)-21 b(ec)o(\023)g(edan)o(t)19 b(le)j(premier)g(pas)75 1447 y(de)c(calcul)g(de)g(la)f(r)o(\023)-21 b(ecurrence.)17 b(p)o(our)g(tout)f fm(k)h ff(2)f fq([1)8

b fm(:)g(:)g(:)d(n)p fq(],)17 b(nous)g(d)o(\023)-21 b(esignons)17 b(par)g ff(d)1525 1454 y fw(k)1563 1447 y fq(l'ensem)o(ble)h(des)75 1504 y(indices)f(r)o(\023)-21 b(ef)o(\023)g(erenc)o(\023)g(es)13 b(par)i(une)h(d)o(\023)-21 b(ep)q(endance)16 b fk(g)870 1511 y fw(k)906 1504 y fq(\022)-23 b(a)14 b(l'instan)o(t)i fm(i)1138 1511 y fg(1)1155 1515 y fd(min)1220 1504 y fq(:)412 1622 y ff(d)447 1629 y fw(k)480 1622 y fq(=)d(\()p fm(t)562 1629 y fg(0)581 1622 y fm(;)8 b(f)624 1629 y fw(k)q fg(2)660 1622 y fq(\()p fm(i)694 1629 y fg(1)711 1633 y fd(min)769 1622 y fm(;)g(i)806 1629 y fg(2)824 1622 y fm(;)g(:)g(:)g(:)t(;)g(i)941 1629 y fw(m)972 1622 y fq(\))p fm(;)g(:)g(:)g(:)t(;)g(f)1113 1629 y fw(k)q(m)1163 1622 y fq(\()p fm(i)1197 1629 y fg(1)1214 1633 y fd(min)1272 1622 y fm(;)g(i)1309 1629 y fg(2)1326 1622 y fm(;)g(:)g(:)g(:)d(;)j(i) 1444 1629 y fw(m)1474 1622 y fq(\)\))75 1741 y(l'ensem)o(ble)14 b(des)f(indices)i(r)o(\023)-21 b(ef)o(\023)g(erenc)o(\023)g(es)10 b(\022)-23 b(a)12 b(l'instan)o(t)h fm(i)963 1748 y fg(1)994 1741 y fq(=)g fm(i)1058 1748 y fg(1)1075 1752 y fd(min)1146 1741 y fq(est)f(alors)h ff(d)h fq(=)f ff(d)1453 1748 y fg(1)1477 1741 y ff([)5 b fm(:)j(:)g(:)s ff([)d(d)1640 1748 y fw(n)1663 1741 y fq(.)12 b(t)l(ous)h(les)74 1797 y(\023)-21 b(el)o(\023)g(emen)o(ts)9 b(de)j fk(y)e fq(appartenan)o(t)g (\022)-23 b(a)11 b ff(d)h fq(doiv)o(en)o(t)f(donc)f(^)-21 b(etre)10 b(initialis)o(\023)-21 b(es)12 b(a)o(v)o(ec)e(les)i(v)m (aleurs)g(corresp)q(ondan)o(tes)75 1854 y(du)k(c)o(hamp)f(initial)i fk(x)p fq(.)146 1916 y(dans)23 b(la)i(suite)g(de)g(la)f(r)o(\023)-21 b(ecurrence,)24 b(nous)h(consid)o(\023)-21 b(erons)24 b(les)h(deux)g(cas)e(\023)-21 b(ev)o(o)q(qu)o(\023)g(es)23 b(pr)o(\023)-21 b(ec)o(\023)g(edem-)75 1972 y(men)o(t)7 b(:)19 b(le)h(c)o(hamp)f(initial)i fk(x)e fq(ne)g(doit)h(pas)f(appara) -5 b(^)-18 b(\020tre)18 b(dans)h(l'instruction)i(si)e(il)i(n'est)d (utilis)o(\023)-21 b(e)20 b(que)75 2029 y(par)k(le)h(premier)g(calcul)g (de)g(la)f(r)o(\023)-21 b(ecurrence,)24 b(ce)h(qui)g(se)f(traduit)g (par)g(la)g(condition)i fm(dom)p fq(\()p fm(v)1766 2036 y fr(x)1783 2029 y fq(\))15 b ff(\\)75 2085 y fq(\()p fm(dom)p fq(\()p fk(g)221 2092 y fg(1)238 2085 y fq(\))h ff([)g fm(:)8 b(:)g(:)14 b ff([)i fm(dom)p fq(\()p fk(g)561 2092 y fw(n)583 2085 y fq(\)\))27 b(=)g ff(fg)p fq(.)c(dans)h(le)g (deuxi)o(\022)-21 b(eme)25 b(cas)f(de)g(\014gure)g(qui)h(se)f(traduit)g (par)75 2142 y(\()p fm(dom)p fq(\()p fk(g)221 2149 y fg(1)238 2142 y fq(\))14 b ff([)g fm(:)8 b(:)g(:)k ff([)i fm(dom)p fq(\()p fk(g)553 2149 y fw(n)575 2142 y fq(\)\))21 b ff(\032)h fm(dom)p fq(\()p fm(v)815 2149 y fr(x)832 2142 y fq(\),)e(il)i(appara)-5 b(^)-18 b(\020t)20 b(dans)h (l'instruction.)h(dans)e(les)i(instruc-)75 2198 y(tions)f(hpf)g(suiv)m (an)o(tes)g(donnan)o(t)g(la)g(forme)g(g)o(\023)-21 b(en)o(\023)g(erale) 19 b(de)i(la)h(traduction,)e(le)i(c)o(hamp)f fk(x)g fq(est)f(donc)75 2254 y(pr)o(\023)-21 b(esen)o(t)14 b(ou)h(pas)g(selon)h(le)g(cas.)146 2373 y(notons)f(que)h(la)g(restriction)h(don)o(t)e(nous)h(a)o(v)o(ons)f (parl)o(\023)-21 b(e)15 b(en)i(section)f(5.1.1)e(nous)i(oblige)h(\022) -23 b(a)16 b(utiliser)75 2429 y(des)f(blo)q(cs)h fk(forall)d fq(p)q(our)i(un)g(sc)o(h)o(\023)-21 b(ema)14 b(g)o(\023)-21 b(en)o(\023)g(eral)13 b(de)i(traduction.)f(nous)h(supp)q(osons)g(dans)g (ce)g(qui)g(suit)75 2486 y(que)i(l'ordre)f(d'im)o(brication)h(des)g fk(forall)e fq(est)i(tel)f(qu'un)h(indice)h(d)o(\023)-21 b(ep)q(end)17 b(uniquemen)o(t)h(des)f(indices)75 2542 y(apparaissan)o(t)e(dans)g(les)h fk(forall)e fq(engloban)o(ts.)p eop %%page: 98 100 98 99 bop 75 269 a fq(98)404 b fl(chapitre)16 b(5.)30

b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fk(c)24 b(initialisation)170 453 y(forall)f(\()p fm(i)377 460 y fg(1)408 453 y fq(=)13 b fm(f)483 437 y fx(0)495 453 y fq(\()p fm(i)529 460 y fg(1)546 464 y fd(min)604 453 y fq(\))p fk(\))242 510 y(forall)23 b fq(\()p fm(i)443 517 y fg(2)474 510 y fq(=)13 b ff(d)557 517 y fg(2)575 510 y fk(\))242 566 y fm(:)8 b(:)g(:)314 623 y fk(forall)23 b(\()p fm(i)521 630 y fw(m)564 623 y fq(=)13 b ff(d)647 630 y fw(m)679 623 y fk(\))170 679 y(&)191 b(y\()p fm(i)449 686 y fg(1)467 679 y fm(;)8 b(i)504 686 y fg(2)522 679 y fm(;)g(:)g(:)g(:)d(;)j(i)640 686 y fw(m)670 679 y fk(\))24 b(=)f(x\()p fm(i)829 686 y fg(1)847 679 y fm(;)8 b(i)884 686 y fg(2)902 679 y fm(;)g(:)g(:)g(:)d(;)j(i)1020 686 y fw(m)1050 679 y fk(\))242 736 y fm(:)g(:)g(:)242 792 y fk(endforall)170 849 y(endforall)75 905 y(!)24 b(recurrence)170 962 y(do)g fm(i)258 969 y fg(1)300 962 y fk(=)g fm(i)364 969 y fg(1)381 973 y fd(min)439 962 y fk(,)p fm(i)479 969 y fg(1)496 973 y fd(max)242 1018 y fk(forall)f(\()p fm(i)449 1025 y fg(2)480 1018 y fq(=)13 b fm(dom)p fq(\()p fm(f)654 1025 y fg(12)689 1018 y fq(\))p fm(;)8 b(:)g(:)g(:)t(;)g(i)824 1025 y fw(m)867 1018 y fq(=)13 b fm(dom)p fq(\()p fm(f)1041 1025 y fw(n)p fg(2)1080 1018 y fq(\))c ff(\\)i fm(dom)p fk(\(fc\)\))242 1074 y fm(:)d(:)g(:)314 1131 y fk(forall)23 b fq(\()p fm(i)515 1138 y fw(m)558 1131 y fq(=)13 b fm(dom)p fq(\()p fm(f)732 1138 y fg(1)p fw(m)780 1131 y fq(\))d ff([)g fm(:)e(:)g(:)g ff([)i fm(dom)p fq(\()p fm(f)1077 1138 y fw(nm)1129 1131 y fq(\))p fk(\))170 1187 y(&)191 b(y\()p fm(i)449 1194 y fg(1)467 1187 y fm(;)8 b(i)504 1194 y fg(2)522 1187 y fm(;)g(:)g(:)g(:)d(;)j(i)640 1194 y fw(m)670 1187 y fk(\))24 b(=)170 1244 y(&)191 b(fc\(x\()p fm(i)521 1251 y fg(1)539 1244 y fm(;)8 b(i)576 1251 y fg(2)593 1244 y fm(;)g(:)g(:)g(:)d(;)j(i)711 1251 y fw(m)742 1244 y fk(\),y\(g)862 1251 y fg(1)879 1244 y fk(\()p fm(i)919 1251 y fg(1)938 1244 y fm(;)g(i)975 1251 y fg(2)992 1244 y fm(;)g(:)g(:)g(:)d(;)j(i)1110 1251 y fw(m)1140 1244 y fk(\)\))p fm(;)g(:)g(:)g(:)d(;)23 b fk(y\(g)1377 1251 y fw(n)1399 1244 y fk(\()p fm(i)1439 1251 y fg(1)1458 1244 y fm(;)8 b(i)1495 1251 y fg(2)1512 1244 y fm(;)g(:)g(:)g(:)d(;)j (i)1630 1251 y fw(m)1660 1244 y fk(\)\)\))314 1300 y fm(:)g(:)g(:)242 1357 y fk(endforall)170 1413 y(enddo)146 1557 y fq(nous)13 b(p)q(ouv)o(ons)h(illustrer)i(simplemen)o(t)f(ce)f (sc)o(h)o(\023)-21 b(ema)13 b(de)h(traduction)g(a)o(v)o(ec)f(l'exemple) i(de)g(la)f(di\013u-)75 1614 y(sion)e(discr)o(\022)-21 b(ete)12 b(pr)o(\023)-21 b(esen)o(t)o(\023)g(e)10 b(sur)i(la)g (\014gure)g(5.2.)e(de)i(l')o(\023)-21 b(equation)12 b(\(2\),)e(page)i (86,)f(p)q(our)h(laquelle)i(on)d(a)h(trouv)o(\023)-21 b(e)75 1670 y(un)18 b(cadencemen)o(t,)h(nous)f(dev)o(ons)g(construire)g (une)h(b)q(oucle)g(s)o(\023)-21 b(equen)o(tielle)20 b(a)o(y)o(an)o(t)c (p)q(our)i(indice)i fk(t)p fq(,)e(et)75 1727 y(donc)e(faire)f(de)h fk(t)f fq(une)g(v)m(ariable)i(ind)o(\023)-21 b(ep)q(endan)o(te.)16 b(ici,)g(les)g(expressions)g(de)g fk(t)f fq(et)g(de)g fk(p)g fq(deviennen)o(t)9 b(:)245 1894 y fx(\()303 1937 y fq(1)j ff(\024)h fk(t)d ff(\000)h fk(p)h ff(\024)h fm(n)303 1994 y fq(1)f ff(\024)h fk(p)g ff(\024)g fm(n)630 1966 y fq(c'est-\022)-23 b(a-dire)901 1894 y fx(\()959 1937 y fq(2)13 b ff(\024)g fk(t)f ff(\024)h fq(2)p fm(n)959 1994 y(max)p fq(\(1)p fm(;)8 b fk(t)h ff(\000)h fm(n)p fq(\))j ff(\024)g fk(p)f ff(\024)h fm(min)p fq(\()p fm(n;)8 b fk(t)h ff(\000)i fq(1\))75 2204 y(l'instan)o(t)16 b(initial)i(est)e

(donn)o(\023)-21 b(e)16 b(par)g(la)g(b)q(orne)h(inf)o(\023)-21 b(erieure)17 b(de)f(l'in)o(terv)m(alle)j(des)d(v)m(aleurs)h(de)g (l'indice)h fk(t)p fq(,)75 2260 y(c'est-\022)-23 b(a-dire)15 b fk(t)p fq(=2,)f(et)h(l'instan)o(t)f(pr)o(\023)-21 b(ec)o(\023)g(eden) o(t)14 b(est)g fm(t)938 2267 y fg(0)970 2260 y fq(=)e(1.)i(au)h (premier)h(pas)e(de)h(la)g(r)o(\023)-21 b(ecurrence)14 b(on)h(fait)75 2316 y(r)o(\023)-21 b(ef)o(\023)g(erence)17 b(aux)h(indices)i(\()p fm(t)545 2323 y fg(0)564 2316 y fq(,)d fk(p-1)p fq(\))h(et)f(\()p fm(t)791 2323 y fg(0)810 2316 y fq(,)h fk(p)p fq(\))f(p)q(our)i fm(max)p fq(\(1)p fm(;)8 b(t)1179 2323 y fg(0)1196 2316 y fq(\))17 b ff(\024)h fk(p)f ff(\024)h fm(min)p fq(\()p fm(n;)8 b(t)1543 2323 y fg(0)1561 2316 y fq(\)\),)17 b(soit)h fk(p)g fq(=)g(1.)75 2373 y(p)o(our)12 b(les)h(deux)g(fonctions)g fk(g1)f fq(et)h fk(g2)p fq(,)f(on)g(doit)h(donc)g(initialiser)i(la)e(v)m (ariable)h fk(y)e fq(aux)h(indices)h ff(f)p fq(\(1)p fm(;)8 b fq(1\))p ff(g)d([)75 2429 y(f)p fq(\(1)p fm(;)j fq(0\))p ff(g)p fq(.)j(si,)i(du)h(reste)g(de)g(l')o(\023)-21 b(enonc)o(\023)g(e,)12 b(nous)h(d)o(\023)-21 b(eduisons)14 b(que)g fm(dom)p fq(\()p fm(v)1246 2436 y fr(x)1263 2429 y fq(\))7 b ff(\\)g fq(\()p fm(dom)p fq(\()p fk(g1)n fq(\))g ff([)g fm(dom)p fq(\()p fk(g2)n fq(\)\))12 b(=)h ff(fg)75 2486 y fq(alors)20 b(l'instruction)h(in)o(terne)g(\022)-23 b(a)20 b(la)h(b)q(oucle)g(sera)f fk(y'\(t,p\))j(=)h (f\(y'\(t-1,p-1\),y'\(t-1,p\)\))o fq(.)17 b(si,)75 2542 y(par)e(con)o(tre)g(\()p fm(dom)p fq(\()p fk(g1)n fq(\))10 b ff([)h fm(dom)p fq(\()p fk(g2)o fq(\)\))h ff(\032)h fm(dom)p fq(\()p fm(v)904 2549 y fr(x)920 2542 y fq(\)\),)h(alors)h(l') o(\023)-21 b(equation)15 b(\(2\))g(est)f(traduite)i(par)7 b(:)p eop %%page: 99 101 99 100 bop 75 269 a fl(5.5.)29 b(exemple)15 b(1)7 b(:)483 257 y(\023)477 269 y(elimina)l(tion)17 b(de)e(ga)o(uss)758 b fq(99)195 397 y fk(forall)23 b(\(t=1,p=0:1\))f(y'\(t,p\))h(=)h (x\(t,p\))195 453 y(do)g(t)f(=)h(2,2*n)243 510 y(forall)f (\(p=max\(1,t-n\):min\(n,t-1\)\))171 566 y(&)120 b(y'\(t,p\))23 b(=)h(f\(x'\(t,p\),y'\(t-1,p-1\),y)o('\(t-1,p)o(\)\))195 623 y(enddo)75 810 y ft(5.5)66 b(exemple)24 b(1)11 b(:)21 b(\023)-32 b(elimination)24 b(de)e(gauss)146 912 y fq(nous)12 b(reprenons)h(ici)i(l'algorithme)e(pr)o(\023)-21 b(esen)o(t)o(\023)g(e) 11 b(sous)h(la)h(forme)f(du)i(programme)d fn(pei)i fq(de)g(la)g (section)75 968 y(4.1.)21 b(il)j(nous)e(p)q(ermet)h(notammen)o(t)e (d'appliquer)j(la)f(m)o(\023)-21 b(etho)q(de)22 b(de)g(traduction)h (des)g(r)o(\023)-21 b(ecurrences)75 1025 y(exp)q(os)o(\023)g(ee)17 b(pr)o(\023)-21 b(ec)o(\023)g(edemmen)o(t.)16 b(p)o(our)h(cet)h (exemple,)h(nous)e(constatons)g(que)h(l'union)h(des)f(domaines)g(de)75 1081 y(v)m(aleurs)13 b(des)f(c)o(hamps)g(initiaux)h fk(x)f fq(et)g fk(y)f fq(est)h(disjon)o(te)g(de)g(l'union)h(des)f(domaines)h (des)f(routages)f(utilis)o(\023)-21 b(es)75 1138 y(dans)15 b(la)g(r)o(\023)-21 b(ecurrence.)15 b(la)f(r)o(\023)-21 b(ecurrence)15 b(n'utilise)i(donc)e(des)g(v)m(aleurs)h(initiales)h (qu'au)e(premier)g(pas)g(de)75 1194 y(calcul,)h(et)f(seul)i(le)f(c)o (hamp)f(d)o(\023)-21 b(e\014ni)16 b(r)o(\023)-21 b(ecursiv)o(emen)o(t) 14 b(appara)-5 b(^)-18 b(\020t)15 b(dans)g(le)h(calcul.)146 1251 y(dans)g(un)i(but)g(de)f(concision,)i(nous)e(a)o(v)o(ons)f (fusionn)o(\023)-21 b(e)18 b(les)g fk(forall)e fq(sur)h(les)h (dimensions)h fk(i)e fq(et)g fk(j)p fq(,)75 1307 y(ces)e(deux)h (indices)h(\023)-21 b(etan)o(t)13 b(ind)o(\023)-21 b(ep)q(endan)o(ts)16 b(l'un)g(par)f(rapp)q(ort)g(\022)-23 b(a)14 b(l'autre.)242 1413 y fk(program)23 b(gauss)242 1470 y(real)g(c\(1:n,1:n+1,0:n\),)f

(xn\(1:n,1:n,1:n\),)g(yn\(1:n,n+1:n+1,1:n\))242 1526 y(real)h(x\(1:n,1:n,0:0\),)f(y\(1:n,n+1:n+1,0:0\))242 1583 y(real)h(a\(1:n,1:n\),)g(b\(1:n\))75 1695 y(!hpf$)47 b(template)23 b(t_c\(1:n,1:n+1,0:n\))75 1752 y(!hpf$)47 b(align)23 b(xn\(i,j,k\),yn\(i,j,k\),)e(c\(i,j,k\))i(with)g (t_c\(i,j,k\))75 1808 y(!hpf$)47 b(align)23 b(a\(i,j\))g(with)h (t_c\(i,j,0\))75 1865 y(!fpf$)47 b(align)23 b(b\(i\))h(with)f (t_c\(i,n+1,0\))242 1978 y(forall)g(\(i=1:n,j=1:n\))f (x\(i,j,0\)=a\(i,j\))242 2034 y(forall)h(\(i=1:n\))g (y\(i,n+1,0\)=b\(i\))75 2147 y(c)h(initialisation)e(recurrence)242 2204 y(forall)h(\(k=0,i=1:n,j=1:n\))f(c\(i,j,k\))g(=)i(x\(i,j,k\))242 2260 y(forall)f(\(k=0,i=1:n,j=n+1\))f(c\(i,j,k\))g(=)i(y\(i,j,k\))75 2316 y(c)g(recurrence)242 2373 y(do)g(k=1,n-1)314 2429 y(forall)f(\(i=k:n,j=k:n\))242 2486 y(&)143 b(c\(i,j,k\))23 b(=)h(c\(i,j,k-1\)-c\(i,k,k-1\)*c)o(\(k,j,k-)o(1\)/c\(k,k)o(,k-1\))242 2542 y(enddo)p eop %%page: 100 102 100 101 bop 75 269 a fq(100)381 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)242 453 y fk(forall)23 b (\(i=1:n,j=1:n,k=1:n,k)e(.eq.)j(i-1\))f(xn\(i,j,k\))f(=)i(c\(i,j,k\)) 242 510 y(forall)f(\(i=1:n,j)g(.eq.)g(n+1)h(.and.)f(k)h(.eq.)f(i-1\))g (yn\(i,j,k\))g(=)h(c\(i,j,k\))242 566 y(end)75 709 y ft(5.6)66 b(exemple)24 b(2)11 b(:)22 b fb(gaxpx)146 811 y fq(nous)g(traduisons)h(ici)h(les)f(programmes)e fk(gaxpx1)h fq(et)g fk(gaxpx4)g fq(de)h(la)g(section)g(4.6.)e(ces)i(pro-)75 867 y(grammes)16 b(illustren)o(t)j(plus)f(particuli)o(\022)-21 b(eremen)o(t)19 b(la)e(traduction)h(des)f(r)o(\023)-21 b(eductions)18 b(g)o(\023)-21 b(eom)o(\023)g(etriques.)15 b(no-)75 924 y(tons)21 b(que)g(nous)g(a)o(v)o(ons)f(instanci)o(\023)-21 b(e)22 b(le)g(param)o(\022)-21 b(etre)19 b(de)i(structure)g(du)h(probl) o(\022)-21 b(eme)21 b(\()p fm(n)h fq(=)h(5\))d(car)h(les)75 980 y(tableaux)e(n)o(\023)-21 b(ecessaires)19 b(\022)-23 b(a)18 b fk(sum)p 612 980 15 2 v 17 w(scatter)f fq(ne)i(doiv)o(en)o(t)g (con)o(tenir)g(que)g(des)g(constan)o(tes.)e(la)i(premier)75 1036 y(programme)h(est)h(la)h(traduction)g(de)f(l'algorithme)h (initial,)h(qui)f(place)h(le)f(v)o(ecteur)f fk(x)g fq(sur)h(la)f(pre-) 75 1093 y(mi)o(\022)-21 b(ere)19 b(ligne)i(de)f(la)g(matrice,)g(et)f(n) o(\023)-21 b(ecessite)20 b(ensuite)h(de)f(transp)q(oser)f(le)i(v)o (ecteur)e(r)o(\023)-21 b(esulat)19 b fk(c0)g fq(p)q(our)75 1149 y(l'additionner)e(a)o(v)o(ec)e(le)h(v)o(ecteur)f(original)h fk(x)p fq(.)242 1253 y fk(program)23 b(gaxpx1)242 1310 y(real)g(a\(0:4,0:4\),)g(b\(0:4,0:4\),)f(p\(0:4,0:4\))242 1366 y(real)h(c0\(0:4,0:4\),)g(c1\(0:4,0:4\),)f(c\(0:4,0:4\))242 1422 y(real)h(x\(0:4\),b0\(0:4,0:4\))75 1479 y(!hpf$)47 b(template)23 b(t_b0\(0:4,0:4\))75 1592 y(!hpf$)47 b(align)23 b(a\(i,j\),)g(b\(i,j\),)g(p\(i,j\),)g(c1\(i,j\),)g(c0\(i,j\),)f (c\(i,j\))i(with)f(t_b0\(i,j\))75 1648 y(!hpf$)47 b(align)23 b(x\(j\))h(with)f(t_b0\(0,j\))242 1761 y(forall)g(\(j=0:4\))g (b0\(0,j\)=x\(j\))242 1818 y(forall)g(\(i=0:4,j=0:4\))f(b\(i,j\))h(=)h (b0\(0,j\))242 1874 y(forall)f(\(i=0:4,j=0:4,true\))e(p\(i,j\))j(=)f (\(a\(i,j\)*b\(i,j\)\))242 1931 y(c0)h(=)f(sum_scatter\(p,0,)242 1987 y(&)95 b(\([[0,1,2,3,4],[0,1,2,3,4],[0,)o(1,2,3,4)o(],)242 2043 y(&)167 b([0,1,2,3,4],[0,1,2,3,4]]\),)242 2100 y(&)95 b(\([[4,4,4,4,4],[4,4,4,4,4],[4,)o(4,4,4,4)o(],)242 2156 y(&)167 b([4,4,4,4,4],[4,4,4,4,4]]\)\))242 2213 y(forall)23 b(\(j=0:4,i)g(.eq.)g(0\))h(c1\(i,j\))f(=)g(c0\(j,4\))242 2269 y(forall)g(\(i=0:4,j=0:4,true\))e(c\(i,j\))j(=)f (b0\(i,j\)+c1\(i,j\))242 2326 y(end)146 2486 y fq(dans)c(ce)h(deuxi)o (\022)-21 b(eme)20 b(programme,)e(on)h(constate)g(que)h(la)g(phase)g

