Lecture 2 Numerical Analysis
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1e5i 9or Converye.rte', , T \e fl +fui,A -T \\e- 6 e,Iu,ence v*w61. be- C-o^ve-'r3e/rLe-e'' 3 ene.,ro1tai
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fir\q*0Jn
t|."t"- wll'!-be'V*g
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etttn<,n+O"4- flw" s 1t',a,ae.
Con&!,,,rnr i (Ior Conv.enettze\ p irzon en i tsr-€.'\1ve 3 c \'e-'-<- 't-n., -J(ln)
rc'at d o{ !l Ll , 'bLe wotr d & (onvecSs@tke Q\em g, wi"(t *l',2 *t5+ ba fu.,;J Co^'+aii2 t\'
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Tkere rrn't an &
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cuhe. "-
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gct-ctm p\ e.i
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bJ recttre"',u'
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a-^\t
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bg r-=6(t) w e
qt-it-+\=o
--3L'Y-^) Loe- co(n obt"ti^ t"'--\ttn'"t)
ea i"ito+[ 'Auesb b o. e di€lerenr' ^ . , -,2 6bt)Lti =0"1b1 g' 6b7 it'' . r) 'r-,-- Li'-t \t8: ,-o -ru--u f h 1 4 { l e r\ -o o c
we trb
'lt{-' U 590 ' 1CZ'-tt 4\l ,
\a5 t o=Zr)'' \" b 67-"t" -' 1-.,. !Lzzn t]*ct t
trelotlton ia
\^- ^ +-
a"d t aVer^2 't=Z -\ l*
jn "t ',
)za.lr"wt"vce'
o L ^ - ,- 3 - $
-{ eqJs t o
2' 615),-
Co ^veYyeEo;
\' z 6 o t { > - - u . Z L z ' L l z o ' 5 7
C on\.ter1 z Ert bnt^q\k'c
?.0-
to 'z-\
tLat
etc.
t ! . " ,t
t\"
Seo1tp'.c <-
Q-,uu,., 1a.b7o ,2"6aot
i i ; C a " v e u V i 2t o t h e
\aner rac4,
1e5i 9or Converye.rte', , T \e fl +fui,A -T \\e- 6 e,Iu,ence v*w61. be- C-o^ve-'r3e/rLe-e'' 3 ene.,ro1tai
f\e
q.. rt-z-I * > e i Q e ^c e
Ig $nb 3 e1.te-,-'cev o,"ut rutp\dgg.t"itk'r
t5 Tet "r.t\a*b
fir\q*0Jn
t|."t"- wll'!-be'V*g
di {l-e,
etttn<,n+O"4- flw" s 1t',a,ae.
Con&!,,,rnr i (Ior Conv.enettze\ p irzon en i tsr-€.'\1ve 3 c \'e-'-<- 't-n., -J(ln)
rc'at d o{ !l Ll , 'bLe wotr d & (onvecSs@tke Q\em g, wi"(t *l',2 *t5+ ba fu.,;J Co^'+aii2 t\'
t\r!t
r-,rI r\e
-{-o* i"**'1,.\ b^e55 +o 1a 5 elee-*eA
i'rt*-'",1
ra,n i,
ft)40 py0ut 44 d.
Lt, b-]
&e-rl-".-\-^'e. o ^ Lc", b1
14.t1 $ Lo.: a Co,rt-qeq5
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Fra*jle 2.