(de)g(transp)q(osition)g(a)f(disparu)75 2542 y(\(a)o(v)m(an)o(t)c (dernier)h fk(forall)f fq(de)h fk(gaxpx1)p fq(\),)f(ainsi)h(que)g(le)h (tableau)f(in)o(term)o(\023)-21 b(ediaire)16 b fk(c1)p fq(.)f(des)h(conclusions)p eop %%page: 101 103 101 102 bop 75 269 a fl(5.7.)29 b(ad)o(apt)l(a)l(tion)509 257 y(\022)502 269 y(a)15 b(d'a)o(utres)g(langa)o(ges)g(d)o(a)l(t)l (a-p)l(arall)1407 257 y(\022)1401 269 y(eles)262 b fq(101)75 397 y(de)16 b(la)h(section)f(4.6,)f(nous)h(sa)o(v)o(ons)e(que)j(les)f (v)o(ecteurs)g(r)o(\023)-21 b(esultats)15 b(de)h(ces)g(deux)h (programmes)e(son)o(t)g(les)75 453 y(m)o(^)-21 b(emes)14 b(et)h(son)o(t)g(plac)o(\023)-21 b(es)15 b(sur)g(la)g(premi)o(\022)-21 b(ere)16 b(ligne)g(dans)g fk(gaxpx1)e fq(et)h(sur)g(la)h(diagonale)g (dans)f fk(gaxpx2)p fq(.)242 604 y fk(real)23 b(a\(0:4,0:4\),)g (b\(0:4,0:4\),)f(p\(0:4,0:4\))242 660 y(real)h(c0\(0:4,0:4\),)g (c\(0:4,0:4\))242 717 y(real)g(x\(0:4\),b0\(0:4,0:4\))75 773 y(!hpf$)47 b(template)23 b(t_b0\(0:4,0:4\))75 886 y(!hpf$)47 b(align)23 b(a\(i,j\),)g(b\(i,j\),)g(p\(i,j\),)g(c0\(i,j\),) g(c\(i,j\))g(with)g(t_b0\(i,j\))75 943 y(!hpf$)47 b(align)23 b(x\(j\))h(with)f(t_b0\(0,j\))242 1056 y(forall)g(\(j=0:4\))g (b0\(j,j\)=x\(j\))242 1112 y(forall)g(\(i=0:4,j=0:4\))f(b\(i,j\))h(=)h (b0\(j,j\))242 1168 y(forall)f(\(i=0:4,j=0:4,true\))e(p\(i,j\))j(=)f (\(a\(i,j\)*b\(i,j\)\))242 1225 y(c0)h(=)f(sum_scatter\(p,0,)242 1281 y(&)95 b(\([[0,1,2,3,4],[0,1,2,3,4],[0,)o(1,2,3,4)o(],)242 1338 y(&)167 b([0,1,2,3,4],[0,1,2,3,4]]\),)242 1394 y(&)95 b(\([[0,1,2,3,4],[0,1,2,3,4],[0,)o(1,2,3,4)o(],)242 1451 y(&)167 b([0,1,2,3,4],[0,1,2,3,4]]\)\))242 1507 y(forall)23 b(\(i=0:4,j=0:4,true\))e(c\(i,j\))j(=)f(b0\(i,j\)+c0\(i,j\))242 1564 y(end)75 1749 y ft(5.7)66 b(adaptation)23 b(\022)-33 b(a)22 b(d'autres)g(langages)f(data-parall)o(\022)-32 b(eles)146 1865 y fq(le)15 b(sc)o(h)o(\023)-21 b(ema)14 b(de)i(traduction)f(pr)o(\023)-21 b(esen)o(t)o(\023)g(e)13 b(pro)q(duit)j(des)f(programmes)f(hpf)h(d'un)h(st)o(yle)f(tr)o(\022)-21 b(es)13 b(parti-)75 1921 y(culier,)18 b(don)o(t)d(la)i(s)o(\023)-21 b(eman)o(tique)16 b(des)g(indices)i(de)f(tableau)g(est)f(la)g(m)o(^)-21 b(eme)16 b(que)g(celles)i(des)f(programmes)75 1978 y fn(pei)7 b fq(:)17 b(les)h(indices)h(d)o(\023)-21 b(esignen)o(t)17 b(les)g(p)q(oin)o(ts)h(d'un)f(domaine)h(de)f(r)o(\023)-21 b(ef)o(\023)g(erence)16 b(comm)o(un)h(aux)g(c)o(hamps)g(du)75 2034 y(programme.)f(ce)i(domaine)g(de)g(r)o(\023)-21 b(ef)o(\023)g(erence)17 b(repr)o(\023)-21 b(esen)o(te)17 b(la)h(mac)o(hine)g(virtuelle)i(utilis)o(\023)-21 b(ee)19 b(par)e(le)i(pro-)75 2091 y(gramme)d(data-parall)o(\022)-21 b(ele.)16 b(nous)h(a)o(v)o(ons)e(vu)i(que)g(cette)g(s)o(\023)-21 b(eman)o(tique)15 b(\023)-21 b(etait)15 b(celle)j(de)f(la)g(plupart)g (des)75 2147 y(langages)i(data-parall)o(\022)-21 b(eles)20 b(\(c)620 2131 y fw(?)639 2147 y fq([54)o(],)e(mpl[38)o(],)h fn(hyper-c)p fq([45],)g(dpce[53)n(],)g fj(etc)p fq(\),)g(et)g(par)g (cons)o(\023)-21 b(e-)75 2204 y(quen)o(t)14 b(le)i(co)q(de)e(pro)q (duit)i(p)q(eut)f(facilemen)o(t)f(^)-21 b(etre)13 b(adapt)o(\023)-21 b(e)13 b(\022)-23 b(a)14 b(n'imp)q(orte)h(quel)g(langage)f(data-parall) o(\022)-21 b(ele)75 2260 y(autorisan)o(t)15 b(la)h(d)o(\023)-21 b(eclaration)16 b(de)g(mac)o(hines)h(virtuelles.)g(les)f(mo)q (di\014cations)h(\022)-23 b(a)16 b(app)q(orter)f(concernen)o(t)75 2316 y(principalemen)o(t)k(le)e(remplacemen)o(t)g(des)g(alignemen)o(ts) h(par)e(des)h(instructions)g(de)g(comm)o(unication,)75 2373 y(comme)e(nous)g(l'a)o(v)o(ons)f(\023)-21 b(ev)o(o)q(qu)o(\023)g (e)13 b(dans)j(la)f(traduction)g(des)h(c)o(hangemen)o(ts)f(de)g(base)h fn(pei)p fq(.)e(egalemen)o(t,)75 2429 y(la)g(r)o(\023)-21 b(eduction)13 b(g)o(\023)-21 b(eom)o(\023)g(etrique)12

b(de)i fn(pei)f fq(n)o(\023)-21 b(ecessite)14 b(une)g(primitiv)o(e)h (de)f(r)o(\023)-21 b(eduction)14 b(tr)o(\022)-21 b(es)12 b(g)o(\023)-21 b(en)o(\023)g(erale)12 b(dans)i(le)75 2486 y(langage)i(cible,)h(et)f(seul)g(hpf)g(\022)-23 b(a)15 b(notre)h(connaissance)h(prop)q(ose)f(un)g(op)o(\023)-21 b(erateur)14 b(assez)i(g)o(\023)-21 b(en)o(\023)g(eral)14 b(p)q(our)75 2542 y(exprimer)i(toutes)f(ces)g(r)o(\023)-21 b(eductions.)p eop %%page: 102 104 102 103 bop 75 269 a fq(102)381 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y ft(5.8)66 b(p)n(ersp)r(ectiv)n(es) 146 500 y fq(nous)15 b(donnons)h(dans)g(cette)f(section)h(quelques)h (pistes)f(p)q(our)g(am)o(\023)-21 b(eliorer)15 b(la)h(traduction)g(pr)o (\023)-21 b(esen-)75 557 y(t)o(\023)g(ee)18 b(pr)o(\023)-21 b(ec)o(\023)g(edemmen)o(t.)17 b(cette)h(am)o(\023)-21 b(elioration)18 b(consiste)h(\022)-23 b(a)18 b(pro\014ter)h(des)g (alignemen)o(ts)g(de)g(hpf)f(p)q(our)75 613 y(abstraire)e(les)g (expressions)h(de)f(tableaux)h(de)f(leur)h(signi\014cation)g(g)o(\023) -21 b(eom)o(\023)g(etrique)15 b(par)g(rapp)q(ort)h(au)f(r)o(\023)-21 b(e-)75 669 y(f)o(\023)g(eren)o(tiel)15 b(repr)o(\023)-21 b(esen)o(tan)o(t)14 b(la)h(mac)o(hine)i(virtuelle.)146 727 y(nous)22 b(p)q(ouv)o(ons)g(main)o(tenan)o(t)f(sc)o(h)o(\023)-21 b(ematiser)21 b(l'utilisation)j(t)o(ypique)f(des)f(c)o(hamps)g(de)h (donn)o(\023)-21 b(ees)75 783 y(dans)18 b(les)g(\023)-21 b(enonc)o(\023)g(es)16 b fn(pei)8 b fq(:)17 b(\022)-23 b(a)18 b(partir)g(des)g(c)o(hamps)g(en)g(en)o(tr)o(\023)-21 b(ee,)17 b(d'autres)g(c)o(hamps)h(son)o(t)f(d)o(\023)-21 b(e\014nis)18 b(p)q(our)75 840 y(repr)o(\023)-21 b(esen)o(ter)11 b(le)h(placemen)o(t)g(sur)f(le)i(r)o(\023)-21 b(ef)o(\023)g(eren)o (tiel)10 b(des)i(v)m(aleurs)h(en)e(en)o(tr)o(\023)-21 b(ee.)11 b(ces)g(d)o(\023)-21 b(e\014nitions)12 b(utilisen)o(t)h(des)75 896 y(op)o(\023)-21 b(erations)12 b(g)o(\023)-21 b(eom)o(\023)g (etriques)11 b(et)i(des)g(c)o(hangemen)o(ts)g(de)g(base)g(\(alignemen)o (t,)g(duplication\))i(a\014n)e(d'ame-)75 953 y(ner)j(les)h(v)m(aleurs)g (en)g(tous)e(les)i(p)q(oin)o(ts)g(du)f(r)o(\023)-21 b(ef)o(\023)g(eren) o(tiel)16 b(o)q(\022)-24 b(u)16 b(auron)o(t)f(lieu)j(des)e(calculs.)h (nous)f(app)q(elons)75 1009 y(cette)i(derni)o(\022)-21 b(ere)19 b(cat)o(\023)-21 b(egorie)18 b(de)h(c)o(hamp,)f(les)h fj(champs)h(plac)o(\023)-22 b(es)p fq(.)18 b(dans)g(notre)g(sc)o(h)o (\023)-21 b(ema)17 b(de)i(traduction)75 1066 y(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t,)15 b(ce)h(son)o(t)g(les)h(c)o(hamps)f(plac)o (\023)-21 b(es)17 b(qui)g(apparaissen)o(t)f(dans)g(les)h(calculs,)h(et) e(par)g(cons)o(\023)-21 b(equen)o(t)75 1122 y(les)16 b(indices)i(des)d(tableaux)h(traduisan)o(t)f(ces)h(c)o(hamps)g(d)o (\023)-21 b(esignen)o(t)15 b(des)h(pro)q(cesseurs)g(virtuels,)g(comme) 75 1178 y(nous)g(l'a)o(v)o(ons)f(soulign)o(\023)-21 b(e)16 b(en)h(section)f(5.2.)f(l'id)o(\023)-21 b(ee)16 b(est)f(donc)i(de)f (substituer)g(dans)g(le)h(programme)e(les)75 1235 y(c)o(hamps)k(plac)o (\023)-21 b(es)19 b(par)g(les)h(c)o(hamps)f(en)g(en)o(tr)o(\023)-21 b(ee)19 b(m)o(unis)g(de)h(l'alignemen)o(t)g(corresp)q(ondan)o(t.)f (ainsi,)h(le)75 1291 y(programme)d(hpf)g(r)o(\023)-21 b(esultat)16 b(de)i(la)g(traduction)g(ne)g(comp)q(orte)f(plus)i(les)f (\\d)o(\023)-21 b(eformations")16 b(dues)i(au)75 1348 y(placemen)o(t)13 b(don)o(t)f(nous)g(a)o(v)o(ons)f(parl)o(\023)-21 b(e.)12 b(p)o(our)f(cela,)i(nous)f(dev)o(ons)g(am)o(\023)-21 b(eliorer)12 b(la)h(g)o(\023)-21 b(en)o(\023)g(eration)10 b(des)i(direc-)75 1404 y(tiv)o(es)k(d'alignemen)o(t)h(de)f(fa\030)-20 b(con)16 b(\022)-23 b(a)15 b(prendre)i(en)f(compte)g(les)g (duplications,)i(ainsi)f(que)f(la)g(traduction)75 1461

y(des)f(\023)-21 b(equations)14 b(r)o(\023)-21 b(ecurren)o(tes.)75 1587 y fw(5.8.1)55 b(t)-5 b(raduction)19 b(des)g(di\013usions)f(par)h (des)f(alignemen)n(ts)146 1675 y fq(les)d(alignemen)o(ts)g(hpf)f(que)h (nous)f(a)o(v)o(ons)g(pris)h(en)g(compte)f(jusqu'\022)-23 b(a)14 b(pr)o(\023)-21 b(esen)o(t)14 b(son)o(t)f(uniquemen)o(t)75 1731 y(d)o(\023)-21 b(eduits)22 b(des)g(c)o(hangemen)o(ts)g(de)g(base)g (de)h(l')o(\023)-21 b(enonc)o(\023)g(e)21 b fn(pei)p fq(,)g(et)h(repr)o(\023)-21 b(esen)o(ten)o(t)20 b(par)i(cons)o(\023)-21 b(equen)o(t)21 b(des)75 1788 y(bijections)13 b(en)o(tre)f(la)g(repr)o (\023)-21 b(esen)o(tation)11 b(des)i(v)m(aleurs)g(de)f(tableau)g(et)g (la)g(repr)o(\023)-21 b(esen)o(tation)11 b(des)i(v)m(aleurs)g(sur)75 1844 y(un)h(template.)g(cep)q(endan)o(t,)g(la)h(directiv)o(e)g fk(align)e fq(nous)h(p)q(ermet)g(de)h(prendre)f(en)h(compte)d(\023)-21 b(egalemen)o(t)75 1901 y(des)18 b(duplications)h(de)f(v)m(aleurs.)f(d)o (\023)-21 b(eterminer)17 b(quand)h(une)f(v)m(aleur)i(in)o(tiale)f(est)f (dupliqu)o(\023)-21 b(ee)19 b(revien)o(t)f(\022)-23 b(a)75 1957 y(iden)o(ti\014er)17 b(dans)e(l')o(\023)-21 b(enonc)o(\023)g(e)14 b fn(pei)p fq(,)h(le)h(motif)7 b(:)859 2058 y fk(x)15 b fq(=)h fk(h)7 b(::)h(y)859 2114 y(z)15 b fq(=)h fk(y)f fa(c)g fk(g)75 2215 y fq(o)q(\022)-24 b(u)14 b fk(x)f fq(est)h(un)g(c)o(hamp)g(de)g(donn)o(\023)-21 b(ees)14 b(en)g(en)o(tr)o(\023)-21 b(ee,)13 b fk(h)7 b fq(:)14 b fe(z)979 2199 y fw(n)1011 2215 y ff(7!)f fe(z)1103 2199 y fw(p)1119 2215 y fq(,)g(et)h fk(g)7 b fq(:)14 b fe(z)1288 2199 y fw(n)1320 2215 y ff(7!)f fe(z)1412 2199 y fw(n)1445 2215 y fq(est)h(une)g(fonction)g(non)75 2272 y(injectiv)o(e,)19 b(a)o(v)o(ec)f fm(img)r fq(\()p fk(g)p fq(\))f(=)h fm(dom)p fq(\()p fk(h)p fq(\).)f(supp)q(osons)i(les) g(c)o(hamps)f fk(x)p fq(,)g fk(y)g fq(et)g fk(z)h fq(resp)q(ectiv)o (emen)o(t)f(\023)-21 b(egaux)75 2328 y(aux)17 b(paires)g(de)h (fonctions)f(\()p fm(v)595 2335 y fr(x)620 2328 y fq(:)g fm(\033)676 2335 y fr(x)693 2328 y fq(\),)f(\()p fm(v)780 2335 y fr(y)805 2328 y fq(:)h fm(\033)861 2335 y fr(y)878 2328 y fq(\))g(et)g(\()p fm(v)1008 2335 y fr(z)1033 2328 y fq(:)f fm(\033)1088 2335 y fr(z)1106 2328 y fq(\).)g(de)h(la)g(d)o (\023)-21 b(e\014nition)18 b(des)f(op)o(\023)-21 b(erations,)16 b(les)75 2385 y(deux)f(\023)-21 b(equations)14 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(tes)14 b(nous)i(donnen)o(t)8 b(:)635 2485 y(\()p fm(v)675 2492 y fr(x)700 2485 y fq(:)15 b fm(\033)754 2492 y fr(x)772 2485 y fq(\))49 b(=)h(\()p fm(v)964 2492 y fr(y)991 2485 y ff(\016)10 b fk(h)1048 2469 y fq(-1)1093 2485 y fq(:)15 b fm(\033)1147 2492 y fr(y)1174 2485 y ff(\016)10 b fk(h)1231 2469 y fq(-1)1269 2485 y fq(\))635 2542 y(\()p fm(v)675 2549 y fr(z)700 2542 y fq(:)15 b fm(\033)754 2549 y fr(z)772 2542 y fq(\))49 b(=)h(\()p fm(v)964 2549 y fr(y)991 2542 y ff(\016)10 b fk(g)e fq(:)15 b fm(\033)1110 2549 y fr(y)1127 2542 y fq(\))p eop %%page: 103 105 103 104 bop 75 269 a fl(5.8.)29 b(perspectives)1248 b fq(103)75 397 y(nous)16 b(c)o(herc)o(hons)h(main)o(tenan)o(t)f(la)g (relation)h(en)o(tre)g(les)g(v)m(aleurs)g(du)g(c)o(hamp)f(en)o(tr)o (\023)-21 b(ee)15 b fk(x)h fq(et)h(les)g(v)m(aleurs)75 453 y(du)23 b(c)o(hamp)g fk(z)p fq(.)f(nous)h(a)o(v)o(ons)f fm(v)638 460 y fr(x)678 453 y fq(=)h fm(v)758 460 y fr(y)785 453 y ff(\016)10 b fk(h)842 437 y fq(-1)902 453 y ff(\))23 b fm(v)992 460 y fr(x)1020 453 y ff(\016)10 b fk(h)22 b fq(=)h fm(v)1179 460 y fr(y)1207 453 y ff(\016)10 b fk(h)1264 437 y fq(-1)1311 453 y ff(\016)g fk(h)23 b fq(soit)f fm(v)1506 460 y fr(y)1524 453 y ff(j)1537 460

y fw(dom)p fg(\()p fw(h)p fg(\))1670 453 y fq(=)h fm(v)1750 460 y fr(x)1777 453 y ff(\016)10 b fk(h)p fq(.)75 510 y(comme)j fm(dom)p fq(\()p fm(v)366 517 y fr(y)383 510 y fq(\))f ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\),)g fm(v)655 517 y fr(y)686 510 y fq(=)h fm(v)757 517 y fr(x)785 510 y ff(\016)c fk(h)p fq(.)j(alors)h fm(v)1009 517 y fr(z)1040 510 y fq(=)g fm(v)1111 517 y fr(x)1139 510 y ff(\016)c fk(h)g ff(\016)f fk(g)p fq(.)14 b(cette)f(relation)h(nous)g (donne)h(le)75 566 y(placemen)o(t)h(des)f(v)m(aleurs)i(initiales)g(sur) e(le)h(c)o(hamp)f fk(z)p fq(.)g(p)o(ar)f(exemple,)i(le)g(motif)f(d')o (\023)-21 b(equations)7 b(:)537 655 y fk(x)12 b fq(=)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(i)p ff(\024)p fk(n)7 b(&)h(j)p fq(=)p fk(i\).\(i\))f(::)g(y)537 711 y(z)12 b fq(=)h fk(y)i fa(c)h fm(\025)p fk(\(i,j\))p ff(j)p fk(\(1)p ff(\024)p fk(j)p ff(\024)p fk(n)7 b(&)g(1)p ff(\024)p fk(j)p ff(\024)p fk(n\).\(j,j\))75 800 y fq(donne)16 b fm(v)231 807 y fr(z)262 800 y fq(=)d fm(v)332 807 y fr(x)360 800 y ff(\016)d fm(\025)p fk(\(i,j\).\(j\))n fq(,)15 b(ce)h(qui)g(signi\014e)h(in)o(tuitiv)o(emen)o(t,)f(que)g(les)g (v)m(aleurs)g(de)g(co)q(ordonn)o(\023)-21 b(ees)75 857 y(\()p fm(i;)8 b(j)s fq(\))i(de)j fk(z)f fj(\()-9 b(\()p fq(viennen)o(t)p fj(\))g(\))14 b fq(des)f(p)q(oin)o(ts)f(de)h(co)q (ordonn)o(\023)-21 b(ees)12 b fm(j)j fq(de)d fk(x)p fq(,)g(ce)h(qui)g (p)q(eut)g(se)f(traduire)h(par)f(la)g(direc-)75 913 y(tiv)o(e)k fk(align)23 b(x\(j\))g(with)g(z\(*,j\))p fq(.)15 b(en)g(alignan)o(t)h (ces)g(deux)g(tableaux)g(sur)f(un)h(template)g(comm)o(un,)75 969 y(et)f(en)h(renomman)o(t)e(les)i(indices,)h(nous)e(a)o(v)o(ons)7 b(:)147 1055 y fk(align)23 b(x\(i\))71 b(with)23 b(t\(*,i\))147 1112 y(align)g(z\(i,j\))g(with)g(t\(i,j\))75 1232 y fw(5.8.2)55 b(substitution)18 b(des)h(c)n(hamps)f(plac)o(\023)-27 b(es)18 b(par)h(les)f(c)n(hamps)g(en)n(tr)o(\023)-27 b(ee)146 1318 y fq(il)17 b(est)g(imp)q(ortan)o(t)f(de)i(noter)e(que)h (l'alignemen)o(t)h(de)g fk(x)e fq(sur)h(le)h(template)f(est)f(p)q (ermanen)o(t,)h(c'est-)75 1374 y(\022)-23 b(a-dire)17 b(que)f(toute)g(mo)q(di\014cation)h(d'un)f(\023)-21 b(el)o(\023)g(emen) o(t)14 b fk(x\(i\))i fq(duran)o(t)g(l'ex)o(\023)-21 b(ecution)16 b(est)g(rep)q(ort)o(\023)-21 b(ee)15 b(sur)h(tous)75 1431 y(ses)e(\023)-21 b(el)o(\023)g(emen)o(ts)14 b(dupliqu)o(\023)-21 b(es)17 b(\([32)n(],)e(page)g(118\).)e(ainsi)k(les)f(instructions)428 1516 y fk(x\(1\))24 b(=)f(v)61 b fq(et)f fk(forall)23 b(\(i)p fq(=)p fk(1:n,j)p fq(=)p fk(1\))h(z\(i,j\))f(=)g(v)75 1602 y fq(r)o(\023)-21 b(ealisen)o(t)16 b(des)h(a\013ectations)f (\(resp)q(ectiv)o(emen)o(t)h(aux)g(v)m(ariables)h fk(x)e fq(et)h fk(y)p fq(\))f(par)g(le)i(m)o(^)-21 b(eme)15 b(ensem)o(ble)j(de)75 1659 y(pro)q(cesseurs)e(virtuels.)146 1715 y(cette)g(caract)o(\023)-21 b(eristique)16 b(p)q(ermet,)h(dans)g (certaines)g(conditions,)h(de)f(remplacer)h(une)f(traduction)75 1772 y(faisan)o(t)h(in)o(terv)o(enir)h(la)f(v)m(ariable)i fk(z)p fq(,)d(par)h(une)h(traduction)f(o)q(\022)-24 b(u)19 b(n'in)o(tervien)o(t)f(que)h(la)f(v)m(ariable)i fk(x)p fq(.)e(la)75 1828 y(condition,)d(p)q(our)f(r)o(\023)-21 b(ealiser)13 b(une)i(telle)g(substitution,)f(dans)g(le)h(cas)e(d'une)i (duplication,)g(est)f(que)g(toute)75 1885 y(op)o(\023)-21 b(eration)12 b(faisan)o(t)g(r)o(\023)-21 b(ef)o(\023)g(erence)12 b(aux)h(v)m(aleurs)h(de)f fk(z)p fq(,)f(le)i(fait)f(p)q(our)g(l'ensem)o (ble)h(des)g(v)m(aleurs)f(dupliqu)o(\023)-21 b(ees.)75 1941 y(p)o(ar)9 b(con)o(tre,)h(si)h(dans)f(l')o(\023)-21 b(enonc)o(\023)g(e)10 b fn(pei)p fq(,)f(une)i(expression)h(fait)e(r)o (\023)-21 b(ef)o(\023)g(erence)9 b(\022)-23 b(a)10 b(une)h(partie)f (seulemen)o(t)i(des)e(v)m(a-)75 1998 y(leurs)i(sur)f(la)g(premi)o(\022)

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gr polyline 293 132 m 346 92 l 438 92 l 385 132 l gs col-1 s gr polyline 356 109 m 356 188 l gs col-1 s gr 358.000 180.000 m 356.000 188.000 l 354.000 180.000 l gs 2 setlinejoin col-1 s gr % polyline n 244 224 m 244 132 l 297 92 l 297 184 l 244 224 l gs 0.50 setgray fill gr gs col-1 s gr % polyline n 270 158 m 379 158 l gs col-1 s gr n 371.000 156.000 m 379.000 158.000 l 371.000 160.000 l gs 2 setlinejoin col-1 s gr % polyline n 438 92 m 438 184 l 385 224 l gs col-1 s gr % polyline n 385 224 m 385 132 l 293 132 l 293 224 l clp gs col-1 s gr $f2psend %%enddocument @endspecial 1274 1898 a fe(i)1300 1760 y(k)1349 1835 y(j)584 1823 y(a)966 1786 y(b)338 2387 y fn(fig.)g fq(5.5)f(-)30 b fj(pr)n(o)n(duit)17 b(matriciel)f(sur)g(un)h(cub)n(e)f(de)g(pr)n(o)n (c)n(esseur)f(virtuels)146 2486 y fq(le)h(c)o(hamp)g fk(a')f fq(est)h(ensuite)h(d)o(\023)-21 b(e\014ni)17 b(comme)e(la)h(di\013usion)h(de)g(la)f(matrice)g fm(a)g fq(selon)g(les)h(lignes)g(du)75 2542 y(cub)q(e,)e(et)e fk(b')h fq(comme)g(la)g(di\013usion)h(de)f fm(b)j fq(selon)d(les)h (colonnes)g(du)f(m)o(^)-21 b(eme)13 b(cub)q(e.)i(les)f(deux)h (di\013usions)p eop %%page: 105 107 105 106 bop 75 269 a fl(5.8.)29 b(perspectives)1248 b fq(105)75 397 y(p)q(ermetten)o(t)17 b(de)h(calculer)h(l'ensem)o(ble)f (des)g fm(n)872 380 y fg(3)908 397 y fq(pro)q(duits)g(en)g(une)g(op)o (\023)-21 b(eration)16 b(fonctionnelle.)j(en\014n,)75 453 y(les)f(pro)q(duits)g(son)o(t)e(somm)o(\023)-21 b(es)16 b(le)h(long)h(de)f(la)h(troisi)o(\022)-21 b(eme)16 b(dimension,)j(et)e (le)h(r)o(\023)-21 b(esulat)16 b fk(r)h fq(est)g(le)h(dernier)75 510 y(plan)e(de)g(cette)f(dimension.)75 656 y fk(matmult)23 b(:)h(\(a,b\))14 b ff(7!)i fk(r)75 672 y fx(8)75 709 y(>)75 721 y(>)75 734 y(>)75 746 y(>)75 759 y(>)75 771 y(>)75 784 y(>)75 796 y(>)75 809 y(>)75 821 y(>)75 833 y(>)75 846 y(>)75 858 y(>)75 871 y(>)75 883 y(<)75 958 y(>)75 970 y(>)75 983 y(>)75 995 y(>)75 1008 y(>)75 1020 y(>)75 1033 y(>)75 1045 y(>)75 1058 y(>)75 1070 y(>)75 1083 y(>)75 1095 y(>)75 1107 y(>)75 1120 y(>)75 1132 y(:)137 706 y fk(a)f fq(=)g fk(align1)7 b(::)g(a0)137 763 y(b)15 b fq(=)g fk(align2)7 b(::)g(b0)137 819 y(a0)15 b fq(=)g fk(dim)7 b(::)h(a)137 875 y(b0)15 b fq(=)g fk(dim)7 b(::)h(b)137 932 y(a')15 b fq(=)g fk(a0)g fa(c)h fk(spread1)137 988 y(b')f fq(=)g fk(b0)g fa(c)h fk(spread2)137 1045 y(p)f fq(=)g fk(mult)g fa(b)g fk(\(a')24 b(/&/)7 b(b'\))137 1101 y(c)15 b fq(=)g fk(add)g fa(b)h fk(\(p)23 b(/&/)8 b(c)23 b fa(c)i fk(prev)o(\))137 1158 y(r)15 b fq(=)g fk(c)g fa(c)h fk(last)100 1230 y(align1)73 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(0)p ff(\024)p fk(i)p fm(<)p fk(n)6 b(&)i(j)p fq(=)p fk(0)g(&)g(0)p ff(\024)p fk(k)p fm(<)p fk(n\).\(i,k\))100 1287 y(align2)73 b fq(=)50 b fm(\025)p fk(\(i,j,k\))p ff(j)p fk(\(i)p fq(=)p fk(0)6 b(&)i(0)p ff(\024)p fk(j)p fm(<)p fk(n)g(&)g(0)p % n % n n

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(\023)-21 b(ep)q(endance)14 b(sur)g(l'axe)f fm(k)q fq(.)g(la)g(b)q (oucle)i(s)o(\023)-21 b(equen)o(tielle)15 b(repr)o(\023)-21 b(esen)o(tan)o(t)12 b(le)i(temps)f(p)q(eut)h(par)75 736 y(cons)o(\023)-21 b(equen)o(t)11 b(utiliser)j fk(k)e fq(comme)g(indice.)h(apr)o(\022)-21 b(es)12 b(substitution)g(de)h fk(p)e fq(par)h(sa)g(d)o(\023)-21 b(e\014nition,)12 b(la)g(traduction) 75 792 y(du)k(calcul)g(des)g(pro)q(duits)g(selon)g(ce)f(pro)q(c)o(\023) -21 b(ed)o(\023)g(e)14 b(est)8 b(:)147 877 y fk(forall)23 b(\(i=0:n-1,j=0:n-1,k=0\))e(c\(i,j,k\))h(=)i(a\(i,k\)*b\(k,j\))147 933 y(do)f(k=1,n-1)242 990 y(forall)g(\(i=0:n-1,j=0:n-1\))f(c\(i,j,k\)) g(=)i(c\(i,j,k-1\))f(+)g(a\(i,k\))g(*)h(b\(k,j\))147 1046 y(enddo)75 1131 y fq(de)16 b(plus,)h(nous)f(sommes)f(dans)h(le)h (cas)f(fa)o(v)o(orable)f(o)q(\022)-24 b(u)16 b(l'assignation)h(unique)g (sur)f fk(c)g fq(p)q(eut)g(^)-21 b(etre)14 b(lev)o(\023)-21 b(ee)75 1187 y(facilemen)o(t.)15 b(la)f(seule)h(autre)f(d)o(\023)-21 b(e\014nition)15 b(faisan)o(t)e(r)o(\023)-21 b(ef)o(\023)g(erence)13 b(\022)-23 b(a)14 b(une)h(o)q(ccurrence)g(de)f fk(c\(i,j,k\))e fq(\023)-21 b(etan)o(t)75 1244 y(la)18 b(s)o(\023)-21 b(election)17 b fk(last)g fq(du)h(dernier)h(plan)f(de)g(la)f(dimension) i fm(k)g fq(nous)e(p)q(ouv)o(ons)h(tranformer)e(le)i(tableau)75 1300 y(tri-dimensionnel)23 b fk(c)c fq(en)h(en)g(tableau)g (bi-dimensionnel)j(et)c(supprimer)i(la)f(derni)o(\022)-21 b(ere)18 b(\023)-21 b(equation.)19 b(le)75 1357 y(probl)o(\022)-21 b(eme)14 b(se)g(p)q(ose)g(de)g(sa)o(v)o(oir)g(commen)o(t)f(doit)g(^)-21 b(etre)13 b(align)o(\023)-21 b(e)14 b(ce)g(tableau)g(bi-dimensionel)k (sur)c(le)g(cub)q(e)75 1413 y(de)k(pro)q(cesseurs)g(virtuels.)h(p)o (our)e(restituer)h(les)g(comm)o(unications)h(de)f(v)o(oisins)g(\022)-23 b(a)18 b(v)o(oisins)g(pr)o(\023)-21 b(esen)o(tes)75 1470 y(dans)17 b(la)g(v)o(ersion)g(\022)-23 b(a)16 b(assignation)h(unique,)g (utilis)o(\023)-21 b(ee)18 b(p)q(our)f(la)g(somme)f(des)h(pro)q(duits,) g(une)h(p)q(ossiblit)o(\023)-21 b(e)75 1526 y(est)13 b(de)g(placer)h(le)g(tableau)f fk(c)g fq(\022)-23 b(a)12 b(c)o(haque)i(it)o(\023)-21 b(eration)12 b(de)h(la)h(b)q(oucle)g fk(do)p fq(,)e(sur)h(le)h(plan)g(corresp)q(ondan)o(t)f(\022)-23 b(a)12 b(la)75 1583 y(v)m(aleur)h(de)f(l'indice)i fk(k)p fq(.)e(cet)f(alignemen)o(t)i(dynamique)g(du)f(tableau)g(s'exprimerait)g (alors)g(par)g fk(realign)75 1639 y(c\(i,j\))23 b(with)g(t\(i,j,k\))p fq(.)14 b(cep)q(endan)o(t,)i(le)h(co)q(^)-24 b(ut)15 b(en)h(temps)f(du)h(r)o(\023)-21 b(ealignemen)o(t)16 b(est)f(probablemen)o(t)75 1695 y(imp)q(ortan)o(t)h(et)h(il)g(sem)o (ble)h(qu'un)f(alignemen)o(t)g(statique)g(arbitraire)g(sur)f(l'un)h (des)g(plans)h(de)f(la)g(troi-)75 1752 y(si)o(\022)-21 b(eme)16 b(dimension)h(est)f(pr)o(\023)-21 b(ef)o(\023)g(erable.)14 b(nous)i(c)o(hoisissons)g(ainsi)h(d'aligner)g fk(c)e fq(sur)h(le)g(plan)h fk(k)p fq(=)p fm(n)11 b ff(\000)g fq(1,)k(et)75 1808 y(notre)g(traduction)g(est)g(\014nalemen)o(t)9 b(:)242 1893 y fk(real)23 b(a\(0:n-1,0:n-1\),)f(b\(0:n-1,0:n-1\),)g (c\(0:n-1,0:n-1\))75 1949 y(!hpf$)47 b(template)23 b (t\(0:n-1,0:n-1,0:n-1\))75 2006 y(!hpf$)47 b(align)23 b(a\(i,k\))g(with)h(t\(i,*,k\))75 2062 y(!hpf$)47 b(align)23 b(b\(k,j\))g(with)h(t\(*,j,k\))75 2119 y(!hpf$)47 b(align)23 b(c\(i,j\))g(with)h(t\(i,j,n-1\))242 2232 y(forall)f (\(i=0:n-1,j=0:n-1,k=0\))e(c\(i,j\))i(=)h(a\(i,k\))f(*)h(b\(k,j\))242 2288 y(do)g(k=1,n-1)337 2345 y(forall\(i=0:n-1,j=0:n-1\))d(c\(i,j\))i (=)h(c\(i,j\))f(+)h(a\(i,k\))f(*)h(b\(k,j\))242 2401 y(enddo)146 2486 y fq(ce)15 b(texte)g(de)h(programme)e(est)h(bien)i(s)q (^)-24 b(ur)15 b(b)q(eaucoup)h(plus)h(in)o(tuitif)f(que)g(la)g(v)o (ersion)f(utilisan)o(t)i(les)75 2542 y(c)o(hamps)e(plac)o(\023)-21 b(es.)p eop %%page: 107 109

107 108 bop 75 269 a fl(5.9.)29 b(envir)o(onnement)16 b(logiciel)g(a)o(utour)g(de)f fn(pei)541 b fq(107)75 397 y ft(5.9)66 b(en)n(vironnemen)n(t)25 b(logiciel)f(autour)f(de)g fs(pei)146 498 y fq(la)15 b(\014gure)h(5.7)e(pr)o(\023)-21 b(esen)o(te)15 b(les)h(outils)h(d)o(\023)-21 b(ej\022)e(a)14 b(d)o(\023)-21 b(ev)o(elopp)o(\023)g(es)15 b(ou)g(en)h(cours)g(de)g(d)o (\023)-21 b(ev)o(elopp)q(emen)o(t)16 b(\(traits)75 555 y(pleins\),)c(o)q(\022)-24 b(u)12 b(les)g(pro)s(jets)e(de)h(d)o(\023) -21 b(ev)o(elopp)q(emen)o(ts)12 b(\(traits)e(hac)o(h)o(ur)o(\023)-21 b(es\))10 b(autour)h(du)h(formalisme)f fn(pei)p fq(.)g(nous)75 611 y(a)o(v)o(ons)i(d)o(\023)-21 b(ej\022)e(a)13 b(pr)o(\023)-21 b(esen)o(t)o(\023)g(e)13 b(en)h(section)h(3.5)e(et)h(3.4,)f(le)i(mo)q (dule)g(de)f(transformation)f(des)g(\023)-21 b(enonc)o(\023)g(es)13 b fn(pei)p fq(,)h(et)75 668 y(celui)20 b(de)e(conception)h(graphique)g fn(vpei)p fq(.)e(le)h(mo)q(dule)h(d'in)o(terpr)o(\023)-21 b(etation)17 b(fonctionnelle)j(est)e(d)o(\023)-21 b(ecrit)75 724 y(dans)16 b([57)o(])g(et)f(implan)o(te)i(l'in)o(terpr)o(\023)-21 b(etation)16 b(fonctionnelle)i(des)e(c)o(hamps)g(de)g(donn)o(\023)-21 b(ees)16 b(telle)h(que)f(nous)75 781 y(l'a)o(v)o(ons)f(d)o(\023)-21 b(e\014nie)16 b(en)g(section)g(2.3.5.)d(la)j(traduction)f(du)h (programme)e fn(pei)i fq(en)f(un)h(co)q(de)g fn(caml)f fq(nous)75 837 y(p)q(ermet)20 b(indirectemen)o(t)h(d'ex)o(\023)-21 b(ecuter)19 b(le)h(programme)f fn(pei)p fq(,)g(et)g(ce)h(mo)q(dule)h(p) q(ourrait)e(^)-21 b(etre)18 b(in)o(t)o(\023)-21 b(egr)o(\023)g(e)75 894 y(dans)15 b(l'en)o(vironnemen)o(t)g(graphique)h(a\014n)f(de)g (visualiser)h(quelles)g(son)o(t,)e(et)g(o)q(\022)-24 b(u)15 b(son)o(t)f(les)h(v)m(aleurs)g(dans)75 950 y(les)h(c)o(hamps,)f (en)g(fonction)h(de)f(c)o(hamps)h(en)f(en)o(tr)o(\023)-21 b(ee.)213 2243 y @beginspecial @setspecial %%begindocument: logiciel.pstex /$f2psdict 200 dict def $f2psdict begin $f2psdict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /col8 {.68 .85 .9 setrgbcolor} bind def /col9 {0 .39 0 setrgbcolor} bind def /col10 {.65 .17 .17 setrgbcolor} bind def /col11 {1 .51 0 setrgbcolor} bind def /col12 {.63 .13 .94 setrgbcolor} bind def /col13 {1 .75 .8 setrgbcolor} bind def /col14 {.7 .13 .13 setrgbcolor} bind def /col15 {1 .84 0 setrgbcolor} bind def /drawellipse {

/endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix } def end /$f2psbegin {$f2psdict begin /$f2psenteredstate save def} def /$f2psend {$f2psenteredstate restore end} def $f2psbegin 0 setlinecap 0 setlinejoin -153.0 315.0 translate 0.900 -0.900 scale 0.500 setlinewidth % ellipse n 303 190 34 34 0 360 drawellipse gs col-1 s gr % ellipse n 220 316 34 34 0 360 drawellipse gs col-1 s gr % ellipse n 450 316 34 34 0 360 drawellipse gs col-1 s gr % ellipse n 534 224 34 34 0 360 drawellipse gs col-1 s gr % ellipse n 505 138 34 34 0 360 drawellipse gs col-1 s gr % ellipse n 444 66 34 34 0 360 drawellipse gs col-1 s gr % polyline n 258 261 m 258 223 l 178 223 l 178 261 l clp gs col-1 s gr % polyline n 373 333 m 373 295 l 292 295 l 292 333 l clp gs col-1 s gr % polyline n 433 159 m 433 125 l 352 125 l 352 159 l clp gs col-1 s gr % polyline n 433 138 m 471 138 l gs col-1 s gr n 463.000 136.000 m 471.000 138.000 l 463.000 140.000 l gs 2 setlinejoin gr % polyline n 318 159 m 352 138 l gs col-1 s gr n 344.143 140.502 m 352.000 138.000 l 346.245 143.906 l gs 2 setlinejoin gr % polyline n 373 312 m 416 312 l gs col-1 s gr n 408.000 310.000 m 416.000 312.000 l 408.000 314.000 l gs 2 setlinejoin gr % polyline n 254 312 m 292 312 l gs col-1 s gr n 284.000 310.000 m 292.000 312.000 l 284.000 314.000 l gs 2 setlinejoin gr % polyline n 216 261 m 216 282 l gs col-1 s gr n 218.000 274.000 m 216.000 282.000 l 214.000 274.000 l gs 2 setlinejoin gr % polyline n 267 189 m 220 223 l gs col-1 s gr n 227.654 219.931 m 220.000 223.000 l 225.310 216.691 l gs 2 setlinejoin

col-1 s

col-1 s

col-1 s

col-1 s

col-1 s

col-1 s

gr polyline 463 227 m 463 185 l 390 185 l 390 227 l clp gs col-1 s gr polyline 390 202 m 335 185 l gs col-1 s gr 342.053 189.273 m 335.000 185.000 l 343.234 185.452 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % polyline n 463 269 m 463 231 l 390 231 l 390 269 l clp gs col-1 s gr [] 0 setdash % polyline n 501 214 m 463 210 l gs col-1 s gr n 470.747 212.826 m 463.000 210.000 l 471.165 208.848 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % polyline n 463 248 m 501 231 l gs col-1 s gr [] 0 setdash n 492.881 232.441 m 501.000 231.000 l 494.514 236.093 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % polyline n 339 248 m 390 248 l gs col-1 s gr [] 0 setdash n 382.000 246.000 m 390.000 248.000 l 382.000 250.000 l gs 2 setlinejoin col-1 s gr % polyline n 305 223 m 305 240 l gs col-1 s gr n 307.000 232.000 m 305.000 240.000 l 303.000 232.000 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % polyline n 305 261 m 305 295 l gs col-1 s gr [] 0 setdash n 307.000 287.000 m 305.000 295.000 l 303.000 287.000 l gs 2 setlinejoin col-1 s gr % polyline n 339 261 m 339 240 l 288 240 l 288 261 l clp gs col-1 s gr % polyline n 267 142 m 267 117 l 169 117 l 169 142 l clp gs col-1 s gr % polyline n 368 83 m 368 49 l 287 49 l 287 83 l clp gs col-1 s gr % polyline n 304 154 m 304 84 l gs col-1 s gr n 302.000 92.000 m 304.000 84.000 l 306.000 92.000 l gs 2 setlinejoin col-1 s gr % polyline n 369 64 m 409 64 l gs col-1 s gr n 401.000 62.000 m 409.000 64.000 l 401.000 66.000 l gs 2 setlinejoin col-1 s gr % interpolated spline n 271 176 m 236.661 175.147 221.767 171.966 212 163 curveto 207.711 160.150 205.584 154.848 203 142 curveto gs col-1 s gr n 202.617 150.237 m 203.000 142.000 l 206.538 149.449 l gs 2 setlinejoin col-1 s gr % interpolated spline n 267 129 m 278.537 126.364 283.856 126.364 288 129 curveto 293.579 133.349 295.707 139.711 297 155 curveto % n % n n

gs col-1 s gr n 298.319 146.860 m 297.000 155.000 l 294.333 147.197 l gs 2 setlinejoin col-1 s gr [4.000000] 0 setdash % interpolated spline n 288 240 m 249.538 210.261 233.581 196.476 224 185 curveto 219.102 177.889 213.783 167.286 203 142 curveto gs col-1 s gr [] 0 setdash n 204.298 150.143 m 203.000 142.000 l 207.978 148.574 l gs 2 setlinejoin col-1 s gr $f2psend %%enddocument @endspecial 1203 2127 a fy(hpf)1203 2067 y(progs.)644 1688 y fa(pei)1506 1733 y fy(dessin)1506 1793 y fa(pei)1424 1497 y(caml)1424 1434 y fy(prog.)340 2067 y(prog.)340 2116 y fa(pei)310 2180 y fy(norm.)12 b(hpf)644 1625 y(enonc)o(\023)-20 b(e)914 1449 y(in)o(terpr)o(\023)g(etation)914 1497 y(fonctionnelle)262 1831 y(f)m(orme)12 b(normale)310 1876 y(hpf)1060 1861 y(t)m(raduction)1060 1910 y(graphique)1060 1688 y(in)o(terpr)o(\023)-20 b(etation)1060 1733 y(graphique)693 2150 y(hpf)228 1415 y(t)m(ransformations)674 1876 y(t)o(yp)q(eur)693 2086 y(t)m(raduction)1188 1209 y(l)1199 1204 y fa(a)1218 1209 y fy(t)1241 1221 y(e)1264 1209 y(x)-613 b(l)693 1204 y fa(a)712 1209 y fy(t)735 1221 y(e)758 1209 y(x)682 1171 y(t)m(raduction)1188 1134 y(source)475 2387 y fn(fig.)15 b fq(5.7)f(-)30 b fj(envir)n(onnement)15 b(lo)n(giciel)g(autour)j(de)e fn(pei)146 2486 y fq(les)j(d)o(\023)-21 b(ev)o(elopp)q(emen)o(ts)19 b(en)h(cours)f(sur)g(le)g(traducteur)g fk(pei2hpf)f fq(on)o(t)h(p)q (our)g(ob)s(jectif)g(imm)o(\023)-21 b(ediat)75 2542 y(d'in)o(t)o(\023)g (egrer)13 b(les)j(informations)e(rendues)h(par)g(le)g(con)o(tr^)-23 b(oleur)14 b(de)h(t)o(yp)q(e.)f(ce)h(con)o(tr^)-23 b(oleur)14 b(de)h(t)o(yp)q(e)g(inf)o(\022)-21 b(ere)p eop %%page: 108 110 108 109 bop 75 269 a fq(108)381 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fq(les)h(domaines)h(de)f(v)m (aleurs)h(et)e(de)h(r)o(\023)-21 b(ef)o(\023)g(erence)15 b(des)h(c)o(hamps)f(\022)-23 b(a)16 b(partir)f(des)h(d)o(\023)-21 b(e\014nitions)17 b(de)f(l')o(\023)-21 b(enonc)o(\023)g(e)15 b(et)75 453 y(v)o(\023)-21 b(eri\014e)12 b(que)h(le)f(programme)f(est)h (bien)h(t)o(yp)o(\023)-21 b(e,)11 b(c'est-\022)-23 b(a-dire)13 b(qu'aucune)f(d)o(\023)-21 b(e\014nition)13 b(n'est)f(en)g(con)o (tradic-)75 510 y(tion)g(a)o(v)o(ec)f(une)i(autre.)d(les)j(calculs)g (ensem)o(blistes)g(n)o(\023)-21 b(ecessaires)12 b(p)q(our)f(ce)i(t)o (yp)q(e)e(de)h(calcul)i(son)o(t)d(r)o(\023)-21 b(ealis)o(\023)g(es)75 566 y(\022)e(a)17 b(l'aide)i(de)g(la)f(librairie)i fn(omega)p fq([30)o(].)e(les)g(informations)g(retourn)o(\023)-21 b(ees)17 b(par)g(ce)h(mo)q(dule)h(devraien)o(t)75 623 y(p)q(ermettre)14 b(la)h(traduction)f(de)g(programmes)f(don)o(t)h(les)h (domaines)g(son)o(t)e(moins)i(simplistes)g(que)g(ceux)75 679 y(actuellemen)o(t)h(accept)o(\023)-21 b(es.)75 825 y ft(5.10)66 b(t)-6 b(ra)n(v)l(aux)24 b(relatifs)75 929 y fw(5.10.1)55 b(con)n(texte)146 1016 y fq(de)21 b(nom)o(breux)h(pro)s (jets)f(de)i(rec)o(herc)o(he)f(visen)o(t)g(\022)-23 b(a)22 b(pro)q(duire)h(du)g(co)q(de)f(parall)o(\022)-21 b(ele)23 b(\022)-23 b(a)21 b(partir)h(de)75 1072 y(sp)o(\023)-21 b(eci\014cations)16 b(plus)g(faciles)g(\022)-23 b(a)15 b(appr)o(\023)-21 b(ehender)16 b(par)e(l'utilisateur.)146

1129 y(la)d(v)o(oie)g(fo)q(calisan)o(t)h(le)g(plus)g(d'e\013orts)e(et)h (la)h(plus)g(am)o(bitieuse)g(est)f(naturellemen)o(t)i(la)e(compilation) 75 1186 y(des)i(programmes)f(s)o(\023)-21 b(equen)o(tiels)13 b(existan)o(ts)g(en)g(programmes)e(parall)o(\022)-21 b(eles,)13 b(a)o(v)o(ec)g(de)g(nom)o(breux)g(pro)s(jets)75 1242 y(comme)e(suif[1],)f(pips[28],)g(p)l(af[18)o(],)g(lo)q(op)o(o[15)o (],)g fj(etc)p fq(.)h(cep)q(endan)o(t,)g(les)h(compilateurs)g(de)f(ce)h (t)o(yp)q(e)75 1299 y(on)o(t)18 b(mon)o(tr)o(\023)-21 b(e)17 b(jusqu'\022)-23 b(a)18 b(pr)o(\023)-21 b(esen)o(t)18 b(des)h(limitations)h(de)f(deux)g(sortes.)f(la)h(premi)o(\022)-21 b(ere)18 b(est)h(la)g(limite)h(du)75 1355 y fj(mo)n(d)o(\022)-22 b(ele)12 b(p)n(olytop)n(e)p fq([36])e(et)i(concerne)g(tous)f(les)h(tra) o(v)m(aux)e(men)o(\023)-21 b(es)11 b(sur)g(les)h(tec)o(hniques)h(de)f (parall)o(\023)-21 b(elisation.)75 1412 y(la)14 b(deuxi)o(\022)-21 b(eme)15 b(est)e(sp)o(\023)-21 b(eci\014que)15 b(au)f(c)o(hoix)g (d'accepter)g(en)h(en)o(tr)o(\023)-21 b(ee)13 b(des)h(sp)o(\023)-21 b(eci\014cations)14 b(form)o(ul)o(\023)-21 b(ees)13 b(dans)75 1468 y(un)j(langage)f(s)o(\023)-21 b(equen)o(tiel)16 b(\(souv)o(en)o(t)e(f)l(ortran)g(77\))g(don)o(t)h(la)h(s)o(\023)-21 b(eman)o(tique)14 b(est)h(tr)o(\022)-21 b(es)14 b(compliqu)o(\023)-21 b(ee.)146 1525 y(av)o(ec)13 b(l'apparition)h(du)f(mo)q(d)o(\022)-21 b(ele)13 b(data-parall)o(\022)-21 b(ele)13 b(est)g(apparue)g(l'id)o (\023)-21 b(ee)14 b(de)f(scinder)h(la)f(compilation)75 1582 y(en)20 b(deux)g(phases)8 b(:)19 b(transformer)f(le)i(co)q(de)g(s) o(\023)-21 b(equen)o(tiel)20 b(en)g(co)q(de)g(data-parall)o(\022)-21 b(ele,)19 b(puis)h(compiler)h(le)75 1638 y(co)q(de)12 b(data-parall)o(\022)-21 b(ele)12 b(en)f(co)q(de)h(\022)-23 b(a)11 b(parall)o(\023)-21 b(elisme)13 b(de)f(con)o(tr^)-23 b(ole,)11 b(pro)q(c)o(he)g(du)h(mo)q(d)o(\022)-21 b(ele)12 b(d'ex)o(\023)-21 b(ecution)11 b(actuel)75 1694 y(des)i(mac)o(hines.)f (le)h(compilateur)g fl(bouclettes)p fq([6)q(])f(appartien)o(t)g(par)g (exemple)i(\022)-23 b(a)12 b(la)g(premi)o(\022)-21 b(ere)12 b(cat)o(\023)-21 b(egorie.)75 1751 y(dans)10 b(la)i(deuxi)o(\022)-21 b(eme)11 b(cat)o(\023)-21 b(egorie,)9 b(de)i(nom)o(breux)g(outils)h(on) o(t)d(\023)-21 b(et)o(\023)g(e)9 b(d)o(\023)-21 b(ev)o(elopp)o(\023)g (es,)10 b(tan)o(t)g(par)g(des)h(univ)o(ersit)o(\023)-21 b(es)75 1807 y(que)15 b(par)g(des)g(en)o(treprises)h(commerciales,)f (parmi)g(lesquels)i(pghpf)e(\(p)o(ortland)f(group\),)g(hpf)o(c[13)n(],) 75 1864 y(adaptor[7)o(],)g fj(etc)p fq(.)146 1921 y(un)f(autre)g(c)o (hoix)h(consiste)g(\022)-23 b(a)13 b(utiliser)i(comme)e(mo)o(y)o(en)f (de)i(sp)o(\023)-21 b(eci\014cation)14 b(un)g(langage)f(dot)o(\023)-21 b(e)12 b(d'une)75 1977 y(s)o(\023)-21 b(eman)o(tique)13 b(claire.)i(le)g(formalisme)f(des)f(\023)-21 b(equations)14 b(r)o(\023)-21 b(ecurren)o(tes)13 b(fait)h(partie)g(de)g(ce)h(t)o(yp)q (e)f(de)g(sp)o(\023)-21 b(eci-)75 2034 y(\014cation,)14 b(et)g(a)f(\023)-21 b(et)o(\023)g(e)13 b(c)o(hoisi)i(p)q(our)g(l'en)o (vironnemen)o(t)g(op)q(era[37])f(par)g(exemple.)h(d'autres)e(pro)s (jets)g(on)o(t)75 2090 y(adopt)o(\023)-21 b(e)15 b(ou)h(in)o(v)o(en)o (t)o(\023)-21 b(e)15 b(des)i(formalismes)f(p)q(ouv)m(an)o(t)g(b)o(\023) -21 b(en)o(\023)g(e\014cier)16 b(des)h(tec)o(hniques)g(d)o(\023)-21 b(ev)o(elopp)o(\023)g(ees)16 b(dans)g(le)75 2147 y(domaine)f(du)g(g)o (\023)-21 b(enie)15 b(logiciel.)h(dans)e(cette)h(optique,)g(plusieurs)h (v)o(oies)f(on)o(t)e(\023)-21 b(et)o(\023)g(e)13 b(explor)o(\023)-21 b(ees.)14 b(p)o(armi)g(les)75 2203 y(id)o(\023)-21 b(ees)12 b(plus)g(r)o(\023)-21 b(epandues,)11 b(on)h(trouv)o(e)f(les)h(pro)s (jets)f(visan)o(t)g(\022)-23 b(a)11 b(extraire)h(le)g(parall)o(\023)-21 b(elisme)13 b(de)f(sp)o(\023)-21 b(eci\014cations)75 2260 y(donn)o(\023)g(ees)11 b(sous)h(forme)f(fonctionnelle,)i(comme)f (dans)g fn(sisal)p fq([41)n(])f(par)h(exemple,)g(ou)g(d'enric)o(hir)h (un)f(lan-)75 2316 y(gage)f(de)h(t)o(yp)q(e)g(fonctionnel)h(a)o(v)o(ec)