?,"+21-il "=-(, lcy ploH-rp f !"e 9,"nc*to n5 l,Je l>.'c 'l'\"tt
c(nd i^-t\e
the
onl6
i,.terv{ {
Ye Qruge,,,e,.ft
"t_.,_utL"r)
Ur-rr3 . ^ d j z ; - 2 " r - - l
f
&'eb ciooeto -cJ..t,'
t-g'?5)-o"LDf
rb 16=. P
fs
I
7
!L'n\ ',- - Lt-tz"\frY -rl/"")/",. 9'L".,), 9-(l l') nrt
gLlqtt"\ teel
a^J
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q [1et-rwtr've-
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ema o-Y-Jt/z r1F7u L*.tfr;\:
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" .- [- Jr4 , ,l'4] --, aeL,[;tt,a] \
/ur.c \3) "t
t
' J
'
*r---'-'-'*'i)l:
o r ' . J ' v . o' - - t J\
)'X-,=-0'59;
o r ^ J a ' - /= - a q 5 3 t t l - , . - > r u o t - i s l L ' q 5 3 7
1r--4.q*36
,^/rte i t\^ e 4 i.rnchpn r^o^otonic on fortf
t> ty,o^o:.oni a la f.tncvun i a
a?\\
ip it i:
czlvL xj5 1", a-rcnbi$6yJeet€<$-"[
0h La,bl ,ii,. grqd;e,\td.resno{- c tvcrrLge srln o^ Lqln itb hr\{r\.{ v^ and rrvi*t".,.{"".. V4L.re> gccqf .,b the ead: og,the
inre,.vo\\l_Jtn, ,af
,_,t 6( gtntqs
e.Lq,.p\e.. 5\s d -IM r 11,,t-e1w'tt oo A tO O t
\-.. tV.e
t - t t
1^t"tv< |
w\cil.e" <,1\av dplh( f
\t^
1n- +tf-- qC.rbrn( k,."S l ,
|
,
And
[,/qrl/.a I iFrq\vg
a n . ,' - - {\ ( ( , s n a . I. ' ' - ' / ; 1 I
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,
d e t e x ^ \ / n r .g V\45
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r
(to^9iJ€f
P1''^; , (z-t-rt)2-.i4 e511
9 t { ) . g i l t r ) 2 - Lc to ) i r : C , f - q t E ) .
. L
= 2.27-L.Z---o
-)-
--2 - + z - L^ - - " n '
,"5+ -,t!)'-
fl,fz+ >+on^.r{,":
there i5 6c poi^+tu;
5 Lo" fl.' e $i.O ache,^c,
?fn) '1gotrni4 + :r u
' ' \ d P. o , \|t , =+--=.E I'U-ztt+)--l'4
-,lt.4
a^d
g'ln).^"
a(('5n11\1' -
= +=-_Orrtlo
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o \ p a i ^ fr " ' F t g t i r i r r l
t\:.:,b
ru e-co ^ cl",^le, t\+
fu, oe cr:,., d ac-\.re
I - 5 i . n r ( . o ( ' 1r-h e
Co\veyc,,!?e
15 "vot fwu.q,rtc.J
wz \c,vrl .
,nnL c,J'l (rc-+!121, $L',.)' " !.} :)'(',) +/ I +# ') r7 r ) ) t / ( ll-('L+t7,)'J \/t -(t*t, 1.-
c\<ar12.1't-)|0,,-_o^ I oni^*e'r"*\ 't')
A(\ ) - c).zb6 , 3(1,,)'o.xlo
ni'no+o"'c an d
b e c " n n s ey C ^ i
\/q:96) I L
/(n1 6,r tt.t ,*{ e'u4l ,ta t or :: |,.dJ be Muro. o4 j Co^siJer L['.) llt -r-l/z) -> . z h(r).
t',thrc \ l b P u S l - i v c O ^ | t L ) t L \ - ,
l
-
li-(r*rt,)qJ \"t"")
i:,.orw.torui L 0^
ha"-,.'-,r."w t*d-,,."S
Lll,4,\/i]
0 CcLry qt
o^h t:
"ff..e e^J
rwfla.r. -,r;.irzr .z-rJ
poi.f S..
\ 1 l t n ) .v . b + 7 , [ ( l / 3 )" ( t . 7 3 7 = > \ e . r ' ' 11] { algebwoall4" z lrr--rl[ )
,1-. =----Yl-LnrV.)L(
v
3- rrt < t- .t( u l7z
(r It
-f-> J - +vt'-t '
{
'=-> 1o (lOttq')L2"u'wa) -*-? (\Jz- r, B\b
'A b - a , 1 a ')tu , 1 U . ? U ' l l 6
t r " { r t p , ' : s i f , }g ,
@ o r
'l - tt'toq4o r . - c [ * Je T\e
:>-.1,#t{.
2 | + o)01)'
( \ 1 , t ,'lsl
<0.59{
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a.J tt.,*-xlre \ u t . 1 1 1 1
rnet"oJ $i5ec't ot'
C - o n adi. r S ( r , ) lg
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fa,^l ,.'..J r*
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