e(des)h(ob)s(jets)f(parall)o(\022)-21 b(eles)12 b(comme)g(dans)g fn(nesl)p fq([3])f(ou)h(dp-hask)o(ell[25)o(].)146 2429 y(une)k(v)o(oie)h(in)o(term)o(\023)-21 b(ediaire)17 b(consiste)g(\022) -23 b(a)16 b(consid)o(\023)-21 b(erer)16 b(d'ab)q(ord)g(les)h(asp)q (ects)g(g)o(\023)-21 b(eom)o(\023)g(etriques)14 b(relatifs)75 2486 y(au)19 b(mo)q(d)o(\022)-21 b(ele)19 b(de)h(programmation)e (parall)o(\022)-21 b(ele,)20 b(\022)-23 b(a)19 b(d)o(\023)-21 b(e\014nir)19 b(des)h(op)o(\023)-21 b(erations)18 b(sur)h(ces)g(ob)s (jets.)f(c'est)h(le)75 2542 y(c)o(hoix)13 b(fait)g(dans)g(les)h (langages)f fn(alpha)p fq(,)g fn(cr)m(yst)m(al)g fq(ou)g fn(pei)p fq(.)f(p)o(ar)g(exemple,)i fn(alpha)f fq(est)g(un)h(langage)p eop %%page: 109 111 109 110 bop 75 269 a fl(5.10.)29 b(tra)-5 b(v)g(a)o(ux)15 b(rela)l(tifs)1113 b fq(109)75 397 y(fonctionnel)22 b(p)q(ermettan)o(t) e(de)g(sp)o(\023)-21 b(eci\014er)21 b(un)g(probl)o(\022)-21 b(eme)20 b(sous)h(la)f(forme)g(d')o(\023)-21 b(equations)20 b(r)o(\023)-21 b(ecurren)o(tes,)75 453 y(p)q(ortan)o(t)14 b(sur)h(des)h(v)m(ariables)g(repr)o(\023)-21 b(esen)o(tan)o(t)14 b(des)i(domaines)f(p)q(oly)o(\023)-21 b(edriques.)146 510 y(ces)10 b(trois)h(pro)s(jets)f(on)o(t)h(en)g(comm)o(un)g(d'a)o(v)o (oir)f(d)o(\023)-21 b(e\014ni)12 b(un)f(formalisme)g(autorisan)o(t)f (les)i(transforma-)75 567 y(tions)k(de)h(programmes.)e(ces)h (transformations)f(p)q(euv)o(en)o(t)h(mo)q(di\014er)h(notammen)o(t)f (les)h(informations)75 623 y(g)o(\023)-21 b(eom)o(\023)g(etriques)13 b(du)j(programme.)75 746 y fw(5.10.2)55 b(comparaison)18 b(a)n(v)n(ec)h fv(alpha)146 832 y fq(les)h(tra)o(v)m(aux)g(men)o(\023) -21 b(es)20 b(\022)-23 b(a)20 b(l'irisa)i(sur)f(le)g(langage)f fn(alpha)h fq(son)o(t)f(pro)q(c)o(hes)g(de)h(ceux)g(en)o(trepris)75 888 y(autour)15 b(de)g fn(pei)p fq(.)75 1009 y fo(le)j(langage)h fn(alpha)146 1096 y(alpha)i fq(est)g(un)h(langage)f(con\030)-20 b(cu[39])21 b(\022)-23 b(a)21 b(l'origine)h(p)q(our)g(la)f (programmation)f(d'arc)o(hitectures)75 1152 y(parall)o(\022)-21 b(eles)16 b(sync)o(hrones,)f(en)h(p)q(ermettan)o(t)f(d'exprimer)h(un)g (probl)o(\022)-21 b(eme)15 b(sous)g(la)h(forme)f(d'un)h(syst)o(\022)-21 b(eme)75 1208 y(d')o(\023)g(equations)14 b(r)o(\023)-21 b(ecurren)o(tes)14 b(a\016nes)h(sur)g(des)h(domaines)f(p)q(oly)o(\023) -21 b(edriques.)16 b(un)f(programme)f fn(alpha)i fq(est)75 1265 y(constitu)o(\023)-21 b(e)21 b(de)h(v)m(ariables)h(en)f(en)o(tr)o (\023)-21 b(ee)20 b(et)i(en)g(sortie,)f(et)g(d'un)h(syst)o(\022)-21 b(eme)21 b(d')o(\023)-21 b(equations)20 b(de)i(la)g(forme)75 1321 y fm(x)f fq(=)d fm(exp)p fq(,)g(o)q(\022)-24 b(u)18 b fm(x)k fq(est)c(une)h(v)m(ariable)h(d)o(\023)-21 b(eclar)o(\023)g(ee) 17 b(sur)i(un)f(domaine)h(p)q(oly)o(\023)-21 b(edrique)20 b fm(d)q(om)p fq(\()p fm(x)t fq(\),)c(app)q(el)o(\023)-21 b(e)75 1378 y(domaine)16 b(spatial,)f(et)g fm(exp)g fq(est)g(une)h (fonction)g(d)o(\023)-21 b(e\014nie)16 b(sur)f fm(d)q(om)p fq(\()p fm(x)t fq(\).)146 1434 y(les)h(exp)q(essions)h(son)o(t)d(form)o (\023)-21 b(ees)15 b(de)h(v)m(ariables)h(et)e(d')p fj(op)o(\023)-22 b(er)n(ateurs)p fq(.)16 b(les)g(op)o(\023)-21 b(erateurs)14 b(son)o(t)h(de)h(deux)75 1491 y(sortes)7 b(:)143 1585 y({)23 b(les)17 b(op)o(\023)-21 b(erateurs)15 b(immmobiles)j(g)o(\023) -21 b(en)o(\023)g(eralisen)o(t)15 b(la)i(notion)g(usuelle)i(d'op)o (\023)-21 b(erateur)15 b(aux)h(domaines)189 1642 y(spatiaux.)g(ils)h(p) q(ermetten)o(t)g(l'application)h(d'op)o(\023)-21 b(erateurs)15 b(arithm)o(\023)-21 b(etiques)17 b(ou)f(logiques)i(\022)-23 b(a)16 b(une)189 1698 y(collection)g(de)e(v)m(aleurs.)h(l'op)o(\023)-21 b(erateur)13 b(immobile)j(est)e(sem)o(blable)h(\022)-23 b(a)14 b(l'op)o(\023)-21 b(eration)13 b(fonctionnelle)189 1755 y(de)i fn(pei)p fq(.)143 1849 y({)23 b(les)16 b(op)o(\023)-21

b(erateurs)13 b(spatiaux)8 b(:)243 1944 y({)23 b(l')p fo(op)o(\023)-25 b(erateur)24 b(de)f(restriction)f fq(p)q(ermet)f(de)g (restreindre)h(le)f(domaine)h(de)f(d)o(\023)-21 b(e\014nition)289 2000 y(d'une)16 b(expression.)g(le)g(domaine)g(de)g(l'expression)g fm(e)i fq(d)o(\023)-21 b(e\014nie)16 b(par)f fm(e)g fq(=)f fm(d)1610 1984 y fx(0)1634 2000 y fq(:)f fm(x)19 b fq(\()p fm(x)f fq(est)289 2057 y(restrein)o(te)d(au)g(domaine)h fm(d)774 2040 y fx(0)785 2057 y fq(\))f(est)g fm(d)q(om)p fq(\()p fm(e)s fq(\))c(=)i fm(d)1161 2040 y fx(0)1183 2057 y ff(\\)d fm(d)q(om)p fq(\()p fm(x)t fq(\).)243 2130 y({)23 b(l')p fo(op)o(\023)-25 b(erateur)17 b(case)p fq(,)e(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t)13 b(utilis)o(\023)-21 b(e)16 b(en)f(conjonction)g(a)o(v)o(ec)g(l'op)o(\023)-21 b(erateur)13 b(de)j(res-)289 2187 y(triction,)j(p)q(ermet)h(de)g(d)o (\023)-21 b(e\014nir)19 b(di\013)o(\023)-21 b(eremmen)o(t)19 b(une)h(v)m(ariables)h(sur)e(des)h(sous-domaines)289 2243 y(disjoin)o(ts.)15 b(une)g(expression)g fm(e)i fq(d)o(\023)-21 b(e\014nie)15 b(par)g(cas)f(est)h(not)o(\023)-21 b(ee)13 b fm(e)k fq(=)e fo(case)g fm(exp)1613 2250 y fg(1)1639 2243 y fq(;)f fm(:)8 b(:)g(:)e fq(;)14 b fm(exp)1824 2250 y fw(n)289 2299 y fo(esac)p fq(;)h(et)g fm(d)q(om)p fq(\()p fm(e)s fq(\))c(=)i fm(d)q(om)p fq(\()p fm(exp)885 2306 y fg(1)903 2299 y fq(\))d ff([)g fm(:)e(:)g(:)g ff([)j fm(d)q(om)p fq(\()p fm(exp)1264 2306 y fw(n)1286 2299 y fq(\).)243 2373 y({)23 b(l')p fo(op)o(\023)-25 b(erateur)19 b(de)g(d)o(\023)-25 b(ep)q(endance)18 b fq(exprime)f(une)g(fonction)g(en)o(tre)g(domaines)g(spatiaux.)289 2429 y(l'expression)f fm(e)s(:)p fq(\()p fm(dep)p fq(\))c(d)o(\023)-21 b(enote)15 b(la)g(comp)q(osition)h(de)f fm(e)j fq(et)c fm(dep)p fq(,)h(o)q(\022)-24 b(u)15 b fm(dep)p fq(,)f(app)q(el)o(\023) -21 b(ee)16 b fj(fonc-)289 2486 y(tion)i(de)g(d)o(\023)-22 b(ep)n(endanc)n(e)p fq(,)16 b(est)h(une)h(fonction)g(a\016ne)f(don)o(t) g(l'image)h(est)f(le)h(domaine)g(de)g fm(e)s fq(.)289 2542 y(cette)d(d)o(\023)-21 b(ep)q(endance)16 b(p)q(eut)h(aussi)f (s'exprimer)f(\022)-23 b(a)16 b(l'aide)g(d'une)g(notation)f(\023)-21 b(equiv)m(alen)o(te)16 b(de)p eop %%page: 110 112 110 111 bop 75 269 a fq(110)381 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)289 397 y fq(t)o(yp)q(e)21 b(tableau)8 b(:)21 b(ainsi)i(les)f(expressions)h fm(e)1051 380 y fx(0)1085 397 y fq(=)g fm(e)s(:)p fq(\()p fm(z)h ff(!)g fm(f)5 b fq(\()p fm(z)r fq(\)\))21 b(et)g fm(e)1546 380 y fx(0)1557 397 y fq([)p fm(z)r fq(])h(=)i fm(e)s fq([)p fm(f)5 b fq(\()p fm(z)r fq(\)],)289 453 y fm(z)25 b ff(2)e fm(d)q(om)p fq(\()p fm(f)5 b fq(\),)20 b(son)o(t)f(\023)-21 b(equiv)m(alen)o(tes.)22 b(l'op)o(\023)-21 b(erateur)20 b(de)i(d)o(\023)-21 b(ep)q(endance)22 b(est)f(sem)o(blable)i(\022)-23 b(a)289 510 y(l'op)o(\023)i(eration)14 b(g)o(\023)-21 b(eom)o(\023)g(etrique)13 b(de)j fn(pei)p fq(.)243 585 y({)23 b(l')p fo(op)o(\023)-25 b(erateur)19 b(de)h(r)o(\023)-25 b(eduction)18 b fq(est)f(sem)o(blable)h(\022)-23 b(a)17 b(la)g(r)o(\023)-21 b(eduction)17 b(g)o(\023)-21 b(eom)o(\023)g (etrique)16 b(de)h fn(pei)p fq(.)289 641 y(etan)o(t)i(donn)o(\023)-21 b(e)19 b(une)i(expression)f fm(e)j fq(et)c fm(f)25 b fq(une)c(fonction)f(a\016ne)g(d)o(\023)-21 b(e\014nie)21 b(sur)f fm(d)q(om)p fq(\()p fm(e)s fq(\),)289 698 y(l'expression)f fo(reduce)p fq(\()p ff(\010)p fm(;)8 b(f)r(;)g(e)s fq(\))16 b(a)j(p)q(our)f(v)m(aleur)i(au)e(p)q(oin)o(t)h fm(z)r fq(,)g fm(z)h ff(2)e fm(img)r fq(\()p fm(f)5 b fq(\),)17

b(la)i(v)m(aleur)289 754 y ff(\010)324 761 y fw(z)341 753 y fm(0)352 761 y fx(2)p fw(d)q(om)p fg(\()p fw(e)r fg(\))p fx(j)p fw(f)s fg(\()p fw(z)558 753 y fm(0)571 761 y fg(\)=)p fw(z)629 754 y fm(e)663 761 y fw(z)680 753 y fm(0)692 754 y fq(,)c(o)q(\022)-24 b(u)15 b fm(e)817 761 y fw(z)834 753 y fm(0)862 754 y fq(d)o(\023)-21 b(enote)15 b(la)g(v)m(aleur)i(de)e fm(e)j fq(au)d(p)q(oin)o(t)g fm(z)1511 738 y fx(0)1523 754 y fq(.)143 852 y({)23 b(t)l (ransformations.)14 b(les)i(transformations)e(du)i(langage)f fn(alpha)h fq(s'exprimen)o(t)g(sous)f(la)h(forme)189 909 y(de)h(r)o(\022)-21 b(egles)17 b(de)g(r)o(\023)-21 b(e)o(\023)g(ecritures)16 b(du)i(programme.)d(p)o(armi)i(les)h(r)o (\022)-21 b(egles)16 b(de)i(transformation)e(les)i(plus)189 965 y(utilis)o(\023)-21 b(ees,)18 b(on)g(trouv)o(e)f(le)i(c)o(hangemen) o(t)f(de)g(base.)f(un)i(c)o(hangemen)o(t)e(de)i(base)f fm(f)24 b fq(sur)18 b(une)g(v)m(a-)189 1022 y(riable)d fm(a)g fq(d)o(\023)-21 b(e\014nie)15 b(sur)f fm(d)q(om)p fq(\()p fm(a)p fq(\))f(est)h(p)q(ossible)j(si)d(il)i(existe)f(une)g (fonction)f fm(g)g fq(in)o(v)o(erse)h(de)g fm(f)21 b fq(sur)189 1078 y fm(d)q(om)p fq(\()p fm(a)p fq(\).)14 b(la)h(r)o(\023)-21 b(e)o(\023)g(ecriture)14 b(du)i(programme)e(se)h (fait)g(de)h(la)f(mani)o(\022)-21 b(ere)15 b(suiv)m(an)o(te)8 b(:)p 189 1161 577 2 v 188 1218 2 57 v 214 1201 a fm(a)f fq(:)91 b fm(d)q(om)p fq(\()p fm(a)p fq(\))p 765 1218 v 188 1274 v 214 1257 a fk(let)p 765 1274 v 188 1330 v 359 1313 a(a)24 b(=)f(...)h(a)f(...;)p 765 1330 v 188 1387 v 359 1370 a(...)g(=)h(...)f(a)h(...;)p 765 1387 v 188 1443 v 214 1426 a(tel;)p 765 1443 v 189 1445 577 2 v 841 1314 a ff(!)p 963 1161 696 2 v 962 1218 2 57 v 987 1201 a fm(a)8 b fq(:)91 b fm(d)q(om)p fq(\()p fm(a)p fq(\))p fm(:f)p 1658 1218 v 962 1274 v 987 1257 a fk(let)p 1658 1274 v 962 1330 v 1133 1313 a(a)23 b(=)h(\(...)f(a.)p fm(f)31 b fk(...\).)p fm(g)p fk(;)p 1658 1330 v 962 1387 v 1133 1370 a(...)23 b(=)h(...)f(a.)p fm(f)30 b fk(...;)p 1658 1387 v 962 1443 v 987 1426 a(tel;)p 1658 1443 v 963 1445 696 2 v 75 1550 a fo(di\013)o(\023)-25 b(erences)17 b(en)o(tre)g fn(pei)g fo(et)h fn(alpha)146 1638 y fq(les)c(programmes)f fn(alpha)h fq(son)o(t)g(fortemen)o(t)f(t)o(yp)o(\023)-21 b(es,)12 b(c)o(haque)i(v)m(ariable)h(\023)-21 b(etan)o(t)12 b(explicitemen)o(t)k(d)o(\023)-21 b(e-)75 1694 y(clar)o(\023)g(ees)11 b(sur)h(un)h(domaine)f(\()o(\023)-21 b(ev)o(en)o(tuellemen)o(t)12 b(param)o(\023)-21 b(etr)o(\023)g(e\).)8 b(nous)k(a)o(v)o(ons)f(vu)h (qu'en)g fn(pei)g fq(les)g(domaines)75 1751 y(de)i(v)m(aleurs)g(ou)g (de)g(r)o(\023)-21 b(ef)o(\023)g(erence)12 b(des)i(c)o(hamps)f(de)h (donn)o(\023)-21 b(es)13 b(p)q(ouv)m(aien)o(t)h(^)-21 b(etre)12 b(non-b)q(orn)o(\023)-21 b(es.)13 b(nous)h(laissons)75 1807 y(au)j(con)o(tr^)-23 b(oleur)18 b(de)f(t)o(yp)q(e)h(la)f(c)o (harge)g(d'inf)o(\023)-21 b(erer)17 b(les)h(informations)g(manquan)o (tes)f(sur)g(ces)g(domaines,)75 1864 y(et)e(de)h(v)o(\023)-21 b(eri\014er)15 b(leurs)h(v)m(alidit)o(\023)-21 b(es.)146 1978 y(outre)17 b(les)g(limites)i(sur)e(la)g(forme)f(des)i(domaines)f (et)g(les)h(fonctions)f(de)g(d)o(\023)-21 b(ep)q(endances,)18 b(la)f(di\013)o(\023)-21 b(e-)75 2034 y(rence)14 b(ma)s(jeure)e(en)o (tre)h(ces)g(deux)h(langages)f(son)o(t)f(les)i(con)o(train)o(tes)e(imp) q(os)o(\023)-21 b(ees)13 b(p)q(our)g(sp)o(\023)-21 b(eci\014er)14 b(l')o(\023)-21 b(egalit)o(\023)g(e)75 2091 y(de)17 b(deux)h (expressions.)f(en)g fn(pei)p fq(,)g(les)h(con)o(train)o(tes)e(son)o(t) g(pro)q(c)o(hes)i(de)f(celles)i(des)e(langages)g(\022)-23 b(a)16 b(paral-)75 2147 y(l)o(\023)-21 b(elisme)16 b(de)g(donn)o(\023) -21 b(ees)14 b(don)o(t)h(la)g(s)o(\023)-21 b(eman)o(tique)15

b(des)g(indices)i(repr)o(\023)-21 b(esen)o(te)14 b(des)i(pro)q (cesseurs)f(virtuels.)h(on)75 2204 y(retrouv)o(e)h(par)f(exemple)j(la)e (notion)h(de)f(conformit)o(\023)-21 b(e)16 b(pr)o(\023)-21 b(esen)o(te)17 b(dans)g(ces)h(langages)7 b(:)17 b(une)h(op)o(\023)-21 b(eration)75 2260 y(fonctionnelle)15 b(o)q(\022)-24 b(u)14 b(g)o(\023)-21 b(eom)o(\023)g(etrique)11 b(en)j fn(pei)f fq(ne)g(p)q(eut)g(^)-21 b(etre)12 b(appliqu)o(\023)-21 b(ee)14 b(que)g(sur)f(des)h(c)o(hamps)f(d)o(\023)-21 b(e\014nis)14 b(sur)75 2316 y(m)o(^)-21 b(eme)11 b(espace)i(et)g(don)o (t)f(un)g(des)h(domaines)g(de)g(r)o(\023)-21 b(ef)o(\023)g(erence)11 b(inclut)j(ceux)f(des)g(autres.)e(cette)h(con)o(train)o(te)75 2373 y(oblige)i(le)g(programmeur)f(\022)-23 b(a)12 b(aligner)j(les)f(c) o(hamps)f(du)g(programme)g(\022)-23 b(a)12 b(l'aide)j(de)e(l'op)o(\023) -21 b(eration)13 b(de)g(c)o(han-)75 2429 y(gemen)o(t)19 b(de)h(base,)f(qui)h(est)f(une)h(op)o(\023)-21 b(eration)18 b(\022)-23 b(a)19 b(part)g(en)o(ti)o(\022)-21 b(ere)19 b(du)h(langage)f fn(pei)p fq(.)g(les)h(informations)75 2486 y(d'alignemen)o(ts)c(son)o(t)f(donc)h(donn)o(\023)-21 b(ees)16 b(d)o(\022)-21 b(es)15 b(la)h(phase)g(initiale)i(de)e (conception.)g(les)g(transformations)75 2542 y(donnan)o(t)c(des)g (programmes)f(fortemen)o(t)f(\023)-21 b(equiv)m(alen)o(ts)13 b(se)f(fon)o(t)f(\023)-21 b(egalemen)o(t)11 b(par)h(c)o(hangemen)o(t)f (de)i(base.)p eop %%page: 111 113 111 112 bop 75 269 a fl(5.10.)29 b(tra)-5 b(v)g(a)o(ux)15 b(rela)l(tifs)1113 b fq(111)75 397 y(cette)18 b(op)o(\023)-21 b(eration)17 b(\023)-21 b(etan)o(t)17 b(la)i(seule)g(\022)-23 b(a)19 b(mo)q(di\014er)g(la)g(fonction)g fm(\033)h fq(d'un)f(c)o(hamp)g (de)g(donn)o(\023)-21 b(ees,)18 b(qui)h(ex-)75 453 y(prime)14 b(la)f(fa\030)-20 b(con)13 b(don)o(t)g(les)h(v)m(aleurs)g(seron)o(t)f (plac)o(\023)-21 b(ees)13 b(en)h(\014n)f(de)h(compte)f(sur)g(les)h(pro) q(cesseurs)g(virtuels,)75 510 y(les)j(alignemen)o(ts)g(sp)o(\023)-21 b(eci\014)o(\023)g(es)16 b(dans)g(l')o(\023)-21 b(enonc)o(\023)g(e)15 b(original)j(son)o(t)d(conserv)o(\023)-21 b(es)16 b(tout)f(au)h(long)h (de)f(ce)h(t)o(yp)q(e)f(de)75 566 y(transformation.)e(en)j fn(alpha)p fq(,)f(par)g(con)o(tre)g(aucune)h(con)o(train)o(te)f(de)g (ce)h(t)o(yp)q(e)f(n'est)g(imp)q(os)o(\023)-21 b(ee)7 b(:)16 b(l'ali-)75 623 y(gnemen)o(t)i(est)h(l')o(\023)-21 b(etap)q(e)18 b(ultime)h(de)g(transformation,)e(et)i(consiste)g(\022) -23 b(a)18 b(d)o(\023)-21 b(ecider)19 b(que)g(certains)g(indices)75 679 y(repr)o(\023)-21 b(esen)o(ten)o(t)14 b(les)i(dimensions)h (spatiales)f(de)f(l'arc)o(hitecture)h(virtuelle.)75 866 y fo(d)o(\023)-25 b(ev)o(elopp)q(emen)o(ts)146 956 y fq(une)19 b(des)h(directions)g(de)g(rec)o(herc)o(he)f(comm)o(une)h (\022)-23 b(a)19 b(ces)g(deux)h(langages,)e(est)h(la)h(traduction)f(de) 75 1012 y(programmes)10 b(exprim)o(\023)-21 b(es)10 b(dans)h(ces)g (formalismes,)g(v)o(ers)f(les)h(principaux)i(mo)q(d)o(\022)-21 b(eles)11 b(de)g(programmation)75 1069 y(existan)o(ts.)146 1127 y(un)i(traducteur)h(quali\014)o(\023)-21 b(e)14 b(de)g(na)-5 b(\177)-18 b(\020f)14 b([61)o(])f(a)f(\023)-21 b(et)o(\023)g(e)12 b(con\030)-20 b(cu)14 b(p)q(our)f(\023)-21 b(ev)m(aluer)14 b(les)g(v)m(ariables)h(r)o(\023)-21 b(esultats)12 b(d'un)75 1184 y(programme)19 b fn(alpha)p fq(.)g(les)i(programmes)d fn(alpha)i fq(p)q(euv)o(en)o(t)g(^)-21 b(etre)18 b(vus)i(comme)g(des)g (programmes)75 1240 y(fonctionnels,)g(et)e(l')o(\023)-21 b(ev)m(aluation)19 b(des)g(di\013)o(\023)-21 b(eren)o(tes)18 b(v)m(aleurs)i(d'une)f(v)m(ariable)h(p)q(eut)g(se)f(faire)f(selon)i (les)75 1297 y(d)o(\023)-21 b(ep)q(endances)20 b(causales)f(en)o(tre)g (les)h(v)m(aleurs)g(des)f(v)m(ariables.)h(ce)f(princip)q(e)j(est)c(le)i (m)o(^)-21 b(eme)18 b(que)i(celui)75 1353 y(implan)o(t)o(\023)-21

b(e)21 b(dans)g(le)g(mo)q(dule)h(d'in)o(terpr)o(\023)-21 b(etation)20 b(fonctionnelle)j(de)e fn(pei)p fq(,)f(o)q(\022)-24 b(u)21 b(c)o(hacune)h(des)f(v)m(aleurs)75 1409 y(d'un)16 b(c)o(hamp)f(de)g(sortie)g(est)g(une)h(fonction)g(des)f(v)m(aleurs)h (de)g(c)o(hamps)f(en)g(en)o(tr)o(\023)-21 b(ee.)14 b(une)i (optimisation)75 1466 y(r)o(\023)-21 b(ealis)o(\023)g(ee)15 b(est)h(la)g(m)o(\023)-21 b(emorisation)16 b(des)g(v)m(aleurs)h(d)o (\023)-21 b(ej\022)e(a)15 b(calcul)o(\023)-21 b(ees)17 b(a\014n)f(d')o(\023)-21 b(eviter)16 b(les)g(r)o(\023)-21 b(e-)o(\023)g(ev)m(aluations.)15 b(un)75 1522 y(autre)e(probl)o(\022) -21 b(eme)12 b(rencon)o(tr)o(\023)-21 b(e)12 b(est)h(le)h(parcours,)e (au)h(mo)o(y)o(en)f(d'un)h(nid)h(de)g(b)q(oucles,)g(du)f(domaine)h(p)q (oly-)74 1579 y(\022)-21 b(edrique)16 b(d'une)g(v)m(ariable)h(r)o(\023) -21 b(esultat)15 b(p)q(our)g(demander)i(l')o(\023)-21 b(ev)m(aluation)16 b(des)g(v)m(aleurs)g(plac)o(\023)-21 b(ees)16 b(en)g(c)o(hacun)75 1635 y(des)i(p)q(oin)o(ts)h(en)o(tiers)f (du)g(p)q(oly)o(\022)-21 b(edre.)18 b(dans)f([50],)g(il)i(est)e(prop)q (os)o(\023)-21 b(e)17 b(un)i(autre)e(ordonnancemen)o(t)h(p)q(our)75 1692 y(l'ex)o(\023)-21 b(ecution)13 b(du)g(programme)f(s)o(\023)-21 b(equen)o(tiel,)13 b(que)g(celui)i(induit)f(par)e(les)i(d)o(\023)-21 b(ep)q(endances)13 b(causales.)g(l'id)o(\023)-21 b(ee)75 1748 y(est)15 b(d'utiliser)i(les)f(tec)o(hniques)h(d'ordonnancemen)o(t) e(et)g(de)h(placemen)o(t)g(utilis)o(\023)-21 b(ees)17 b(en)e(parall)o(\023)-21 b(elisation)75 1805 y(automatique.)16 b(l'ordonnancemen)o(t)g(fournit)g(un)h(ordre)f(d')o(\023)-21 b(ev)m(aluation)17 b(des)f(v)m(aleurs)h(coh)o(\023)-21 b(eren)o(t,)15 b(tout)75 1861 y(en)g(\023)-21 b(evitan)o(t)14 b(les)i(c)o(hangemen)o(ts)f(de)h(con)o(texte)f(co)q(^)-24 b(uteux)16 b(qu'implique)i(l')o(\023)-21 b(ev)m(aluation)16 b(fonctionnelle)h(du)75 1918 y(traducteur)d(na)-5 b(\177)-18 b(\020f.)14 b(notons)f(que)h(le)h(sc)o(h)o(\023)-21 b(ema)13 b(de)h(traduction)h(pro)q(c)o(\022)-21 b(ede,)13 b(de)h(mani)o(\022)-21 b(ere)14 b(analogue)g(\022)-23 b(a)14 b(celui)75 1974 y(exp)q(os)o(\023)-21 b(e)18 b(dans)g(le)h(c)o(hapitre)g(pr)o(\023)-21 b(ec)o(\023)g(eden)o(t,)17 b(par)h(des)g(mises)h(sous)f(formes)g (normales)g(des)h(programmes)75 2030 y fn(alpha)d fq(par)f (l'application)i(con\015uen)o(te)f(de)f(r)o(\022)-21 b(egles)15 b(de)g(r)o(\023)-21 b(e)o(\023)g(ecriture.)146 2145 y(la)23 b(pr)o(\023)-21 b(ec)o(\023)g(eden)o(te)23 b(m)o(\023)-21 b(etho)q(de)23 b(est)g(\023)-21 b(etendue)24 b([49)o(])f(p)q(our)h(la)g(g)o(\023)-21 b(en)o(\023)g(eration)22 b(de)i(programmes)f(data-)75 2202 y(parall)o(\022)-21 b(eles)21 b(imp)o(\023)-21 b(eratifs.)19 b(le)i(m)o(^)-21 b(eme)19 b(princip)q(e)k(d'ordonnancemen)o(t)d(est)g(conserv)o(\023)-21 b(e.)19 b(p)o(ar)h(comparai-)75 2258 y(son,)15 b(la)h(m)o(\023)-21 b(etho)q(de)15 b(exp)q(os)o(\023)-21 b(ee)16 b(p)q(our)g(notre)f (traducteur)h(utilise)h(un)g(ordonnancemen)o(t)e(h)o(ybride)9 b(:)15 b(nous)75 2315 y(utilisons)j(l'ordre)f(induit)h(des)f(d)o(\023) -21 b(ep)q(endances)18 b(causales)f(p)q(our)g(les)f(\023)-21 b(equations)16 b(simples,)i(et)e(nous)h(cal-)75 2371 y(culons)j(un)f(cadencemen)o(t)h(p)q(our)f(les)g(d)o(\023)-21 b(e\014nitions)20 b(r)o(\023)-21 b(ecursiv)o(es)18 b(quand)i(aucune)f (dimension)i(ne)e(p)q(eut)75 2427 y(repr)o(\023)-21 b(esen)o(ter)14 b(le)i(temps)f(de)h(mani)o(\022)-21 b(ere)14 b(\023)-21 b(eviden)o(te.)146 2486 y(un)16 b(probl)o(\022)-21 b(eme)16 b(suppl)o(\023)-21 b(emen)o(taire)17 b(est)f(l'alignemen)o(t)h(des)g (donn)o(\023)-21 b(ees)16 b(sur)g(la)h(grille)h(de)e(pro)q(cesseurs)75 2542 y(virtuels,)d(de)h(fa\030)-20 b(con)12 b(\022)-23 b(a)13 b(minimiser)h(le)g(co)q(^)-24 b(ut)12 b(des)i(comm)o (unications.)f(il)g(est)g(par)f(cons)o(\023)-21 b(equen)o(t)13

b(n)o(\023)-21 b(ecessaire)p eop %%page: 112 114 112 113 bop 75 269 a fq(112)381 b fl(chapitre)16 b(5.)30 b fn(pei)15 b fl(et)g(les)g(langa)o(ges)g(d)o(a)l(t)l(a-p)l(arall)1735 257 y(\022)1729 269 y(eles)75 397 y fq(de)23 b(trouv)o(er)f(un)h(\\b)q (on")f(c)o(hangemen)o(t)h(de)g(base)f(p)q(our)h(l'alignemen)o(t)h(des)f (v)m(ariables.)g(a)g(la)g(di\013)o(\023)-21 b(e-)75 453 y(rence)20 b(d'un)h(programme)d fn(alpha)p fq(,)i(un)g(programme)f fn(pei)h fq(comp)q(orte)f(les)i(informations)e(d'aligne-)75 510 y(men)o(t)d(des)g(c)o(hamps.)f(ce)h(son)o(t)f(ces)h(alignemen)o (ts,)g(rep)o(\023)-21 b(erables)16 b(par)g(l'application)i(d'un)e(c)o (hangemen)o(t)75 566 y(de)c(base,)f(don)o(t)f(nous)i(nous)f(serv)o(ons) g(dans)g(la)h(traduction)f(des)h(programmes)e fn(pei)h fq(en)g(hpf.)g(l'optique)75 623 y(d')p fn(alpha)k fq(p)q(eut)g(s'a)o(v) o(\023)-21 b(erer)12 b(tr)o(\022)-21 b(es)13 b(in)o(t)o(\023)-21 b(eressan)o(te)14 b(d)o(\022)-21 b(es)13 b(lors)i(que)g(des)g(outils)g (de)g(placemen)o(t)g(automatique)75 679 y(seraien)o(t)h(e\016caces.)g (dans)g(ces)g(conditions,)h(on)f(p)q(ourrait)g(cac)o(her)g(la)h(phase)f (d'alignemen)o(t)h(\022)-23 b(a)15 b(l'utili-)75 736 y(sateur.)i(cep)q(endan)o(t,)i(nous)f(a)o(v)o(ons)g(d)o(\023)-21 b(ej\022)e(a)17 b(soulign)o(\023)-21 b(e)18 b(la)h(di\016cult)o(\023) -21 b(e)19 b(du)g(probl)o(\022)-21 b(eme)7 b(:)18 b(darte)f(et)h(rob)q (ert)75 792 y([14)o(])d(on)o(t)g(notammen)o(t)f(mon)o(tr)o(\023)-21 b(e)14 b(que)h(le)i(probl)o(\022)-21 b(eme)15 b(de)h(l'alignemen)o(t)g (sur)f(une)h(grille)h(de)f(pro)q(cesseurs)75 849 y(virtuels,)i(des)g (calculs)h(et)e(des)h(comm)o(unications)g(de)g(mani)o(\022)-21 b(ere)18 b(\022)-23 b(a)17 b(minimiser)i(les)f(comm)o(unications,)75 905 y(est)f(np-complet.)i(ce)e(probl)o(\022)-21 b(eme)18 b(n)o(\023)-21 b(ecessite)18 b(donc)g(de)g(recourir)g(\022)-23 b(a)17 b(des)i(heuristiques,)g(qui)f(p)q(euv)o(en)o(t)74 962 y(^)-21 b(etre)14 b(fond)o(\023)-21 b(ees)15 b(sur)g(le)h(v)o (olume)f(des)h(comm)o(unications)g(induites)h(par)e(un)g(alignemen)o(t) h([51)o(].)p eop %%page: 113 115 113 114 bop 1778 269 a fq(113)75 716 y fi(chapitre)32 b(6)75 932 y fp(conclusion)146 1163 y fq(il)16 b(est)g(classique)h(de)f (consid)o(\023)-21 b(erer)16 b(qu'un)g(programme)f(est)g(une)i (fonction)f(sur)g(des)g(ob)s(jets)f(t)o(yp)o(\023)-21 b(es)6 b(:)75 1219 y(cette)13 b(fonction)f(d)o(\023)-21 b(e\014nit)13 b fj(c)n(e)h(que)g(c)n(alcule)e fq(le)h(programme.)e (d'un)i(p)q(oin)o(t)g(de)g(vue)g(plus)g(op)o(\023)-21 b(erationnel,)13 b(on)75 1276 y(doit)h(en)g(outre)f(d)o(\023)-21 b(e\014nir)14 b fj(c)n(omment)f fq(ces)h(calculs)h(son)o(t)e(organis)o (\023)-21 b(es;)12 b(et)h(lorsque)h(l'implan)o(tation)h(supp)q(ose)75 1332 y(l'utilisation)20 b(d'une)e(mac)o(hine)h(parall)o(\022)-21 b(ele,)18 b(on)f(d)o(\023)-21 b(e\014nit)18 b(seulemen)o(t)h(un)f (ordre)g(partiel,)g(qui)h(p)q(eut)f(\(ou)75 1389 y(doit,)e(selon)h(les) h(mo)q(d)o(\022)-21 b(eles)16 b(d')o(\023)-21 b(ex)o(\023)g(ecution\)) 14 b(^)-21 b(etre)15 b(compl)o(\023)-21 b(et)o(\023)g(e)15 b(en)i(disan)o(t)g fj(o)q(\022)-24 b(u)17 b fq(son)o(t)f(op)o(\023)-21 b(er)o(\023)g(es)14 b(ces)j(calculs)h(et)75 1445 y fj(o)q(\022)-24 b(u)16 b fq(son)o(t)e(plac)o(\023)-21 b(ees)16 b(les)f(donn)o(\023)-21 b(ees)15 b(impliqu)o(\023)-21 b(ees.)146 1566 y(cette)11 b(derni)o(\022)-21 b(ere)13 b(appro)q(c)o(he)g(conduit)g(naturellemen)o (t)h(\022)-23 b(a)12 b(l'in)o(tro)q(duction)h(d'une)g(certaine)g(\\g)o (\023)-21 b(eom)o(\023)g(e-)75 1623 y(trie")16 b(dans)f(l'expression)i (des)f(programmes,)e(compl)o(\023)-21 b(emen)o(taire)16 b(de)g(l'asp)q(ect)g(fonctionnel)h(habituel.)75 1679 y(c'est)d(ce)i(qui)g(est)f(\022)-23 b(a)15 b(la)g(base)h(des)f (concepts)h(de)f fn(pei)p fq(.)146 1744 y fn(pei)f fq(a)f(\023)-21

b(et)o(\023)g(e)12 b(con\030)-20 b(cu)15 b(\022)-23 b(a)14 b(une)g(\023)-21 b(ep)q(o)q(que)15 b(o)q(\022)-24 b(u)14 b(p)q(eu)h(de)g(langages)f(incluaien)o(t)j(cette)d(notion)h(de)g(g)o (\023)-21 b(eom)o(\023)g(etrie)75 1800 y(comme)20 b(caract)o(\023)-21 b(eristique)19 b(\022)-23 b(a)20 b(part)f(en)o(ti)o(\022)-21 b(ere)20 b(du)h(langage.)e(les)i(formalismes)f(comparables,)g(autori-) 75 1857 y(san)o(t)f(\023)-21 b(egalemen)o(t)19 b(le)i(ra\016nemen)o(t)f (et)g(la)g(transformation)f(de)i(programmes,)e(comme)h fn(alpha)p fq([39)o(],)75 1913 y fn(cr)m(yst)m(al)p fq([12)o(])d(ou)g fn(unity)p fq([10)n(])g(par)g(exemple,)h(n'in)o(t)o(\022)-21 b(egren)o(t)15 b(pas,)i(\022)-23 b(a)17 b(ce)g(p)q(oin)o(t,)g(les)h (con)o(train)o(tes)f(g)o(\023)-21 b(eo-)75 1970 y(m)o(\023)g(etriques) 17 b(sur)h(le)g(placemen)o(t)g(des)g(v)m(aleurs)h(pr)o(\023)-21 b(esen)o(tes)17 b(dans)g fn(pei)p fq(.)g(nous)h(a)o(v)o(ons)f(soulign)o (\023)-21 b(e)18 b(dans)f(la)75 2026 y(comparaison)11 b(a)o(v)o(ec)g(le)i(langage)e fn(alpha)p fq(,)g(que)h(cette)g(di\013)o (\023)-21 b(erence)11 b(mo)q(di\014ait)i(l'appro)q(c)o(he)f(de)g(la)g (concep-)75 2082 y(tion)h(d'un)g(programme)6 b(:)12 b(l')o(\023)-21 b(ecriture)13 b(d'un)f(programme)g fn(pei)g fq(imp)q(ose)h(en)g (e\013et)f(de)h(concev)o(oir)g fj(a)h(priori)75 2139 y fq(la)i(mani)o(\022)-21 b(ere)14 b(don)o(t)h(les)h(donn)o(\023)-21 b(ees)15 b(son)o(t)g(align)o(\023)-21 b(ees,)15 b(\022)-23 b(a)14 b(l'aide)j(de)e(l'op)o(\023)-21 b(eration)15 b(de)g(c)o (hangemen)o(t)g(de)h(base.)146 2260 y(notre)f(th)o(\022)-21 b(ese)15 b(est)h(que)g fn(pei)g fq(p)q(ermet)g(de)g(mo)q(d)o(\023)-21 b(eliser)17 b(les)f(concepts)h(des)f(langages)g(\022)-23 b(a)15 b(parall)o(\023)-21 b(elisme)75 2316 y(de)16 b(donn)o(\023)-21 b(ees)15 b(don)o(t)f(la)i(s)o(\023)-21 b(eman)o(tique)15 b(des)g(indices)j(des)d(v)m(ariables)i(est)e(celle)i(des)e(pro)q (cesseurs)h(virtuels.)75 2373 y(citons)22 b(par)g(exemple)i(c)530 2356 y fw(?)549 2373 y fq([54)o(],)e(mpl[38)o(],)f fn(hyperc)p fq([45],)h(et)g(dpce[53)n(].)g(le)h(langage)f ff(l)p fq([4])g(qui)75 2429 y(mo)q(d)o(\023)-21 b(elise)21 b(cette)e(appro)q (c)o(he)h(in)o(t)o(\022)-21 b(egre)19 b(une)i(certaine)f(forme)g(de)g (g)o(\023)-21 b(eom)o(\023)g(etrie,)17 b(limit)o(\023)-21 b(ee)21 b(au)f(minim)o(um,)75 2486 y(c'est-\022)-23 b(a-dire)17 b(\022)-23 b(a)16 b(la)g(d)o(\023)-21 b(e\014nition)18 b(de)e(v)o(ecteurs)g(et)h(\022)-23 b(a)16 b(la)g(r)o(\023)-21 b(ef)o(\023)g(erence)15 b(\022)-23 b(a)16 b(l'indice)j(couran)o(t)d (des)g(pro)q(cesseurs)75 2542 y(virtuels)i(asso)q(ci)o(\023)-21 b(es.)16 b(de)h(fait,)g(dans)g(ces)g(langages,)g(l'asp)q(ect)g(g)o (\023)-21 b(eom)o(\023)g(etrique)15 b(est)i(omnipr)o(\023)-21 b(esen)o(t)17 b(par)f(la)p eop %%page: 114 116 114 115 bop 75 269 a fq(114)1068 b fl(chapitre)15 b(6.)30 b(conclusion)75 397 y fq(notion)17 b(m)o(^)-21 b(eme)16 b(de)h(v)m(ariables)i(parall)o(\022)-21 b(eles)8 b(:)16 b(le)i(programmeur)e(doit)h(explicitemen)o(t)i(aligner,)f(par)e(des)75 453 y(comm)o(unications,)21 b(les)g(v)m(ariables)g(de)g(son)f (programme)g(de)h(fa\030)-20 b(con)20 b(\022)-23 b(a)20 b(ce)h(que)f(tous)g(les)h(calculs)h(ne)75 510 y(s'e\013ectuen)o(t)d (que)g(sur)g(des)h(v)m(aleurs)g(lo)q(cales)g(\022)-23 b(a)19 b(c)o(haque)h(pro)q(cesseur)f(virtuel.)h(nous)f(p)q(ensons)h (donc)75 566 y(que)d(b)q(eaucoup)g(de)g(tec)o(hniques)g(formelles)g(d)o (\023)-21 b(ev)o(elopp)o(\023)g(ees)16 b(dans)g fn(pei)h fq(p)q(euv)o(en)o(t)e(^)-21 b(etre)15 b(appliqu)o(\023)-21 b(ees)18 b(aux)75 623 y(langages)d(\022)-23 b(a)16 b(parall)o(\023)-21 b(elisme)17 b(de)f(donn)o(\023)-21 b(ees,)15 b(et)g(son)o(t)g(compl)o (\023)-21 b(emen)o(taires)15 b(d'autres)g(tra)o(v)m(aux)g(th)o(\023)-21 b(eoriques,)75 679 y(comme)15 b(ceux)h(sur)f(la)g(preuv)o(e)h(de)g

(programmes)e(data-parall)o(\022)-21 b(eles)15 b([5].)146 792 y(cep)q(endan)o(t,)d(pr)o(\023)-21 b(ealablemen)o(t)11 b(\022)-23 b(a)12 b(l'application)i(des)e(tec)o(hniques)h fn(pei)e fq(aux)h(langages)f(de)i(program-)75 849 y(mation)f(existan)o (ts,)g(il)g(\023)-21 b(etait)11 b(n)o(\023)-21 b(ecessaire)13 b(de)f(p)q(oursuivre)i(le)f(tra)o(v)m(ail)f(d'in)o(v)o(estigation)h (sur)f(la)h(d)o(\023)-21 b(e\014nition)75 905 y(du)19 b(formalisme)g(lui-m)o(^)-21 b(eme.)18 b(nous)h(a)o(v)o(ons)e(pr)o (\023)-21 b(esen)o(t)o(\023)g(e)17 b(dans)h(le)i(premier)f(c)o(hapitre) g(la)f(d)o(\023)-21 b(e\014nition)20 b(ac-)75 962 y(tuelle)h(du)e (formalisme.)g(nous)g(a)o(v)o(ons)f(ensuite)h(\023)-21 b(etendu)19 b(le)g(calcul)i(de)e(ra\016nemen)o(t)g(par)f(rapp)q(ort)h (\022)-23 b(a)75 1018 y(sa)18 b(d)o(\023)-21 b(e\014nition)19 b(originale,)g(de)g(fa\030)-20 b(con)18 b(\022)-23 b(a)18 b(p)q(ouv)o(oir)g(statuer)g(sur)g(des)h(relations)f(plus)i(\014nes)f (en)o(tre)f(pro-)75 1074 y(grammes,)13 b(que)j(nous)f(app)q(elons)f fj(\023)-22 b(e)n(quivalenc)n(e)15 b(faible)p fq(.)f(le)h(troisi)o (\022)-21 b(eme)14 b(c)o(hapitre)i(traite)e(des)h(strat)o(\023)-21 b(egies)75 1131 y(de)14 b(transformations)e(d')o(\023)-21 b(enonc)o(\023)g(es)13 b fn(pei)p fq(.)g(ces)g(strat)o(\023)-21 b(egies)12 b(son)o(t)h(une)i(application)g(directe)g(du)f(calcul)h(de) 75 1187 y(ra\016nemen)o(t,)i(et)g(mon)o(tren)o(t)f(commen)o(t)i (utiliser)h(les)f(r)o(\022)-21 b(egles)17 b(de)h(ra\016nemen)o(t)f(p)q (our)h(repr)o(\023)-21 b(esen)o(ter)16 b(des)75 1244 y(transformations)h(couran)o(tes.)h(notammen)o(t,)g(on)h(est)f(souv)o (en)o(t)h(amen)o(\023)-21 b(e)18 b(\022)-23 b(a)18 b(substituer)i(des)f (comm)o(u-)75 1300 y(nications)h(de)g(v)o(oisin)f(\022)-23 b(a)19 b(v)o(oisin)h(par)f(des)g(di\013usions)h(\(ou)f(vice-v)o (ersa\),)g(ou)g(par)g(des)g(op)o(\023)-21 b(erations)18 b(de)75 1357 y(r)o(\023)-21 b(eductions)15 b(lorsqu'un)h(op)o(\023)-21 b(erateur)14 b(est)h(appliqu)o(\023)-21 b(e)16 b(sur)f(les)h(v)m (aleurs)h(comm)o(uniqu)o(\023)-21 b(ees.)15 b(les)h(strat)o(\023)-21 b(egies,)75 1413 y(s'appuy)o(an)o(t)15 b(sur)g(le)h(calcul)h(de)f (ra\016nemen)o(t,)f(devraien)o(t)h(\022)-23 b(a)15 b(terme,)g(p)q (ermettre)g(de)h(passer)f(imm)o(\023)-21 b(edia-)75 1470 y(temen)o(t)14 b(d'une)h(v)o(ersion)f(\022)-23 b(a)14 b(l'autre,)f(et)i(mon)o(trer)e(la)h(relation)h(en)o(tre)f(les)h(deux)g (v)o(ersions.)f(deux)g(autres)75 1526 y(strat)o(\023)-21 b(egies)12 b(son)o(t)h(\023)-21 b(egalemen)o(t)13 b(d)o(\023)-21 b(ev)o(elopp)o(\023)g(ees)13 b(dans)h(ce)h(c)o(hapitre.)f(la)g(premi)o (\022)-21 b(ere)14 b(concerne)g(la)h(mo)q(di\014ca-)75 1583 y(tion)j(du)g(sc)o(h)o(\023)-21 b(ema)16 b(de)i(comm)o(unication)h (d'un)e(programme,)g(et)g(la)h(deuxi)o(\022)-21 b(eme,)18 b(la)g(mo)q(di\014cation)g(des)75 1639 y(alignemen)o(ts)d(de)g(donn)o (\023)-21 b(ees.)13 b(cette)h(derni)o(\022)-21 b(ere)15 b(transformation)e(illustre)j(un)e(cas)g(d')o(\023)-21 b(equiv)m(alence)16 b(faible)75 1695 y(en)o(tre)f(programmes.)f(il)i (est)f(probable)h(que)f(l')o(\023)-21 b(etude)15 b(de)h(cas)e (pratiques)i(plus)g(nom)o(breux)f(p)q(ourraien)o(t)75 1752 y(nous)j(inspirer)h(d'autres)e(id)o(\023)-21 b(ees)18 b(de)g(transformations,)e(que)j(nous)e(p)q(ourrons)h(ten)o(ter)f(de)i (g)o(\023)-21 b(en)o(\023)g(eraliser)75 1808 y(en)16 b(strat)o(\023)-21 b(egies.)146 1921 y(la)13 b(pratique)g(mon)o(tre)f (que)i(la)f(manipulation)i(d'un)e(formalisme)g(comme)g fn(pei)p fq(,)g(devien)o(t)h(tr)o(\022)-21 b(es)11 b(vite)75 1978 y(complexe)19 b(et)f(fastidieuse,)h(en)f(raison)g(des)g(nom)o (breuses)g(con)o(train)o(tes)g(qu'imp)q(ose)g(la)h(pr)o(\023)-21 b(ecision)18 b(du)75 2034 y(calcul)h(de)f(ra\016nemen)o(t.)f(p)o(our)g (aider)h(le)h(programmeur,)d(les)i(outils)h(de)f(d)o(\023)-21 b(ev)o(elopp)q(emen)o(ts)18 b(se)g(mon-)75 2091 y(tren)o(t)g(rapidemen)

o(t)h(indisp)q(ensabl)q(es.)i(en)e fn(pei)p fq(,)f(ces)h(outils)h(de)f (base)g(son)o(t)7 b(:)18 b(un)i(en)o(vironnemen)o(t)f(de)75 2147 y(transformations)e(p)q(ermettan)o(t)g(d'appliquer)j(les)f(r)o (\022)-21 b(egles)18 b(de)h(ra\016nemen)o(t,)e(un)i(traducteur)f(de)h (pro-)75 2204 y(grammes)14 b fn(pei)g fq(en)i(programmes)d fn(caml)i fq(bas)o(\023)-21 b(e)14 b(sur)h(l'in)o(terpr)o(\023)-21 b(etation)14 b(fonctionnelle,)j(et)d(un)i(outil)f(de)75 2260 y(conception)i(graphique)h(baptis)o(\023)-21 b(e)16 b fn(vpei)p fq(.)f(ce)i(dernier)g(outil)h(d)o(\023)-21 b(ev)o(elopp)o(\023)g(e)15 b(en)i(d)o(\023)-21 b(ebut)16 b(de)h(th)o(\022)-21 b(ese,)15 b(visait)75 2316 y(\022)-23 b(a)15 b(mon)o(trer)g(l'in)o(t)o(\023)-21 b(er)o(^)g(et)14 b(d'une)i(repr)o(\023)-21 b(esen)o(tation)15 b(graphique)h(des)g(v)m (ariables)h(parall)o(\022)-21 b(eles)17 b(et)e(de)h(leurs)h(ali-)75 2373 y(gnemen)o(ts,)e(lors)g(de)g(la)h(conception)g(du)g(programme.)146 2486 y(en\014n,)c(p)q(our)h(amorcer)f(l'application)i(des)f(tec)o (hniques)h fn(pei)e fq(aux)h(langages)f(data-parall)o(\022)-21 b(eles)13 b(exis-)75 2542 y(tan)o(ts,)j(il)i(nous)f(a)g(sem)o(bl)o (\023)-21 b(e)17 b(in)o(t)o(\023)-21 b(eressan)o(t)16 b(de)h(r)o(\023)-21 b(ealiser)17 b(un)h(traducteur)f(de)g fn(pei)g fq(v)o(ers)g(le)h(langage)f fj(high)p eop %%page: 115 117 115 116 bop 1778 269 a fq(115)75 397 y fj(performanc)n(e)16 b(f)m(ortr)n(an)p fq([27)n(].)f(ce)g(traducteur,)g(pr)o(\023)-21 b(esen)o(t)o(\023)g(e)13 b(dans)i(le)h(c)o(hapitre)g(5,)e(a)h(deux)h (ob)s(jectifs.)146 453 y(l'ob)s(jectif)h(premier)i(est)f(la)g(compr)o (\023)-21 b(ehension)18 b(des)g(probl)o(\022)-21 b(emes)18 b(que)h(p)q(ose)f(le)g(passage)g(d'un)g(for-)75 510 y(malisme)g(\023) -21 b(equationnel)18 b(comme)g fn(pei)g fq(\022)-23 b(a)17 b(un)i(langage)f(data-parall)o(\022)-21 b(ele)18 b(imp)o(\023)-21 b(eratif)18 b(comme)f(hpf.)h(ce)75 566 y(tra)o(v)m(ail)e(nous)f(a)g (amen)o(\023)-21 b(e)14 b(\022)-23 b(a)15 b(isoler)h(un)g(sous-ensem)o (ble)g(des)f(\023)-21 b(enonc)o(\023)g(es)14 b fn(pei)p fq(,)g(qui)i(son)o(t)f(les)h(programmes)75 623 y fn(pei)d fq(traduisibles.)i(nous)e(mon)o(trons)f(que)i(de)f(nom)o(breuses)h (transformations)d(s)o(\023)-21 b(eman)o(tiques)13 b(p)q(euv)o(en)o(t) 74 679 y(^)-21 b(etre)9 b(e\013ectu)o(\023)-21 b(ees)10 b(dans)h(l'en)o(vironnemen)o(t)h(de)f(transformation,)e(p)q(our)i(ne)g (laisser)g(au)g(traducteur)f(qu'un)75 736 y(tra)o(v)m(ail)16 b(de)g(transformation)e(syn)o(taxique.)i(nous)g(soulignons)g(ici)h(les) g(am)o(\023)-21 b(eliorations)15 b(\022)-23 b(a)15 b(app)q(orter)h(au) 75 792 y(traducteur,)k(notammen)o(t)g(dans)h(la)g(traduction)g(des)f (\023)-21 b(equations)20 b(r)o(\023)-21 b(ecurren)o(tes)7 b(:)20 b(la)h(g)o(\023)-21 b(en)o(\023)g(eralit)o(\023)g(e)19 b(des)75 849 y(domaines)d(p)q(oten)o(tiellemen)o(t)h(utilis)o(\023)-21 b(es)16 b(dans)f(une)h(d)o(\023)-21 b(e\014nition)16 b(r)o(\023)-21 b(ecursiv)o(e,)14 b(p)q(ose)i(encore)f(de)h(nom)o(breux) 75 905 y(probl)o(\022)-21 b(emes.)14 b(p)o(ar)h(con)o(tre,)f(il)i(est)f (remarquable)g(que,)g(dans)g(le)h(pro)q(cessus)f(de)g(traduction,)g (les)h(op)o(\023)-21 b(era-)75 962 y(tions)14 b(du)g(formalisme)g(son)o (t)e(traduites)i(de)g(mani)o(\022)-21 b(ere)13 b(tr)o(\022)-21 b(es)12 b(directe,)i(notammen)o(t)f(gr^)-23 b(ace)13 b(\022)-23 b(a)13 b(l'op)o(\023)-21 b(eration)75 1018 y(de)17 b(c)o(hangemen)o(t)g(de)g(base,)f(exprim)o(\023)-21 b(ee)17 b(par)g(un)g(alignemen)o(t)h(en)f(hpf.)f(le)h(deuxi)o(\022)-21 b(eme)18 b(ob)s(jectif,)e(une)75 1074 y(fois)f(le)g(traducteur)g (\014abilis)o(\023)-21 b(e,)15 b(est)g(de)g(faire)g(indirectemen)o(t)h (de)f(la)g(transformation)f(de)h(programmes)75 1131 y(hpf.)j(on)h(p)q (ourrait)f(en)h(e\013et)f(observ)o(er)h(pr)o(\023)-21 b(ecis)o(\023)g(emen)o(t)17 b(l'impact)i(des)g(transformations)e(d')o

(\023)-21 b(enonc)o(\023)g(es)75 1187 y fn(pei)15 b fq(sur)g(leurs)h (traductions)f(hpf.)146 1300 y(ces)21 b(tra)o(v)m(aux)g(sur)h (l'utilisation)h(du)f(formalisme)g fn(pei)g fq(fon)o(t)e(appara)-5 b(^)-18 b(\020tre)22 b(que)g(la)g(t^)-23 b(ac)o(he)21 b(\022)-23 b(a)22 b(ac-)75 1357 y(complir)e(p)q(our)g(in)o(t)o(\023)-21 b(egrer)18 b(ces)h(tec)o(hniques)i(dans)e(un)h(en)o(vironnemen)o(t)g(r) o(\023)-21 b(eel)19 b(de)g(d)o(\023)-21 b(ev)o(elopp)q(emen)o(t)20 b(est)75 1413 y(encore)e(immense.)h(d'une)f(part,)f(il)i(faut)f(in)o(t) o(\023)-21 b(egrer)17 b(et)h(adapter)f(aux)h(outils)h(d)o(\023)-21 b(ev)o(elopp)o(\023)g(es,)17 b(les)i(r)o(\023)-21 b(esul-)75 1470 y(tats)15 b(th)o(\023)-21 b(eoriques)17 b(existan)o(ts.)e(les)i (strat)o(\023)-21 b(egies)15 b(de)i(transformations)e(par)i(exemple,)g (doiv)o(en)o(t)g(dev)o(enir)75 1526 y(des)j(fonctionnalit)o(\023)-21 b(es)21 b(\022)-23 b(a)19 b(part)h(en)o(ti)o(\022)-21 b(ere)20 b(de)g(l'en)o(vironnemen)o(t)h(de)f(transformations.)f (d'autre)g(part,)75 1583 y(d'autres)f(id)o(\023)-21 b(ees,)18 b(comme)g(le)h(test)f(d')o(\023)-21 b(equiv)m(alence)20 b(forte)e(en)o(tre)g(programmes,)f(pr)o(\023)-21 b(esen)o(t)o(\023)g(e) 16 b(en)j(annexe)75 1639 y(b,)c(p)q(ourraien)o(t)f(\023)-21 b(egalemen)o(t)14 b(v)o(enir)h(enric)o(hir)i(l'aide)f(qu'app)q(orte)e (ce)h(t)o(yp)q(e)g(d'outil.)h(d'autres)d(tra)o(v)m(aux)75 1695 y(resten)o(t)18 b(\022)-23 b(a)18 b(mener)h(p)q(our)f(utiliser)i (la)f(p)q(ossibilit)o(\023)-21 b(e)20 b(qu'o\013re)e(hpf)g(d'exprimer)h (a)o(v)o(ec)f(les)h(indices)h(des)75 1752 y(v)m(ariables)13 b(parall)o(\022)-21 b(eles,)13 b(non)f(plus)h(des)f(pro)q(cesseurs)h (virtuels,)g(mais)f(des)f(\023)-21 b(el)o(\023)g(emen)o(ts)11 b(de)i(tableaux.)f(nous)75 1808 y(a)o(v)o(ons)f(ab)q(ord)o(\023)-21 b(e)11 b(cet)h(asp)q(ect)g(dans)g(le)h(dernier)g(c)o(hapitre,)f(et)g (nous)g(esp)o(\023)-21 b(erons)11 b(p)q(ouv)o(oir)i(am)o(\023)-21 b(eliorer)11 b(la)h(tra-)75 1865 y(duction)j(a\014n)g(de)g(pro)q(duire) g(des)g(textes)f(de)h(programmes)e(utilisan)o(t)j(cette)e(notation)g (plus)h(abstraite.)75 1921 y(d'un)20 b(programme)g fn(pei)g fq(don)o(t)g(les)h(v)m(ariables)h(son)o(t)e(align)o(\023)-21 b(ees)21 b(de)f(mani)o(\022)-21 b(ere)21 b(\022)-23 b(a)20 b(rendre)h(les)g(comm)o(u-)75 1978 y(nications)e(lo)q(cales,)f(nous)g (obtiendrions)h(alors)f(un)g(programme)f(hpf)g(don)o(t)g(les)i (alignemen)o(ts)f(son)o(t)75 2034 y(presque)c(cac)o(h)o(\023)-21 b(es)13 b(\022)-23 b(a)13 b(l'utilisateur,)h(n'apparaissan)o(t)f(que)h (sous)f(forme)g(de)g(directiv)o(es)i(\022)-23 b(a)13 b(l'ext)o(\023)-21 b(erieur)13 b(des)75 2091 y(instructions)j(de)g (traitemen)o(ts.)p eop %%page: 116 118 116 117 bop 75 269 a fq(116)1068 b fl(chapitre)15 b(6.)30 b(conclusion)p eop %%page: 117 119 117 118 bop 1778 269 a fq(117)75 729 y fp(bibliographie)98 952 y fq([1])21 b(saman)g(p)l(.)f(amarasinghe,)g(jennifer)i(m.)e (anderson,)g(monica)h(s.)f(lam,)g(and)h(c.w.)e(tseng.)168 1008 y(the)c(suif)h(compiler)g(for)e(scalable)i(parallel)g(mac)o (hines.)j(in)d fj(pr)n(o)n(c)n(e)n(e)n(dings)d(of)j(the)g(seventh)f (siam)168 1064 y(confer)n(enc)n(e)g(on)h(par)n(al)r(lel)f(pr)n(o)n(c)n (essing)f(for)j(scienti\014c)d(computing)p fq(,)h(f)l(ebruary)g(1995.) 98 1153 y([2])21 b(r.)f(j.)g(bac)o(k.)34 b(a)20 b(calculus)h(of)f (re\014nemen)o(ts)g(for)f(program)g(deriv)m(ations.)35 b fj(a)n(cta)20 b(informatic)n(a)p fq(,)168 1210 y(25\(6\):593{624,)11 b(august)16 b(1988.)98 1299 y([3])21 b(guy)e(e.)f(blello)q(c)o(h,)j (siddharta)e(chatterjee,)f(jonathan)g(c.)g(hardwic)o(k,)h(ja)o(y)f(sip) q(elstein,)j(and)168 1355 y(marco)16 b(zagha.)24 b(implemen)o(tation)18 b(of)e(a)h(p)q(ortable)g(nested)g(data-parallel)h(language.)25 b fj(journal)168 1412 y(of)17 b(par)n(al)r(lel)e(and)i(distribute)n(d)f

(computing)p fq(,)f(21\(1\):4{14,)d(1994.)98 1501 y([4])21 b(luc)16 b(boug)o(\023)-21 b(e.)18 b fj(the)e(data)h(par)n(al)r(lel)e (pr)n(o)n(gr)n(amming)g(mo)n(del)7 b(:)16 b(a)g(semantic)f(persp)n(e)n (ctive)p fq(,)e(v)o(olume)168 1557 y(lncs)j(1132,)e(pages)h(4{26.)k (springer)d(v)l(erlag,)f(ma)o(y)f(1996.)98 1646 y([5])21 b(luc)c(boug)o(\023)-21 b(e.)19 b(on)d(the)g(seman)o(tics)f(of)g (languages)h(for)f(massiv)o(ely)h(parallel)h(simd)f(arc)o(hitectures.) 168 1702 y(in)g fj(p)m(arle'91,)f(lncs)p fq(,)f(pages)h(166{183.)e (springer-v)l(erlag,)j(1991.)98 1791 y([6])21 b(pierre)15 b(boulet.)k fj(outils)c(p)n(our)h(la)f(p)n(ar)n(al)r(l)o(\023)-22 b(elisation)14 b(automatique)p fq(.)20 b(phd)13 b(thesis,)1562 1780 y(\023)1558 1791 y(ecole)i(normale)168 1848 y(sup)o(\023)-21 b(erieure)16 b(de)g(ly)o(on,)f(jan)o(vier)h(1996.)98 1937 y([7])21 b(thomas)13 b(brandes.)k(compiling)d(data-parallel)h (programs)d(to)g(message)h(passing)g(programs)f(for)168 1993 y(massiv)o(ely)h(parallel)g(mimd)g(systems.)h(in)f(k.)e(m.)h(dec)o (k)o(er)f(and)h(r.)g(m.)f(rehmann,)i(editors,)f fj(wor-)168 2050 y(king)j(confer)n(enc)n(e)f(on)i(massively)e(par)n(al)r(lel)h(pr)n (o)n(gr)n(amming)g(mo)n(dels)7 b(:)15 b(suitability,)h(r)n(e)n (alization,)168 2106 y(and)h(performanc)n(e)p fq(,)d(septem)o(b)q(er)i (1993.)98 2195 y([8])21 b(l.)f(cannon.)31 b fj(a)20 b(c)n(el)r(lular)f (c)n(omputer)i(to)f(implement)f(the)h(kalman)g(filter)f(a)o(lgorithm)p fq(.)32 b(phd)168 2252 y(thesis,)16 b(mon)o(tana)e(state)g(univ)o (ersit)o(y)l(,)i(bozeman,)f(1969.)98 2340 y([9])21 b(zbigniew)g (chamski.)30 b fj(envir)n(onnement)18 b(lo)n(giciel)h(de)g(pr)n(o)n(gr) n(ammation)h(d'un)g(ac)n(c)o(\023)-22 b(el)o(\023)g(er)n(ateur)20 b(de)168 2397 y(c)n(alcul)c(p)n(ar)n(al)r(l)o(\022)-22 b(ele)p fq(.)19 b(phd)c(thesis,)g(univ)o(ersit)o(\023)-21 b(e)16 b(de)f(rennes)i(i,)e(f)l(ebruary)g(1993.)75 2486 y([10])21 b(k.)14 b(mani)h(chandy)f(and)h(ja)o(y)o(adev)e(misra.)18 b fj(par)n(al)r(lel)d(pr)n(o)n(gr)n(am)g(design)7 b(:)15 b fq(a)f(foundation.)k(addi-)168 2542 y(son)d(w)l(esley)l(,)h(1988.)p eop %%page: 118 120 118 119 bop 75 269 a fq(118)1326 b fl(bibliographie)75 397 y fq([11])21 b(bruce)i(w.)e(char,)g(keith)h(o.)g(geddes,)g(gaston)e (h.)i(gonnet,)f(mic)o(hael)i(b.)e(monagan,)f(and)168 453 y(stephen)i(m.)f(w)l(att.)35 b fj(maple)20 b(r)n(efer)n(enc)n(e)g (manual)p fq(.)37 b(univ)o(ersit)o(y)21 b(of)g(w)l(aterlo)q(o,)f(w)l (aterlo)q(o)168 510 y(maple)c(soft)o(w)o(are,)d(1989.)75 604 y([12])21 b(marina)h(chen,)g(y)l(oung)g(il)h(cho)q(o,)e(and)h (jingk)o(e)g(li.)41 b fj(par)n(al)r(lel)21 b(f)m(unctional)g(l)n (anguages)h(and)168 660 y(compilers)p fq(,)15 b(c)o(hapter)g(chapter)f (7.)20 b(f)l(ron)o(tier)15 b(series.)h(a)o(cm)e(press,)h(1991.)75 755 y([13])21 b(f)l(abien)13 b(co)q(elho.)j fj(contributions)d(to)g (high)h(performanc)n(e)f(f)m(ortr)n(an)g(compilation)p fq(.)h(phd)e(thesis,)168 811 y(ecole)k(des)g(mines)g(de)f(p)o(aris,)g (octobre)g(1996.)75 905 y([14])21 b(alain)13 b(darte)e(and)h(yv)o(es)f (rob)q(ert.)k(a)c(graph)h(theoretic)g(approac)o(h)f(to)g(the)h (alignemen)o(t)g(problem.)168 962 y(t)l(ec)o(hnical)17 b(rep)q(ort,)e(ecole)h(normale)f(sup)o(\023)-21 b(erieure)16 b(de)g(ly)o(on,)f(lip)l(,)g(july)i(1993.)h(rr93-20.)75 1056 y([15])j(nills)f(ellmenreic)o(h,)g(p)o(eter)e(f)l(ab)q(er,)f (martin)h(griebl,)g(rob)q(ert)h(g)q(\177)-24 b(unz,)18 b(harald)g(keimer,)g(w)l(olf-)168 1112 y(gang)c(meisl,)g(sabine)i(w)l (etzel,)e(christian)h(wieninger,)g(and)f(alexander)h(w)q(\177)-24 b(ust.)17 b(lo)q(op)e(paralle-)168 1169 y(lization)i(in)f(the)f(p)q (olytop)q(e)h(mo)q(del.)21 b(t)l(ec)o(hnical)c(rep)q(ort,)e(univ)o (ersit)o(y)h(of)e(p)o(assau,)g(april)j(1996.)75 1263 y([16])k(p)o(aul)i(f)l(eautrier.)42 b(some)23 b(e\016cien)o(t)g

(solution)h(to)e(the)g(a\016ne)h(sc)o(heduling)i(problem,)e(part)f(ii,) 168 1319 y(m)o(ultidimensional)h(time.)33 b fj(international)19 b(journal)h(of)h(par)n(al)r(lel)e(pr)n(o)n(gr)n(amming)p fq(,)g(21\(6\):389{)168 1376 y(420,)14 b(1992.)75 1470 y([17])21 b(p)o(aul)14 b(f)l(eautrier.)j(t)l(o)o(w)o(ards)12 b(automatic)g(distribution.)19 b fj(par)n(al)r(lel)14 b(pr)n(o)n(c)n(essing)e(letters)p fq(,)g(4\(3\):233{)168 1526 y(244,)i(1994.)75 1620 y([18])21 b(p)o(aul)16 b(f)l(eautrier.)j (compiling)e(for)d(massiv)o(ely)i(parallel)g(arc)o(hitectures;)f(a)g(p) q(ersp)q(ectiv)o(e.)21 b fj(micr)n(o-)168 1677 y(pr)n(o)n(gr)n(amming) 16 b(and)h(micr)n(opr)n(o)n(c)n(essors)p fq(,)c(1995.)18 b(t)l(o)d(app)q(ear.)75 1771 y([19])21 b(m.)13 b(j.)f(flynn.)17 b(some)c(computer)g(organizations)f(and)h(their)h(e\013ectiv)o(eness.)i fj(ieee)d(t)m(r)n(ansaction)168 1827 y(on)j(computer)p fq(,)g(21\(9\):948{960)o(,)c(1972.)75 1921 y([20])21 b(a.)13 b(geist,)g(a.)g(beguelin,)i(j.)e(dongarra,)e(w.)i(jiang,)g(r.)g (manc)o(hek,)g(and)h(v.)e(sunderam.)18 b fj(pvm)7 b(:)168 1978 y(par)n(al)r(lel)16 b(virtual)g(machine,)g(a)g(user's)f(guide)i (and)f(t)m(utorial)p fq(.)k(the)15 b(mit)g(press.)75 2072 y([21])21 b(st)o(\023)-21 b(ephane)20 b(genaud.)36 b(t)l(ransformations)19 b(d')o(\023)-21 b(enonc)o(\023)g(es)18 b fn(pei)p fq(.)36 b fj(t)m(e)n(chnique)19 b(et)i(scienc)n(e)e (informa-)168 2128 y(tiques)p fq(,)c(15\(5\):601{618)o(,)d(1996.)75 2222 y([22])21 b(st)o(\023)-21 b(ephane)18 b(genaud,)f(eric)h(violard,) f(and)h(guy-ren)o(\023)-21 b(e)17 b(p)o(errin.)27 b(t)l(ransformations) 16 b(tec)o(hniques)168 2279 y(in)g fn(pei)p fq(.)i(in)d(p)o(eter)f (magn)o(usson,)f(seif)i(haridi,)g(and)g(ali)g(kha)o(yri,)g(editors,)f fj(lncs)p fq(,)e(v)o(olume)j(966,)168 2335 y(pages)f(131{142,)d(sto)q (c)o(kholm,)j(sw)o(eden,)g(august)f(1995.)f(eur)o(o-p)l(ar95)j(p)o (arallel)f(pro)q(cessing,)168 2392 y(springer-v)l(erlag.)75 2486 y([23])21 b(gene)16 b(h.)f(golub)h(and)g(charles)f(f.)g(v)l(an)h (loan.)k fj(matrix)d(computations)p fq(.)j(the)c(john)g(hopkins)168 2542 y(univ)o(ersit)o(y)g(press,)f(2nd)g(edition,)h(1989.)p eop %%page: 119 121 119 120 bop 75 269 a fl(bibliographie)1329 b fq(119)75 397 y([24])21 b(martin)16 b(griebl)h(and)f(jean-f)l(ran\030)-20 b(cois)17 b(collard.)22 b(generation)16 b(of)g(sync)o(hronous)f(co)q (de)i(for)e(auto-)168 453 y(matic)d(parallelization)j(of)c fk(while)g fq(lo)q(ops.)k(in)d(p)o(eter)g(magn)o(usson,)e(seif)j (haridi,)f(and)g(ali)i(kha)o(yri,)168 510 y(editors,)e fj(lncs)p fq(,)f(v)o(olume)h(966,)f(sto)q(c)o(kholm,)h(sw)o(eden,)h (august)f(1995.)f(eur)o(o-p)l(ar95)i(p)o(arallel)168 566 y(pro)q(cessing,)j(springer-v)l(erlag.)75 660 y([25])21 b(jonathan)16 b(m.)f(d.)f(hill.)23 b fj(data-p)n(ar)n(al)r(lel)17 b(lazy)f(functional)g(pr)n(o)n(gr)n(amming)p fq(.)k(phd)c(thesis,)f (queen)168 717 y(mary)e(and)i(w)l(est\014eld)f(college,)h(departmen)o (t)e(of)g(computer)h(science,)h(mile)g(end)f(road,)g(lon-)168 773 y(don,)h(e1)g(4ns,)g(uk,)g(septem)o(b)q(er)h(1994.)75 867 y([26])21 b(w.)14 b(daniel)i(hillis)h(and)d(guy)g(l.)h(steele)g (jr.)k(data)13 b(parallel)j(algorithms.)j fj(communic)n(ations)14 b(of)168 924 y(the)j(a)o(cm)p fq(,)d(29\(12\):1170)o({118)o(3,)e(1986.) 75 1018 y([27])21 b(hpf)d(f)l(orum.)26 b fj(high)18 b(performanc)n(e)g (f)m(ortr)n(an)g(l)n(anguage)f(sp)n(e)n(ci\014c)n(ation)p fq(,)e(1.0)i(edition,)h(ma)o(y)f(3)168 1074 y(1993.)75 1169 y([28])k(f)l(ran\030)-20 b(cois)13 b(irigoin,)g(pierre)f(jouv)o (elot,)g(and)h(r)o(\023)-21 b(emi)12 b(t)l(riolet.)k(seman)o(tical)d (in)o(terpro)q(cedural)g(paral-)168 1225 y(lelization)d(:)15 b(an)h(o)o(v)o(erview)g(of)f(the)h(pips)g(pro)s(ject.)21 b(in)16 b fj(international)f(workshop)j(on)e(c)n(ompilers)168

1281 y(for)h(par)n(al)r(lel)f(computers)p fq(,)f(pages)g(199{212,)d (decem)o(b)q(er)k(1990.)75 1376 y([29])21 b(kenneth)c(e.)d(iv)o(erson.) 20 b fj(a)c(pr)n(o)n(gr)n(amming)g(l)n(anguage)p fq(.)j(john)d(wiley)g (and)f(sons,)g(inc.,)h(1962.)75 1470 y([30])21 b(w)l(a)o(yne)d(kelly)l (,)i(v)l(adim)e(maslo)o(v,)f(william)j(pugh,)e(ev)m(an)g(rosser,)g(t)l (atiana)f(shp)q(eisman,)i(and)168 1526 y(da)o(vid)d(w)l(onnacott.)j fj(the)d(ome)n(ga)g(libr)n(ary)f(-)i(v)m(ersion)e(1.00)p fq(,)g(april)i(1996.)i(in)o(terface)c(guide.)75 1620 y([31])21 b(e.)14 b(knapp.)k(an)c(exercise)h(in)g(the)f(formal)f(deriv) m(ation)j(of)d(parallel)i(programs:)e(maxim)o(um)g(\015o)o(ws)168 1677 y(in)19 b(graphs.)25 b fj(a)o(cm)17 b(t)m(r)n(ansactions)f(on)i (pr)n(o)n(gr)n(amming)g(l)n(anguages)f(and)h(systems)p fq(,)e(12\(2\):203{)168 1733 y(223,)e(april)j(1990.)75 1827 y([32])k(charles)i(h.)e(ko)q(elb)q(el,)j(da)o(vid)e(b.)g(lo)o(v)o (eman,)f(rob)q(ert)h(s.)g(sc)o(hreib)q(er,)h(guy)f(l.)g(steele,)g(and) 168 1884 y(mary)15 b(e.)f(zosel.)21 b fj(the)15 b(high)i(performanc)n (e)f(f)m(ortr)n(an)f(handb)n(o)n(ok)p fq(.)k(the)d(mit)f(press,)f (1994.)75 1978 y([33])21 b(c.)c(p)l(.)g(krusk)m(al,)h(l.)f(rudolph,)h (and)g(m.)e(snir.)26 b(the)18 b(p)q(o)o(w)o(er)e(of)h(parallel)i (pre\014x.)26 b fj(ieee)17 b(t)m(r)n(an-)168 2034 y(sactions)f(on)g(c)n (omputers)p fq(,)f(34\(10\):965{96)o(8,)d(octob)q(er)j(1985.)75 2128 y([34])21 b(vipin)j(kumar,)e(anan)o(th)g(grama,)e(ansh)o(ul)k (gupta,)d(and)h(george)f(karypis.)41 b fj(intr)n(o)n(duction)168 2185 y(to)19 b(p)n(ar)n(al)r(lel)e(c)n(omputing,)h(design)g(and)g (analysis)f(of)h(algorithms)p fq(.)26 b(the)18 b(benjamin/cummings)168 2241 y(publishing)g(compan)o(y)l(,)c(1994.)75 2335 y([35])21 b(vincen)o(t)15 b(lefeb)o(vre.)i(gestion)d(de)g(la)f(m)o(\023)-21 b(emoire)13 b(dans)g(les)h(programmes)f(parall)o(\023)-21 b(elis)o(\023)g(es.)17 b(t)l(ec)o(hnical)168 2392 y(rep)q(ort)f(rr)g (96/17,)d(lab)j(prism)g(-)f(univ)o(ersit)o(\023)-21 b(e)16 b(de)f(v)l(ersailles,)i(august)e(1996.)75 2486 y([36])21 b(christian)d(lengauer.)24 b(lo)q(op)17 b(parallelization)i(in)e(the)f (p)q(olytop)q(e)i(mo)q(del.)25 b(in)17 b(e.)f(best,)g(editor,)168 2542 y fj(concur'93,)g(lncs)e(715)p fq(,)i(pages)f(398{416.)e (springer-v)l(erlag,)j(june)g(1993.)p eop %%page: 120 122 120 121 bop 75 269 a fq(120)1326 b fl(bibliographie)75 397 y fq([37])21 b(v.)e(lo)q(ec)o(hner)g(and)f(c.)g(mongenet.)29 b(a)19 b(to)q(olb)q(o)o(x)f(for)g(a\016ne)g(recurrence)i(equations)e (paralleli-)168 453 y(zation.)29 b(in)19 b(b.)f(hertzb)q(erger)g(and)g (g.)f(serazzi,)i(editors,)f fj(int.)g(conf.)g(on)h(high-performanc)n(e) 168 510 y(computing)d(and)f(networking)p fq(,)e(pages)h(263{268.)e (springer)j(v)l(erlag,)f(ma)o(y)f(1995.)k(milan,)d(italy)l(.)75 607 y([38])21 b(masp)o(ar)11 b(computer)g(corp)q(oration.)i fj(maspar)g(mp-1)g(mpl)f(pr)n(o)n(gr)n(aming)h(manuals)p fq(,)d(decem)o(b)q(er)168 664 y(1991.)75 761 y([39])21 b(christophe)g(mauras.)32 b fn(alpha)8 b fj(:)20 b(un)h(langage)d(\023) -22 b(e)n(quationnel)20 b(p)n(our)h(la)f(c)n(onc)n(eption)f(et)i(la)f (pr)n(o-)168 818 y(gr)n(ammation)e(d'ar)n(chite)n(ctur)n(es)f(p)n(ar)n (al)r(l)o(\022)-22 b(eles)17 b(synchr)n(ones)p fq(.)22 b(phd)16 b(thesis,)h(univ)o(ersit)o(\023)-21 b(e)16 b(de)h(rennes)168 874 y(i,)f(1989.)75 972 y([40])21 b(dror)h(e.)g(ma)o(ydan,)g(saman)g(p) l(.)g(amarasinghe,)g(and)h(monica)g(s.)f(lam.)42 b(arra)o(y)22 b(data-\015o)o(w)168 1028 y(analysis)16 b(and)g(its)f(use)g(in)h(arra)o (y)e(priv)m(atization.)21 b(in)16 b fj(principles)e(of)j(pr)n(o)n(gr)n (amming)e(l)n(anguages)p fq(,)168 1085 y(1993.)75 1182 y([41])21 b(james)12 b(mcgra)o(w,)e(stephen)j(sk)o(edzielewski,)h (stephen)f(allan,)g(ro)q(d)f(oldeho)q(eft,)h(john)f(glauert,)168 1239 y(chris)18 b(kirkham,)f(bill)i(no)o(yce,)e(and)g(rob)q(ert)g (thomas.)25 b fj(sisal:)16 b(str)n(e)n(ams)h(and)h(iter)n(ations)f(in)

168 1295 y(a)i(single)f(assignement)f(language:)i(r)n(efer)n(enc)n(e)e (manual)h(version)h(1.2)p fq(.)28 b(la)o(wrence)18 b(liv)o(ermore)168 1352 y(national)e(lab)q(oratory)l(,)e(liv)o(ermore,)h(ca,)g(marc)o(h)g (1985.)75 1449 y([42])21 b(message)13 b(p)o(assing)h(in)o(terface)f(f)l (orum.)k fj(mpi)7 b(:)14 b(a)h(message-p)n(assing)e(interfac)n(e)g (standar)n(d)p fq(,)h(june)168 1506 y(1995.)75 1603 y([43])21 b(catherine)16 b(mongenet.)k(mappings)15 b(for)g(comm)o(unications)h (minimization)h(using)f(distribution)168 1660 y(and)i(alignmen)o(t.)28 b(in)19 b(lub)q(omir)g(bic,)f(wim)g(b\177)-23 b(ohm,)17 b(p)o(arask)o(ev)m(as)g(evripidiou,)i(and)f(jean-luc)168 1716 y(gaudiot,)d(editors,)g fj(pr)n(o)n(c)n(e)n(e)n(dings)f(of)i(ifip) f(wg)i(10.3)g(working)f(confer)n(enc)n(e)e(on)i(par)n(al)r(lel)g(a)o (r-)168 1772 y(chite)n(ctur)n(es)e(and)g(compilation)f(t)m(e)n (chniques,)f(p)m(a)o(ct'95)p fq(,)e(pages)j(185{193,)d(limassol,)j (cyprus,)168 1829 y(june)k(1995.)c(a)o(cm)i(press.)75 1926 y([44])21 b(carroll)16 b(morgan.)k fj(pr)n(o)n(gr)n(amming)c(fr)n (om)g(sp)n(e)n(ci\014c)n(ations)p fq(.)j(c.a.r.)c(hoare.)f(pren)o(tice) j(hall)f(ed.,)168 1983 y(endlew)o(o)q(o)q(d)h(cli\013s,)e(n.j.,)f (1990.)75 2080 y([45])21 b(nicolas)c(p)o(aris.)i fj(hyp)n(erc)d(do)n (cumentation)p fq(.)k(hyp)q(erparallel)d(t)l(ec)o(hnologies,)f(june)g (1993.)75 2178 y([46])21 b(guy-ren)o(\023)-21 b(e)21 b(p)o(errin.)35 b(p)o(arallel)21 b(programming)e(with)i fn(pei)p fq(.)35 b(in)21 b(christian)f(lengauer,)h(editor,)168 2234 y fj(dagsthul)c(seminar)f(on)g(l)n(o)n(op)g(par)n(al)r(lelization) p fq(,)d(april)k(1996.)75 2332 y([47])k(serge)c(p)o(etiton)f(and)g (nahid)h(emad.)23 b fj(a)16 b(data)i(p)n(ar)n(al)r(lel)f(scienti\014c)e (c)n(omputation)j(intr)n(o)n(duction)p fq(,)168 2388 y(v)o(olume)e(lncs)g(1132,)d(pages)i(45{62.)k(springer)d(v)l(erlag,)f (ma)o(y)f(1996.)75 2486 y([48])21 b(p)o(atrice)11 b(quin)o(ton)g(and)f (vincen)o(t)h(v)l(an)g(dongen.)g(the)g(mapping)g(of)f(linear)h (recurrence)g(equations)168 2542 y(on)16 b(regular)f(arra)o(ys.)j fj(the)e(journal)h(of)f(vlsi)f(signal)g(pr)n(o)n(c)n(essing)p fq(,)d(1:95{113,)h(1989.)p eop %%page: 121 123 121 122 bop 75 269 a fl(bibliographie)1329 b fq(121)75 397 y([49])21 b(p)o(atrice)c(quin)o(ton,)g(sanja)o(y)g(radjopadh)o(y)o (e,)f(and)h(doran)f(wilde.)26 b(on)18 b(deriving)g(data)e(parallel)168 453 y(co)q(de)c(from)e(a)g(functional)i(program.)f(in)g fj(pr)n(o)n(c)n(e)n(e)n(dings)f(of)i(the)g(9th)h(international)e(symp)n (osium)h(on)168 510 y(par)n(al)r(lel)17 b(pr)n(o)n(c)n(essing,)e (ipps'95)p fq(,)h(pages)g(766{773,)e(san)o(ta)i(barbara,)f(ca,)h(april) i(1995.)d(ieee)168 566 y(computer)g(so)q(ciet)o(y)h(press.)75 656 y([50])21 b(p)o(atrice)e(quin)o(ton,)h(doran)e(wilde,)i(and)g (sanja)o(y)e(ra)s(jopadh)o(y)o(e.)31 b(deriving)20 b(imp)q(erativ)o(e)g (co)q(de)168 712 y(from)i(functional)i(programs.)41 b(in)23 b fj(f)m(unctional)f(pr)n(o)n(gr)n(amming)h(l)n(anguages)f(and)h (computer)168 769 y(a)o(r)n(chite)n(ctur)n(es,)16 b(f)o(cp)m(a'95)p fq(,)d(pages)i(36{44.)e(a)o(cm)i(press,)f(june)j(1995.)75 858 y([51])k(sanja)o(y)10 b(ra)s(jopadh)o(y)o(e.)h(analysis)h(of)d (a\016ne)i(comm)o(unications)f(sp)q(eci\014cations.)k(in)d fj(ieee)f(symp)n(o-)168 914 y(sium)15 b(on)g(par)n(al)r(lel)f(and)h (distribute)n(d)h(pr)n(o)n(c)n(essing)p fq(,)11 b(pages)i(530{537.)f (ieee)i(computer)f(so)q(ciet)o(y)168 971 y(press,)i(decem)o(b)q(er)h (1993.)75 1060 y([52])21 b(rob)q(ert)i(s.)f(sc)o(hreib)q(er.)41 b fj(a)o(n)22 b(intr)n(o)n(duction)g(to)h(hpf)p fq(,)e(v)o(olume)i (lncs)f(1132,)f(pages)g(27{44.)168 1117 y(springer)c(v)l(erlag,)e(ma)o (y)f(1996.)75 1206 y([53])21 b(dpce)c(sub)q(committe.)28 b(data-parallel)18 b(c)g(extensions.)g(tec)o(hnical)h(rep)q(ort)e(1.6)f (x3j11/94-080.)168 1263 y(t)l(ec)o(hnical)h(rep)q(ort,)e(numerical)i(c) e(extensions)g(group)g(of)g(x3j11,)f(1994.)75 1352 y([54])21

b(thinking)c(mac)o(hines)f(corp)q(oration.)j fj(c*)d(pr)n(o)n(gr)n (amming)g(guide)p fq(,)f(no)o(v)o(em)o(b)q(er)g(1990.)75 1441 y([55])21 b(thinking)c(mac)o(hines)f(corp)q(oration.)j fj(cm)d(f)m(ortr)n(an)f(r)n(efer)n(enc)n(e)g(manual)p fq(,)f(decem)o(b)q(er)i(1992.)75 1531 y([56])21 b(eric)c(violard.)22 b fj(une)17 b(th)o(\023)-22 b(eorie)17 b(uni\014c)n(atric)n(e)f(p)n (our)h(la)g(c)n(onstruction)f(de)h(pr)n(o)n(gr)n(ammes)g(p)n(ar)n(al)r (l)o(\022)-22 b(eles)168 1587 y(p)n(ar)22 b(des)f(te)n(chniques)g(de)g (tr)n(ansformations)p fq(.)36 b(phd)21 b(thesis,)g(univ)o(ersit)o(\023) -21 b(e)21 b(de)g(f)l(ranc)o(he-com)o(t)o(\023)-21 b(e,)168 1644 y(octobre)16 b(1992.)75 1733 y([57])21 b(eric)e(violard.)31 b(data-parallelism)19 b(v)o(ersus)g(functional)g(programming)7 b(:)18 b(the)h(con)o(tribution)g(of)168 1790 y fn(pei)p fq(.)i(t)l(ec)o(hnical)c(rep)q(ort,)d(icps,)h(jan)o(uary)g(1995.)k (publication)e(in)o(terne)f(95-03.)75 1879 y([58])21 b(eric)e(violard)f(and)g(guy-ren)o(\023)-21 b(e)18 b(p)o(errin.)27 b fn(pei)7 b fq(:)18 b(a)f(single)i(unifying)g(mo)q(del)g(to)d(design)j (parallel)168 1935 y(programs.)38 b(in)22 b(arndt)g(bo)q(de,)g(mik)o(e) g(reev)o(e,)g(and)g(gottfried)f(w)l(olf,)g(editors,)g fj(p)m(arle)g(93,)168 1992 y(lncs)p fq(,)14 b(v)o(olume)h(694,)f(pages) h(500{516.)e(springer-v)l(erlag,)j(june)g(1993.)75 2081 y([59])21 b(eric)15 b(violard)f(and)g(guy-ren)o(\023)-21 b(e)14 b(p)o(errin.)k(reduction)d(in)g fn(pei)p fq(.)i(in)e(bruno)f (buc)o(h)o(b)q(erger)g(and)g(jens)168 2138 y(v)l(olk)o(ert,)k(editors,) g fj(lncs)p fq(,)e(v)o(olume)j(854,)e(pages)h(112{123.)d(conp)l(ar)k (94,)f(springer-v)l(erlag,)168 2194 y(septem)o(b)q(er)e(1994.)75 2284 y([60])21 b(pierre)16 b(w)l(eis)g(and)f(xa)o(vier)h(lero)o(y)l(.)k fj(l)n(e)15 b(langage)h fn(caml)p fq(.)j(in)o(ter)o(\023)-21 b(editions)16 b(-)f(iia,)h(1993.)75 2373 y([61])21 b(doran)14 b(wilde)h(and)f(sanja)o(y)g(ra)s(jopadh)o(y)o(e.)k(the)c(naiv)o(e)g (execution)i(of)d(a\016ne)h(recurrence)h(equa-)168 2429 y(tions.)31 b(in)19 b(p)o(eter)f(capp)q(ello,)i(catherine)f(mongenet,)f (guy-ren)o(\023)-21 b(e)19 b(p)o(errin,)g(p)o(atrice)f(quin)o(ton,)168 2486 y(and)g(yv)o(es)e(rob)q(ert,)h(editors,)g fj(applic)n(ation)g(sp)n (e)n(ci\014c)g(a)o(rr)n(ay)g(pr)n(o)n(c)n(essors,asap'95)p fq(,)d(pages)j(1{)168 2542 y(12.)e(ieee)g(computer)g(so)q(ciet)o(y)h (press,)e(august)h(1995.)p eop %%page: 122 124 122 123 bop 75 269 a fq(122)1326 b fl(bibliographie)75 397 y fq([62])21 b(doran)10 b(wilde)i(and)f(sanja)o(y)f(ra)s(jopadh)o (y)o(e.)i(memory)e(reuse)h(analysis)h(in)f(the)g(p)q(olyhedral)h(mo)q (del.)168 453 y(in)18 b(luc)f(boug)o(\023)-21 b(e,)16 b(pierre)h(f)l(raignaud,)f(anne)i(mignotte,)d(and)i(yv)o(es)g(rob)q (ert,)f(editors,)h fj(eur)o(o-)168 510 y(p)m(ar'96)h(par)n(al)r(lel)g (pr)n(o)n(c)n(essing)p fq(,)c(v)o(olume)k(i,)f(pages)g(389{397.)e (springer-v)l(erlag,)j(lncs)g(1123,)168 566 y(1996.)75 660 y([63])j(doran)13 b(k.)h(wilde.)19 b(a)13 b(library)i(for)e(doing)h (p)q(olyhedral)h(op)q(erations.)j(t)l(ec)o(hnical)d(rep)q(ort,)e (irisa,)168 717 y(decem)o(b)q(er)j(1993.)j(pi)c(785.)p eop %%page: 123 125 123 124 bop 1778 269 a fq(123)75 709 y fi(annexe)33 b(a)75 918 y fp(preuv)m(es)39 b(des)f(propri)m(\023)-56 b(et)m(\023)g(es)75 1141 y fj(pr)n(euve.)16 b(\(pr)n(opri)o(\023)-22 b(et)o(\023)g(e)17 b(1,)g(p)n(age)g(23\))f fq(si)g fk(y)g fq(est)g(une)g(expression)h (fonction)f(de)h fk(x)p fq(,)e(d'apr)o(\022)-21 b(es)15 b(la)h(d)o(\023)-21 b(e\014nition)75 1197 y(des)16 b(op)o(\023)-21 b(erations,)13 b(il)k(existe)e(une)h(bijection)h fm(\014)g fq(telle)g(que)8 b(:)619 1302 y fm(\033)645 1309 y fr(y)675 1302 y fq(=)13 b fm(\033)749 1309 y fr(x)777 1302 y ff(\016)c

fm(\014)837 1284 y fx(\000)p fg(1)892 1302 y ff(^)i fm(dom)p fq(\()p fm(v)1059 1309 y fr(x)1075 1302 y fq(\))i ff(\022)g fm(dom)p fq(\()p fm(\014)r fq(\))75 1407 y(mon)o(trons)h(alors)h(qu'il) h(existe)f(une)h(bijection)h fm(g)f fq(telle)g(que)g(la)f(paire)h(de)f (bijections)i(\()p fm(g)11 b ff(\016)e fm(\014)r(;)f(g)r fq(\))14 b(v)o(\023)-21 b(eri\014e)75 1464 y(les)16 b(trois)f(prop)q (ositions)h(\()p fm(a)p fq(\),)e(\()p fm(b)p fq(\))g(et)h(\()p fm(c)p fq(\))7 b(:)107 1561 y(\()p fm(a)p fq(\))22 b(s')o(\023)-21 b(ecrit)14 b fm(\033)349 1568 y fr(x)376 1561 y ff(\016)c fm(\014)437 1544 y fx(\000)p fg(1)492 1561 y ff(\016)g fm(g)549 1544 y fx(\000)p fg(1)605 1561 y fq(=)j fm(\033)679 1568 y fr(y)707 1561 y ff(\016)d fm(g)764 1544 y fx(\000)p fg(1)808 1561 y fq(,)k(qui)i(est)f(v)o(\023)-21 b(eri\014)o(\023)g(ee) 15 b(quelquesoit)h fm(g)r fq(.)111 1658 y(\()p fm(b)p fq(\))22 b(s')o(\023)-21 b(ecrit)14 b fm(dom)p fq(\()p fm(v)449 1665 y fr(x)466 1658 y fq(\))g ff(\022)f fm(dom)p fq(\()p fm(g)e ff(\016)f fm(\014)r fq(\).)15 b(d'apr)o(\022)-21 b(es)14 b fm(dom)p fq(\()p fm(v)1083 1665 y fr(x)1100 1658 y fq(\))f ff(\022)h fm(dom)p fq(\()p fm(\014)r fq(\),)h(on)g(p)q (eut)i(conclure)g(que)f(\()p fm(b)p fq(\))189 1715 y(est)f(v)o(\023)-21 b(eri\014)o(\023)g(ee)14 b(p)q(our)h(toute)g(bijection)i fm(g)f fq(telle)h(que)e fm(img)r fq(\()p fm(\014)r fq(\))c ff(\022)i fm(dom)p fq(\()p fm(g)r fq(\).)111 1812 y(\()p fm(c)p fq(\))22 b(s')o(\023)-21 b(ecrit)14 b fm(dom)p fq(\()p fm(v)449 1819 y fr(y)466 1812 y fq(\))e ff(\022)h fm(dom)p fq(\()p fm(g)r fq(\).)75 1909 y(on)g(p)q(eut)h(donc)f (conclure)h(que)f(toute)f(bijection)i fm(g)g fq(telle)g(que)g fm(img)r fq(\()p fm(\014)r fq(\))5 b ff([)g fm(dom)p fq(\()p fm(v)1440 1916 y fr(y)1456 1909 y fq(\))12 b ff(\022)h fm(dom)p fq(\()p fm(g)r fq(\))e(satisfait)75 1966 y(les)16 b(conditions.)75 2022 y fa(\003)75 2195 y fj(pr)n(euve.\(pr)n(opri)o(\023)-22 b(et)o(\023)g(e)17 b(2,)g(p)n(age)g(31\))g fq(si)g fm(dom)p fq(\()p fm(\033)859 2202 y fr(x)875 2195 y fq(\))d ff(\022)h fm(img)r fq(\()p fk(h)p fq(\),)f(alors)i(le)h(c)o(hamp)f(de)g(donn)o(\023)-21 b(ees)16 b(\()p fk(h)1705 2179 y fq(-1)1750 2195 y fk(::)7 b(x)p fq(\))75 2252 y(est)k(correctemen)o(t)h(d)o(\023)-21 b(e\014ni,)11 b(car)h fm(dom)p fq(\()p fm(v)742 2259 y fr(x)759 2252 y fq(\))g ff(\022)h fm(dom)p fq(\()p fm(\033)967 2259 y fr(x)984 2252 y fq(\))f ff(\022)h fm(img)r fq(\()p fk(h)p fq(\).)c(en)j(le)g(renomman)o(t)f fk(x)1598 2235 y fx(0)1610 2252 y fq(,)g(nous)h(a)o(v)o(ons)75 2308 y(alors)7 b(:)100 2352 y fk(x)100 2408 y fq(=)15 b fk(h)10 b ff(\016)g fk(h)241 2392 y fq(-1)286 2408 y fk(::)d(x)118 b fq(\()p fm(dom)p fq(\()p fm(\033)631 2415 y fr(x)648 2408 y fq(\))12 b ff(\022)h fm(img)r fq(\()p fk(h)p fq(\)\))100 2465 y(=)i fk(h)8 b(::)f(\(h)285 2448 y fq(-1)330 2465 y fk(::)h(x\))100 2521 y fq(=)15 b fk(h)8 b(::)f(x)261 2505 y fx(0)p eop %%page: 124 126 124 125 bop 75 269 a fq(124)772 b fl(annexe)15 b(a.)31 b(preuves)15 b(des)g(pr)o(opri)1731 257 y(\023)1725 269 y(et)1795 257 y(\023)1789 269 y(es)75 397 y fq(les)h(c)o(hamps)f fk(x)g fq(et)g fk(x)436 380 y fx(0)463 397 y fq(son)o(t)f(fortemen)o(t) f(\023)-21 b(equiv)m(alen)o(ts)16 b(car)f fm(dom)p fq(\()p fm(\033)1207 414 y fr(x)1222 397 y fx(0)s fq(\))e(=)g fm(dom)p fq(\()p fm(\033)1444 404 y fr(x)1470 397 y ff(\016)d fk(h)p fq(\))i ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\).)75 453 y fa(\003)75 629 y fj(pr)n(euve.\(pr)n(opri)o(\023)-22 b(et)o(\023)g(e)16 b(3,)h(32\))e fq(nous)g(utilisons)i(la)e(d)o(\023)

-21 b(e\014nition)16 b(10.)e(soit)i fk(x')f fq(=)g fk(h)10 b ff(\016)g fk(h)e(::)f(x)p fq(.)143 737 y({)23 b fk(x)213 721 y fx(0)234 737 y fq(est)10 b(correctemen)o(t)g(d)o(\023)-21 b(e\014ni.)10 b(en)g(e\013et,)g fk(x)896 721 y fx(0)917 737 y fq(est)g(correctemen)o(t)g(d)o(\023)-21 b(e\014ni)11 b(ssi)f fm(dom)p fq(\()p fm(v)1553 744 y fr(x)1570 737 y fq(\))i ff(\022)h fm(dom)p fq(\()p fk(h)d ff(\016)f fk(h)p fq(\).)189 794 y(p)o(ar)17 b(d)o(\023)-21 b(e\014nition)18 b(de)g(la)g(comp)q(osition)h(de)f(fonctions)g fm(dom)p fq(\()p fk(h)9 b ff(\016)h fk(h)p fq(\))17 b(=)g fm(dom)p fq(\()p fk(h)p fq(\))11 b ff(\\)h fk(h)1617 777 y fq(-1)1654 794 y fq(\()p fm(dom)p fq(\()p fk(h)p fq(\))n(\).)189 850 y(si)i fm(img)r fq(\()p fk(h)p fq(\))d ff(\022)i fm(dom)p fq(\()p fk(h)p fq(\))g(alors)h fm(dom)p fq(\()p fk(h)p fq(\))6 b ff(\\)i fk(h)920 834 y fq(-1)957 850 y fq(\()p fm(dom)p fq(\()p fk(h)p fq(\))o(\))14 b(=)g fm(dom)p fq(\()p fk(h)p fq(\))6 b ff(\\)i fk(h)1415 834 y fq(-1)1453 850 y fq(\()p fm(img)r fq(\()p fk(h)p fq(\))m(\))14 b(=)g fm(dom)p fq(\()p fk(h)p fq(\))o(.)189 906 y(nous)i(a)o(v)o(ons)g (bien)i fm(dom)p fq(\()p fm(v)658 913 y fr(x)675 906 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))f(car)i(nous)g(a)o(v) o(ons)e fm(dom)p fq(\()p fm(v)1358 913 y fr(x)1375 906 y fq(\))g ff(\022)h fm(dom)p fq(\()p fm(\033)1589 913 y fr(x)1606 906 y fq(\))g(\()p fk(x)g fq(est)h(cor-)189 963 y(rectemen)o(t)e(d)o(\023)-21 b(e\014ni\))15 b(et)g fm(dom)p fq(\()p fm(\033)724 970 y fr(x)741 963 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))h(\(h)o(yp)q(oth)o(\022)-21 b(ese\).)143 1071 y({)23 b fk(x)213 1055 y fx(0)240 1071 y fq(est)16 b(fortemen)o(t)d(\023)-21 b(equiv)m(alen)o(t)17 b(\022)-23 b(a)15 b fk(h)8 b(::)f(x)p fq(.)16 b(en)g(e\013et)f fk(x)1110 1055 y fx(0)1137 1071 y fq(est)h(fortemen)o(t)e(\023)-21 b(equiv)m(alen)o(t)16 b(\022)-23 b(a)16 b fk(h)7 b(::)h(x)15 b fq(ssi)189 1127 y fm(dom)p fq(\()p fm(\033)319 1134 y fr(x)346 1127 y ff(\016)9 b fk(h)402 1111 y fq(-1)440 1127 y fq(\))18 b ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))o(.)f(nous)h (a)o(v)o(ons)f(bien)i fm(dom)p fq(\()p fm(\033)1186 1134 y fr(x)1213 1127 y ff(\016)9 b fk(h)1269 1111 y fq(-1)1307 1127 y fq(\))18 b ff(\022)h fm(dom)p fq(\()p fk(h)1525 1111 y fq(-1)1561 1127 y fq(\))g(et)g fm(img)r fq(\()p fk(h)p fq(\))d ff(\022)189 1184 y fm(dom)p fq(\()p fk(h)p fq(\))e(\(h)o(yp)q(oth)o(\022)-21 b(ese\))14 b(impliquen)o(t)j fm(dom)p fq(\()p fm(\033)955 1191 y fr(x)982 1184 y ff(\016)10 b fk(h)1039 1167 y fq(-1)1076 1184 y fq(\))i ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))o(.)143 1292 y({)23 b(soit)f fm(e)p fq(\()p fk(x)352 1276 y fx(0)363 1292 y fq(\))i(=)g fk(h)488 1276 y fq(-1)533 1292 y fk(::)7 b(x')p fq(.)21 b(cette)h(expression)g(est)g(correctemen)o(t)f(d)o(\023)-21 b(e\014nie)23 b(car)f fm(dom)p fq(\()p fm(v)1742 1299 y fr(x)1757 1291 y fm(0)1770 1292 y fq(\))h ff(\022)189 1349 y fm(dom)p fq(\()p fk(h)317 1332 y fq(-1)353 1349 y fq(\),)15 b(c'est-\022)-23 b(a-dire)15 b fm(dom)p fq(\()p fm(v)757 1356 y fr(x)784 1349 y ff(\016)10 b fk(h)841 1332 y fq(-1)888 1349 y ff(\016)g fk(h)945 1332 y fq(-1)982 1349 y fq(\))j ff(\022)g fm(dom)p fq(\()p fk(h)1189 1332 y fq(-1)1225 1349 y fq(\))i(qui)h(est)f(vrai.)189 1431 y(nous)g(dev)o(ons)g(mon)o(trer)f fm(e)p fq(\()p fk(x)689 1414 y fx(0)701 1431 y fq(\))g(=)i fk(h)7 b(::)h(x)p fq(.)14 b(nous)h(a)o(v)o(ons)g(bien)214 1487 y fm(e)p fq(\()p fk(x)277 1471 y fx(0)288 1487 y fq(\))49 b(=)i fk(h)465 1471 y fq(-1)509 1487 y fk(::)8 b(x')355 1544

y fq(=)51 b fk(h)465 1527 y fq(-1)509 1544 y fk(::)8 b(\(h)h ff(\016)h fk(h)e(::)g(x\))49 b fq(\(d)o(\023)-21 b(ef.\))355 1600 y(=)51 b fk(h)465 1584 y fq(-1)512 1600 y ff(\016)10 b fk(h)g ff(\016)f fk(h)f(::)f(x)118 b fq(\(7a\),\(7b\)) 355 1657 y(=)51 b fk(h)7 b(::)g(x)189 1760 y fq(d'autre)14 b(part)214 1816 y fm(e)p fq(\()p fk(h)7 b(::)g(x)p fq(\))49 b(=)i fk(h)540 1800 y fq(-1)584 1816 y fk(::)8 b(\(h)f(::)h(x\))430 1873 y fq(=)51 b fk(h)540 1856 y fq(-1)587 1873 y ff(\016)10 b fk(h)d(::)h(x)430 1929 y fq(=)51 b fk(x)309 b fq(\(car)14 b fm(dom)p fq(\()p fm(v)1068 1936 y fr(x)1085 1929 y fq(\))e ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\))h(et)h fm(dom)p fq(\()p fm(\033)1506 1936 y fr(x)1523 1929 y fq(\))d ff(\022)h fm(dom)p fq(\()p fk(h)p fq(\)\))146 2033 y fa(\003)75 2208 y fj(pr)n(euve.\(pr)n(opri)o(\023)-22 b(et)o(\023)g(e)16 b(4,)h(p)n(age)f(32\))106 2316 y(\(1\))22 b fq(soit)h fk(e)314 2300 y fx(0)348 2316 y fq(le)g(c)o(hamp)g fk(\(h)604 2300 y fq(-1)649 2316 y fk(::)8 b(e)o(\))15 b fa(c)h fk(g)p fq(.)22 b(supp)q(osons)i fk(h)p fq(\()p fm(dom)p fq(\()p fk(g)p fq(\)\))f ff(\022)i fm(dom)p fq(\()p fm(\033)1528 2323 y fr(e)1545 2316 y fq(\))g ff(\022)g fm(img)r fq(\()p fk(h)p fq(\))c(et)189 2373 y fm(dom)p fq(\()p fm(v)315 2380 y fr(e)331 2373 y fq(\))h(=)g fm(dom)p fq(\()p fk(g)p fq(\))f(=)h fk(h)p fq(\()p fm(img)r fq(\()p fk(g)p fq(\)\).)c(alors,)i fk(e)1045 2356 y fx(0)1078 2373 y fq(est)g(bien)i(d)o(\023)-21 b(e\014ni)21 b(car)g fm(dom)p fq(\()p fm(v)1595 2380 y fr(e)1612 2373 y fq(\))g ff(\022)i fm(img)r fq(\()p fk(h)p fq(\))189 2429 y(\(car)18 b fm(dom)p fq(\()p fm(v)412 2436 y fr(e)429 2429 y fq(\))h ff(\022)g fm(dom)p fq(\()p fm(\033)650 2436 y fr(e)667 2429 y fq(\)\),)f fm(dom)p fq(\()p fk(g)p fq(\))g ff(\022)h fk(h)976 2413 y fq(-1)1014 2429 y fq(\()p fm(dom)p fq(\()p fm(\033)1162 2436 y fr(e)1178 2429 y fq(\)\))f(\(car)h fk(h)p fq(\()p fm(dom)p fq(\()p fk(g)p fq(\)\))e ff(\022)j fm(dom)p fq(\()p fm(\033)1738 2436 y fr(e)1754 2429 y fq(\)\))f(et)189 2486 y fm(img)r fq(\()p fk(g)p fq(\))10 b ff(\022)j fk(h)411 2469 y fq(-1)449 2486 y fq(\()p fm(dom)p fq(\()p fm(v)593 2493 y fr(e)609 2486 y fq(\)\))i(\(car)f fm(dom)p fq(\()p fm(v)879 2493 y fr(e)896 2486 y fq(\))e(=)h fm(dom)p fq(\()p fk(g)p fq(\)\).)p eop %%page: 125 127 125 126 bop 1778 269 a fq(125)189 397 y(nous)15 b(a)o(v)o(ons)f(alors)7 b(:)217 441 y fk(e)214 497 y fq(=)15 b fk(h)10 b ff(\016)g fk(h)355 481 y fq(-1)400 497 y fk(::)d(e)465 b fq(\(car)14 b fm(dom)p fq(\()p fm(\033)1167 504 y fr(e)1184 497 y fq(\))e ff(\022)h fm(img)r fq(\()p fk(h)p fq(\)\))214 554 y(=)i fk(h)7 b(::)h(\(h)399 537 y fq(-1)444 554 y fk(::)g(e)o(\))214 610 y fq(=)15 b fk(h)7 b(::)h(\(\(h)423 594 y fq(-1)468 610 y fk(::)f(e\))24 b fa(c)g fk(g)10 b ff(\016)g fk(g)745 594 y fq(-1)782 610 y fk(\))138 b fq(\(car)14 b fm(dom)p fq(\()p fm(v)1163 617 y fr(e)1180 610 y fq(\))e(=)h fm(dom)p fq(\()p fk(g)p fq(\)\))214 666 y(=)i fk(h)7 b(::)h(\(\(\(h)447 650 y fq(-1)492 666 y fk(::)f(e\))24 b fa(c)g fk(g\))g fa(c)g fk(g)833 650 y fq(-1)870 666 y fk(\))50 b fq(\(car)14 b fk(h)p fq(\()p fm(dom)p fq(\()p fk(g)p fq(\)\))d ff(\022)i fm(dom)p fq(\()p fm(\033)1432 673 y fr(e)1449 666 y fq(\)\))214 723 y(=)i fk(h)7 b(::)h(\(e)399 706 y fx(0)434 723 y fa(c)24 b fk(g)517 706 y fq(-1)554 723 y fk(\))189 794 y fq(de)e(la)g(caract)o(\023)-21 b(erisation)21 b(de)i(l')o(\023)-21 b(equiv)m(alence)24 b(faible,)f fk(e)f fq(et)g fk(e')g

fq(son)o(t)g(faiblemen)o(t)g(\023)-21 b(equiv)m(alen)o(ts)189 851 y(car)17 b fm(dom)p fq(\()p fm(v)393 858 y fr(e)408 849 y fm(0)421 851 y fq(\))e(=)i fm(dom)p fq(\()p fk(g)p fq(\))d(=)j fm(img)r fq(\()p fk(g)840 834 y fq(-1)875 851 y fq(\).)g(donc,)f(d'apr)o(\022)-21 b(es)16 b(la)i(d)o(\023)-21 b(e\014nition)18 b(10,)e(nous)h(concluons)189 907 y fk(p)12 b ff(v)h fk(p)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fk(h)6 b(::)h(\(e)24 b fa(c)g fk(g)651 891 y fq(-1)688 907 y fk(\))p fq(])p ff(g)p fq(.)106 1001 y fj(\(2\))e fq(soit)10 b fk(e)301 984 y fx(0)323 1001 y fq(le)h(c)o(hamp)f fk(h)530 984 y fq(-1)575 1001 y fk(::)d(\(e)23 b fa(c)i fk(g)o(\))p fq(.)10 b(supp)q(osons)h fm(dom)p fq(\()p fm(v)1176 1008 y fr(e)1193 1001 y fq(\))h(=)h fm(img)r fq(\()p fk(g)p fq(\))8 b(et)i fm(dom)p fq(\()p fk(g)p fq(\))i ff(\022)h fm(dom)p fq(\()p fm(\033)1803 1008 y fr(e)1819 1001 y fq(\))g ff(\022)189 1057 y fm(img)r fq(\()p fk(h)p fq(\).)k(alors,)i fk(e)520 1041 y fx(0)550 1057 y fq(est)g(correctemen)o(t)g(d)o(\023)-21 b(e\014ni)20 b(car)f fm(dom)p fq(\()p fk(g)p fq(\))f ff(\022)i fm(img)r fq(\()p fk(h)p fq(\))d(\()p fk(h)i fq(est)g(une)h(in)o(v)o(erse)189 1114 y(droite)15 b(de)h fk(g)p fq(\),)e fm(dom)p fq(\()p fk(g)p fq(\))d ff(\022)i fm(dom)p fq(\()p fm(\033)785 1121 y fr(e)802 1114 y fq(\))i(et)g fm(img)r fq(\()p fk(g)p fq(\))c ff(\022)i fm(dom)p fq(\()p fm(v)1213 1121 y fr(e)1230 1114 y fq(\).)189 1189 y(nous)i(a)o(v)o(ons) f(alors)7 b(:)217 1233 y fk(e)214 1289 y fq(=)15 b fk(e)g fa(c)h fk(g)10 b ff(\016)f fk(g)444 1273 y fq(-1)938 1289 y fq(\(car)14 b fm(dom)p fq(\()p fm(v)1157 1296 y fr(e)1174 1289 y fq(\))e(=)h fm(img)r fq(\()p fk(g)p fq(\)\))214 1346 y(=)i fk(\(e)24 b fa(c)g fk(g\))15 b fa(c)g fk(g)532 1329 y fq(-1)927 1346 y fq(\(car)g fm(dom)p fq(\()p fk(g)p fq(\))c ff(\022)i fm(dom)p fq(\()p fm(\033)1356 1353 y fr(e)1373 1346 y fq(\)\))214 1402 y(=)i fk(\(h)10 b ff(\016)g fk(h)379 1386 y fq(-1)424 1402 y fk(::)e(\(e)23 b fa(c)h fk(g\)\))15 b fa(c)h fk(g)772 1386 y fq(-1)934 1402 y fq(\(car)e fm(dom)p fq(\()p fm(\033)1157 1409 y fr(e)1174 1402 y fq(\))e ff(\022)h fm(img)r fq(\()p fk(h)p fq(\)\))214 1459 y(=)i fk(\(h)8 b(::)f(\(h)423 1442 y fq(-1)468 1459 y fk(::)h(\(e)24 b fa(c)g fk(g\)\)\))15 b fa(c)g fk(g)840 1442 y fq(-1)214 1515 y fq(=)g fk(\(h)8 b(::)f(e)399 1498 y fx(0)411 1515 y fk(\))15 b fa(c)g fk(g)524 1498 y fq(-1)189 1586 y fq(de)22 b(la)g(caract)o(\023)-21 b(erisation)21 b(de)i(l')o(\023)-21 b(equiv)m(alence)24 b(faible,)f fk(e)f fq(et)g fk(e')g fq(son)o(t)g(faiblemen)o(t)g(\023) -21 b(equiv)m(alen)o(ts)189 1643 y(puisque)16 b fk(h)p fq(\()p fm(dom)p fq(\()p fm(v)522 1650 y fr(e)537 1641 y fm(0)550 1643 y fq(\)\))c(=)h fk(h)p fq(\()p fk(h)712 1626 y fq(-1)749 1643 y fq(\()p fm(dom)p fq(\()p fk(g)p fq(\)\)\))d(=)j fm(dom)p fq(\()p fk(g)p fq(\))f(=)g fm(img)r fq(\()p fk(g)1334 1626 y fq(-1)1370 1643 y fq(\).)189 1699 y(donc,)i fk(p)f ff(v)g fk(p)p ff(f)p fm(s)s fq([)p fk(e)p ff(n)p fk(\(h)6 b(::)h(e\))15 b fa(c)h fk(g)789 1683 y fq(-1)826 1699 y fq(])p ff(g)f fq(d'apr)o(\022)-21 b(es)14 b(la)h(d)o(\023)-21 b(e\014nition)16 b(10.)146 1793 y fa(\003)p eop %%page: 126 128 126 127 bop 75 269 a fq(126)772 b fl(annexe)15 b(a.)31 b(preuves)15 b(des)g(pr)o(opri)1731 257 y(\023)1725 269 y(et)1795 257 y(\023)1789 269 y(es)p eop %%page: 127 129 127 128 bop 1778 269 a fq(127)75 715 y fi(annexe)33 b(b)75

930 y fp(t)-10 b(est)39 b(d')m(\023)-56 b(equiv)-6 b(alence)39 b(forte)146 1159 y fq(nous)14 b(a)o(v)o(ons)g(vu)h(que)g(la)h (transformation)d(de)i(programmes)f(est)h(souv)o(en)o(t)f(fond)o(\023) -21 b(ee)14 b(sur)h(l'applica-)75 1216 y(tion)j(de)h(c)o(hangemen)o(t)f (de)g(base.)g(sans)f(^)-21 b(etre)17 b(sp)o(\023)-21 b(eci\014que)19 b(\022)-23 b(a)17 b fn(pei)p fq(,)h(mais)g(au)g(con)o (traire)f(p)q(ouv)m(an)o(t)h(^)-21 b(etre)75 1272 y(fertile)21 b(plus)h(g)o(\023)-21 b(en)o(\023)g(eralemen)o(t)18 b(en)j (construction)g(de)f(programmes)g(\(par)f(exemple)j(de)f(programmes)75 1329 y(hpf)g(corresp)q(ondan)o(ts)h(\022)-23 b(a)21 b(des)h(programmes) e fn(pei)i fq(fortemen)o(t)d(\023)-21 b(equiv)m(alen)o(ts\),)22 b(cette)f(constatation)75 1385 y(p)q(eut)c(mener)f(\022)-23 b(a)16 b(des)h(pratiques)f(in)o(t)o(\023)-21 b(eressan)o(tes)16 b(par)f(\\analogies")i(en)o(tre)f(progammes,)f(p)q(our)h(d)o(\023)-21 b(ecider)75 1442 y(que)19 b(deux)h(programmes)e(son)o(t)f(\023)-21 b(equiv)m(alen)o(ts,)19 b(ou)g(bien)i(v)m(alider)f(une)g(in)o(tuition)g (qu'un)g(programme)75 1498 y(donn)o(\023)-21 b(e)15 b(calcule)i(bien)f (ce)g(qu'on)f(p)q(ense.)146 1562 y(nous)g(exp)q(osons)g(ici)i(une)e (solution)h(p)q(ermettan)o(t)f(de)g(d)o(\023)-21 b(eterminer,)14 b(\023)-21 b(etan)o(t)13 b(donn)o(\023)-21 b(e)15 b(deux)h(syst)o(\022) -21 b(emes)75 1618 y(d')o(\023)g(equations)20 b(r)o(\023)-21 b(ecurren)o(tes)20 b(a)o(y)o(an)o(t)f(une)j(structure)d(\023)-21 b(equiv)m(alen)o(te)22 b(\(seules)f(les)h(fonctions)f(de)g(d)o(\023)-21 b(ep)q(en-)75 1674 y(dances)15 b(di\013)o(\022)-21 b(eren)o(t\))13 b(si)i(l'un)g(p)q(eut)f(^)-21 b(etre)13 b(obten)o(u)i(de)f(l'autre)g (par)h(un)f(c)o(hangemen)o(t)h(de)f(base)h(global)g(sur)75 1731 y(le)i(programme.)d(nous)i(ne)h(consid)o(\023)-21 b(erons)15 b(ici)j(que)e(le)h(cas)f(d'un)g(c)o(hangemen)o(t)g(de)g (base)g(exprim)o(\023)-21 b(e)16 b(sous)75 1787 y(la)h(forme)f(d'une)i (fonction)f(a\016ne,)f(ainsi)i(que)f(des)g(d)o(\023)-21 b(ep)q(endances)18 b(a\016nes)f(dans)g(l')o(\023)-21 b(enonc)o(\023)g(e.)15 b(dans)h(ces)75 1844 y(conditions,)h(le)f(probl) o(\022)-21 b(eme)16 b(revien)o(t)g(\022)-23 b(a)16 b(r)o(\023)-21 b(esoudre)15 b(un)h(syst)o(\022)-21 b(eme)15 b(lin)o(\023)-21 b(eaire,)16 b(p)q(our)g(lequel)i(nous)e(utilisons)75 1900 y fn(maple)p fq([11].)146 1964 y(bien)e(que)g(cette)f(id)o(\023) -21 b(ee)14 b(ne)g(soit)f(qu'une)g(\023)-21 b(ebauc)o(he)13 b(de)h(r)o(\023)-21 b(e\015exion)14 b(sur)f(des)h(tests)f(d')o(\023)-21 b(equiv)m(alence)15 b(plus)75 2020 y(g)o(\023)-21 b(en)o(\023)g(eraux,) 15 b(cette)j(fonctionnalit)o(\023)-21 b(e)17 b(p)q(ourrait)g(^)-21 b(etre)16 b(implan)o(t)o(\023)-21 b(ee)18 b(assez)f(facilemen)o(t)i (dans)e(l'en)o(vironne-)75 2077 y(men)o(t)e(de)h(transformations)d fn(pei)p fq(.)146 2197 y(le)21 b(formalisme)f(utilis)o(\023)-21 b(e)22 b(p)q(our)f(repr)o(\023)-21 b(esen)o(ter)19 b(ces)h(\023)-21 b(equations)20 b(imp)q(orte)h(p)q(eu,)g(et)f(nous)h(utilisons)75 2253 y fn(alpha)f fq(dans)h(cet)f(exemple.)h(etan)o(t)e(donn)o(\023)-21 b(e)20 b(les)g(deux)h(programmes)e fk(cholesky1)g fq(et)h fk(cholesky2)75 2309 y fq(suiv)m(an)o(ts,)c(nous)f(c)o(herc)o(hons)h (\022)-23 b(a)15 b(sa)o(v)o(oir)g(si)h(ils)g(son)o(t)e(\023)-21 b(equiv)m(alen)o(ts.)16 b(une)g(condition)g(su\016san)o(te)f(est)h (qu'il)75 2366 y(existe)g(un)f(c)o(hangemen)o(t)g(de)h(base)f(p)q (ermettan)o(t)g(de)g(passer)g(de)h(l'un)g(\022)-23 b(a)15 b(l'autre.)146 2486 y(nous)i(dirons)h(que)f(l')o(\023)-21 b(enonc)o(\023)g(e)16 b fk(cholesky2)g fq(p)q(eut)h(^)-21 b(etre)16 b(obten)o(u)i(de)f fk(cholesky1)p fq(,)f(si)i(il)g(existe)g (une)75 2542 y(bijection)f fm(h)f fq(telle)h(que,)f(p)q(our)g(toute)f (d)o(\023)-21 b(ep)q(endance)17 b fm(f)996 2549 y fw(i)1025 2542 y fq(de)g fk(cholesky1)p fq(,)d fm(f)1352 2549 y fw(i)1379 2542 y fq(=)g fm(h)1454 2526 y fx(\000)p fg(1)1509 2542 y ff(\016)c fm(g)1564 2549 y fw(i)1588 2542 y ff(\016)h

fm(h)p fq(,)k(o)q(\022)-24 b(u)16 b fm(g)1762 2549 y fw(i)1791 2542 y fq(est)p eop %%page: 128 130 128 129 bop 75 269 a fq(128)720 b fl(annexe)16 b(b.)30 b(test)15 b(d')1344 257 y(\023)1338 269 y(equiv)-5 b(alence)15 b(f)o(or)l(te)75 397 y fq(la)h(d)o(\023)-21 b(ep)q(endance)16 b(corresp)q(ondan)o(te)f(dans)g fk(cholesky2)p fq(.)f(de)h(plus,)h(p)q (our)f(v)o(\023)-21 b(eri\014er)16 b(l')o(\023)-21 b(equiv)m(alence)17 b(forte,)75 453 y(une)f(m)o(^)-21 b(eme)14 b(relation)i(doit)f(exister) h(en)o(tre)f(les)h(domaines)g(des)f(v)m(ariables)i(\()p fm(v)d fq(=)f fm(v)1472 437 y fx(0)1493 453 y ff(\016)d fm(h)1552 437 y fx(\000)p fg(1)1597 453 y fq(\).)p 146 529 1704 2 v 145 1396 2 867 v 170 593 a fu(system)21 b(cholesky1)f(:)i({)f(n)h(|)f(n>=1)g(})497 643 y(\()h(a)g(:)f({)h(i,j)f (|)g(1<=i<=j;)g(1<=j<=n)f(})i(of)f(integer)f(\))323 693 y(returns)g(\()i(l)g(:)f({)h(i,j)f(|)g(1<=i<=n;)g(1<=j<=i)f(})i(of)f (integer)f(\);)170 792 y(var)236 842 y(l)43 b(:)22 b({)f(i,j,k)g(|)h (j<=i<=n;)e(-1<=k<=j-1;)f(1<=j)i(})h(of)f(integer;)170 942 y(let)236 992 y(l[i,j,k])42 b(=)21 b(case)497 1041 y({)h(|)g(i=j;)f(k=i-1;)f(1<=i<=n)g(})i(:)g(sqrt\(l[i,j,k-1])o(\);)497 1091 y({)g(|)g(2<=i<=n;)e(1<=j<=i-1;)f(k=j-1)i(})h(:)f(l[i,j,k-1])f(/)i (l[j,j,j-1])d(;)497 1141 y({)j(|)g(2<=i<=n;)e(2<=j<=i;)g(0<=k<=j-2)g(}) h(:)h(l[i,j,k-1])e(-)519 1191 y(l[i,k+1,k])g(*)h(l[j,k+1,k])f(;)497 1241 y({)i(|)g(1<=i<=n;)e(1<=j<=i;)g(k=-1)h(})g(:)h(a[j,i];)497 1290 y(esac;)236 1340 y(l[i,j])e(=)i(l[i,j,j-1];)170 1390 y(tel;)p 1848 1396 v 146 1398 1704 2 v 146 1421 v 145 2238 2 817 v 170 1485 a(system)f(cholesky2)f(:{n)h(|)h(1<=n})497 1535 y(\(a)g(:)f({i,j)g(|)h(1<=i<=j;)e(j<=n})h(of)g(integer\))323 1585 y(returns)f(\(l)i(:)f({i,j)g(|)h(j<=i<=n;)e(1<=j})h(of)g (integer\);)170 1634 y(var)214 1684 y(l)h(:)f({i,j,k)g(|)g (2j+k<=i<=j+k+n;)e(\(k+1,1\)<=j;)h(-1<=k})g(of)i(integer;)170 1734 y(let)214 1784 y(l[i,j,k])e(=)301 1834 y(case)345 1883 y({|)h(i=3k+2;)g(j=k+1;)f(0<=k<=n-1})g(:)h(sqrt\(l[i-1,j,k-1]\);) 345 1933 y({|)g(3k+3<=i<=2k+n+1;)e(j=k+1;)h(0<=k})h(:)h(l[i-1,j,k-1])d (/)i(l[3j-1,j,j-1];)345 1983 y({|)g(2j+k<=i<=j+k+n;)e(k+2<=j;)h(0<=k})h (:)410 2033 y(l[i-1,j,k-1])f(-)h(l[i-j+k+1,k+1,k])e(*)i (l[j+2k+1,k+1,k];)345 2083 y({|)g(2j-1<=i<=j+n-1;)e(1<=j;)i(k=-1})f(:)i (a[j,i-j-k];)301 2132 y(esac;)214 2182 y(l[i,j])f(=)g(l[i+2j-1,j,j-1];) 170 2232 y(tel;)p 1848 2238 v 146 2240 1704 2 v 75 2373 a fq(nous)11 b(p)q(ouv)o(ons)g(exprimer)h(le)g(probl)o(\022)-21 b(eme)11 b(sous)g(forme)g(lin)o(\023)-21 b(eaire.)11 b(nous)f(\023)-21 b(ecriv)o(ons)11 b(la)g(d)o(\023)-21 b(ep)q(endance)13 b(a\016ne)75 2429 y fm(f)21 b fq(d'un)c fm(n)p fq(-uplet)g fm(z)i fq(sous)c(la)i(forme)f fm(f)6 b(z)13 b fq(+)e fm(t)833 2436 y fw(f)855 2429 y fq(,)16 b(o)q(\022)-24 b(u)16 b fm(f)22 b fq(est)16 b(une)h(matrice)f(constan)o (te)g(et)g fm(t)1598 2436 y fw(f)1636 2429 y fq(un)h(v)o(ecteur)75 2486 y(constan)o(t.)j(la)h(bijection)i(que)e(nous)h(c)o(herc)o(hons)f (s')o(\023)-21 b(ecrit)21 b(alors)f fm(h)p fq(\()p fm(z)r fq(\))j(=)g fm(h)t(z)16 b fq(+)e fm(t)1522 2493 y fw(h)1565 2486 y fq(et)21 b(son)g(in)o(v)o(erse)75 2542 y fm(h)101 2526 y fx(\000)p fg(1)146 2542 y fq(\()p fm(z)r fq(\))12 b(=)h fm(h)307 2526 y fx(\000)p fg(1)351 2542 y fm(z)f ff(\000)f fm(h)472 2526 y fx(\000)p fg(1)516 2542 y fm(t)532 2549 y fw(h)554 2542 y fq(.)p eop %%page: 129 131 129 130 bop 1778 269 a fq(129)146 397 y(v)o(\023)-21 b(eri\014er)19 b(qu'il)h(existe)g(un)g(c)o(hangemen)o(t)f(de)g(base)h (en)o(tre)f(deux)h(d)o(\023)-21 b(ep)q(endances)20 b fm(f)k fq(et)c fm(g)g fq(consiste)75 453 y(alors)15 b(\022)-23

b(a)15 b(v)o(\023)-21 b(eri\014er)15 b(l')o(\023)-21 b(egalit)o(\023)g(e)6 b(:)446 552 y fm(f)g(z)13 b fq(+)e fm(t)577 559 y fw(f)640 552 y fq(=)42 b fm(h)759 535 y fx(\000)p fg(1)803 552 y fq(\()p fm(g)p fq(\()p fm(h)t(z)11 b fq(+)f fm(t)1010 559 y fw(h)1032 552 y fq(\))g(+)g fm(t)1121 559 y fw(g)1141 552 y fq(\))g ff(\000)g fm(h)1256 535 y fx(\000)p fg(1)1300 552 y fm(t)1316 559 y fw(h)640 608 y fq(=)42 b fm(h)759 592 y fx(\000)p fg(1)803 608 y fm(gh)t(z)11 b fq(+)g fm(h)1001 592 y fx(\000)p fg(1)1045 608 y fm(gt)1097 615 y fw(h)1129 608 y fq(+)f fm(h)1216 592 y fx(\000)p fg(1)1260 608 y fm(t)1276 615 y fw(g)1306 608 y ff(\000)g fm(h)1393 592 y fx(\000)p fg(1)1437 608 y fm(t)1453 615 y fw(h)75 706 y fq(c'est-\022)-23 b(a-dire,)15 b(v)o(\023)-21 b(eri\014er)15 b(d'une)h(part)7 b(:)802 808 y fm(f)48 b fq(=)42 b fm(h)998 792 y fx(\000)p fg(1)1042 808 y fm(gh)75 911 y fq(et)15 b(d'autre)g(part)7 b(:)624 967 y fm(t)640 974 y fw(f)704 967 y fq(=)42 b fm(h)823 950 y fx(\000)p fg(1)867 967 y fm(gt)919 974 y fw(h)951 967 y fq(+)10 b fm(h)1038 950 y fx(\000)p fg(1)1082 967 y fm(t)1098 974 y fw(g)1128 967 y ff(\000)g fm(h)1215 950 y fx(\000)p fg(1)1259 967 y fm(t)1275 974 y fw(h)75 1050 y fq(le)20 b(syst)o(\022)-21 b(eme)19 b(\022)-23 b(a)20 b(r)o(\023)-21 b(esoudre)19 b(est)g(consitu)o(\023)-21 b(e)20 b(d')o(\023)-21 b(equations)19 b(de)h(ce)g(t)o(yp)q(e)g(p)q(our) h(c)o(hacun)f(des)g(couples)h(de)75 1107 y(d)o(\023)-21 b(ep)q(endances)23 b fm(f)368 1114 y fw(i)382 1107 y fm(;)8 b(g)425 1114 y fw(i)437 1107 y fq(.)22 b(la)g(r)o(\023)-21 b(esolution)21 b(est)h(e\013ectu)o(\023)-21 b(ee)21 b(par)g fn(maple)i fq(qui)g(ten)o(te)e(de)h(d)o(\023)-21 b(eterminer)22 b(les)75 1163 y(v)m(aleurs)16 b(des)g(inconn)o(ues.)g(sur)f(cet)h (exemple)g(de)g(cholesky)l(,)f(le)h(syst)o(\022)-21 b(eme)14 b(a)h(une)h(solution)g(qui)g(est)7 b(:)612 1312 y fm(h)16 b fq(=)714 1215 y fx(0)714 1288 y(b)714 1315 y(@)771 1256 y fq(1)42 b(1)f(1)771 1312 y(0)h(1)f(0)771 1369 y(0)h(0)f(1)943 1215 y fx(1)943 1288 y(c)943 1315 y(a)995 1312 y fq(et)15 b fm(t)1064 1319 y fw(h)1098 1312 y fq(=)1146 1215 y fx(0)1146 1288 y(b)1146 1315 y(@)1204 1256 y fq(3)p fm(x)1215 1312 y(x)1215 1369 y(x)1273 1215 y fx(1)1273 1288 y(c)1273 1315 y(a)75 1461 y fq(le)j(fait)f(que)h(le)g(v)o(ecteur)f (de)h(translation)f fm(t)834 1468 y fw(h)874 1461 y fq(ne)g(soit)g(pas) h(compl)o(\022)-21 b(etemen)o(t)16 b(d)o(\023)-21 b(etermin)o(\023)g(e) 17 b(pro)o(vien)o(t)g(du)75 1518 y(fait)f(que)h(nous)g(ne)g(prenons)g (pas)f(en)h(compte)f(les)h(domaines)g(des)g(v)m(ariables.)h(on)f(p)q (eut)g(v)o(\023)-21 b(eri\014er)16 b(que)75 1574 y(ces)k(domaines)h(v)o (\023)-21 b(eri\014en)o(t)19 b(la)h(relation)h(et)f(conclure)h(que)f (ces)g(deux)h(programmes)d(son)o(t)i(fortemen)o(t)74 1631 y(\023)-21 b(equiv)m(alen)o(ts.)p eop %%page: 130 132 130 131 bop 75 269 a fq(130)720 b fl(annexe)16 b(b.)30 b(test)15 b(d')1344 257 y(\023)1338 269 y(equiv)-5 b(alence)15 b(f)o(or)l(te)p eop %%page: 131 133 131 132 bop 1778 269 a fq(131)75 729 y fp(t)-10 b(able)39 b(des)f(\014gures)143 952 y fq(2.1)46 b(deux)15 b(placemen)o(ts)h (di\013)o(\023)-21 b(eren)o(ts)14 b(d'un)i(m)o(ulti-ensem)o(ble)h(de)f (v)m(aleurs)h fm(:)22 b(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(10)143 1008 y(2.2)46 b(le)15 b(c)o(hamp)h fk(w)f fq(d)o(\023)-21 b(e\014ni)15 b(par)g(un)h(routage)e(des)i(v)m(aleurs)g (de)g fk(v)41 b fm(:)23 b(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h (:)f(:)40 b fq(11)143 1064 y(2.3)46 b(le)15 b(c)o(hamp)h

fk(w)f fq(d)o(\023)-21 b(e\014ni)15 b(par)g(un)h(routage)e (\(di\013usion\))i(des)g(v)m(aleurs)g(de)f fk(v)35 b fm(:)23 b(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(12)143 1121 y(2.4)46 b(le)15 b(c)o(hamp)h fk(w)f fq(d)o(\023)-21 b(e\014ni)15 b(par)g(une)h(op)o(\023)-21 b(eration)14 b(fonctionnelle)k(sur)d fk(v)39 b fm(:)23 b(:)f(:)g(:)h(:)f(:)h(:)f(:)g (:)h(:)f(:)40 b fq(12)143 1177 y(2.5)46 b(le)15 b(c)o(hamp)h fk(y)f fq(d)o(\023)-21 b(e\014ni)15 b(par)g(une)h(r)o(\023)-21 b(eduction)15 b(g)o(\023)-21 b(eom)o(\023)g(etrique)14 b(des)h(v)m(aleurs)i(de)e fk(x)46 b fm(:)22 b(:)g(:)h(:)f(:)40 b fq(13)143 1234 y(2.6)46 b(le)15 b(c)o(hamp)h fk(z)f fq(d)o(\023)-21 b(e\014ni)15 b(par)g fk(z)g fq(=)h fk(x/&/)7 b(w)18 b fm(:)k(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f (:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(13)143 1290 y(2.7)46 b(le)15 b(c)o(hamp)h fk(r)f fq(con)o(tien)o(t)g(les)h(sommes)f (partielles)h(du)g(c)o(hamp)f fk(a)24 b fm(:)e(:)h(:)f(:)g(:)h(:)f(:)h (:)f(:)g(:)h(:)f(:)40 b fq(16)143 1347 y(2.8)46 b(relation)16 b(en)o(tre)f(deux)h(placemen)o(ts)f(\023)-21 b(equiv)m(alen)o(ts)40 b fm(:)22 b(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h (:)f(:)40 b fq(19)143 1403 y(2.9)46 b(equiv)m(alences)17 b(forte)e(et)g(faible)h(en)o(tre)f(c)o(hamps)g(de)h(donn)o(\023)-21 b(ees)29 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(22)143 1460 y(2.10)23 b(deux)15 b(c)o(hamps)g(de)h(donn)o(\023)-21 b(ees)15 b(faiblemen)o(t)g(\023)-21 b(equiv)m(alen)o(ts)15 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:) 40 b fq(25)143 1558 y(3.1)46 b(ra\016nemen)o(t)15 b(par)g(substitution) h(de)g(c)o(hamps)e(\023)-21 b(equiv)m(alen)o(ts)46 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(30)143 1614 y(3.2)46 b(construction)15 b(d'un)g(programme)g(a)o(v) o(ec)f fn(vpei)38 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g (:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(41)143 1712 y(4.1)46 b(t)l(ransformation)14 b(de)i fk(matvect1)e fq(en)h fk(matvect2)f fq(p)q(our)i fm(n)d fq(=)g(3)38 b fm(:)22 b(:)h(:)f(:)g(:)h(:)f(:)h(:)f (:)g(:)h(:)f(:)40 b fq(52)143 1768 y(4.2)46 b(pro)q(duit)18 b(de)g(matrices)g(de)g(cannon)f(\()p fm(n)p fq(=)q(4\))7 b(:)17 b(alignemen)o(t)h(initial)i(des)e(matrices)g(et)248 1825 y(comm)o(unications)c fm(:)23 b(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:) h(:)f(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h (:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(55)143 1881 y(4.3)46 b(compactage)14 b(des)h(c)o(hamps)g fk(a)h fq(et)f fk(b)g fq(dans)g(un)h(carr)o(\023)-21 b(e)13 b(de)j fe(z)1260 1865 y fg(2)1288 1881 y fm(:)23 b(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f (:)g(:)h(:)f(:)40 b fq(59)143 1938 y(4.4)46 b(routage)15 b(r)o(\023)-21 b(ecursif)15 b(des)g(pro)q(duits)32 b fm(:)22 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h (:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(64)143 1994 y(4.5)46 b(\(a\))14 b(algorithme)i(initial)47 b(\(b\))15 b(algorithme)h(am)o(\023)-21 b(elior)o(\023)g(e)14 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(69)143 2092 y(5.1)46 b(t)l(ransp)q(osition)16 b(de)f(matrice)h(en) f(c)852 2076 y fx(\003)886 2092 y fq(et)g(en)h(hpf)h fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g (:)h(:)f(:)40 b fq(84)143 2149 y(5.2)46 b(pro)q(cessus)15 b(de)h(di\013usion)g(discr)o(\022)-21 b(ete)13 b fm(:)22 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)g (:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(85)143 2205 y(5.3)46 b(t)l(ransformations)14 b(syn)o(taxiques)h(r)o(\023)-21 b(ealis)o(\023)g(ees)15 b(par)g fk(normhpf)29 b fm(:)23 b(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)40 b fq(89)143 2262 y(5.4)46 b(t)l(ri)15 b(des)g(\023)-21

b(equations)14 b(selon)i(l'ordre)g(des)f(d)o(\023)-21 b(ep)q(endances)16 b(causales)30 b fm(:)23 b(:)f(:)g(:)h(:)f(:)h(:)f(:) g(:)h(:)f(:)40 b fq(90)143 2318 y(5.5)46 b(pro)q(duit)15 b(matriciel)i(sur)e(un)h(cub)q(e)g(de)g(pro)q(cesseur)f(virtuels)40 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)17 b fq(104)143 2375 y(5.6)46 b(programme)14 b fn(pei)h fq(de)g(pro)q(duit)h(matriciel)28 b fm(:)22 b(:)g(:)h(:)f(:)g(:)h(:)f (:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)17 b fq(105)143 2431 y(5.7)46 b(en)o(vironnemen)o(t)15 b(logiciel)j (autour)d(de)g fn(pei)44 b fm(:)22 b(:)h(:)f(:)g(:)h(:)f(:)g(:)h(:)f(:) g(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)17 b fq(107)p eop %%trailer end userdict /end-hook known{end-hook}if %%eof