1. Solution. A) The Following Is The Program Of A).

1. Solution. a) The following is the program of a). clear; close; format long N=2;err=1;x0=0;x1=1;s1=0; f=@(x) sin(x)./(1+x.^4)+sin(1./x).* x.^2./(1+x.^4); disp(' Step approximation') while err>=10^(-4)/2 h=(x1-x0)/N; s=0*h/2; for i=1:N-1 s=s+h*f(x0+h*i); end s=s+f(x1)*h/2; err=abs(s-s1)/(2^4-1); s1=s; disp([N, s]) N=N*2; end The following is the result: Step

approximation

2.000000000000000 0.542955932230503 4.000000000000000 0.554587142041583 8.000000000000000 0.568766288145437 16.000000000000000 0.569553726654222 32.000000000000000 0.569624349130263

b) The following is the program of b). clear; close; format long N=2;err=1;s1=1; f=@(x) exp(-x)./(1+x).^(5)+x.^3.*exp(-1./x)./(x+1).^5; disp('Step approximation') while err>=10^(-4)/2 x0=0;x1=1; h=(x1-x0)/N; s=1*h/2; for i=1:N-1 s=s+h*f(x0+h*i); end s=s+f(x1)*h/2; err=abs(s1-s)/(2^4-1); s1=s; disp([N, s]) N=N*2; end The following is the result: Step

approximation

2.000000000000000 0.296798162037338 4.000000000000000 0.221110701061077 8.000000000000000 0.199193261253196 16.000000000000000 0.193453482148226 32.000000000000000 0.192000466701357 64.000000000000000 0.191636049869488 1.0e+002 * 1.280000000000000 0.001915448724313

c) The following the program of c). clear; close; format long N=2;err=1;s1=0; f=@(x) exp(-x.^2).*cos(x)+exp(-1./x.^2).*cos(1./x).*(1./x.^2); disp('Step approximation') while err>=10^(-4)/2 x0=0;x1=1; h=(x1-x0)/N; s=1*h/2; for i=1:N-1 s=s+h*f(x0+h*i); end s=s+f(x1)*h/2; err=2*abs(s1-s)/(2^4-1); s1=s; disp([N, 2*s]) N=N*2; end Step

approximation

2.000000000000000 1.351740116024899 4.000000000000000 1.374766532917430

8.000000000000000 1.379371257478457 16.000000000000000 1.380129499702764 32.000000000000000 1.380323735398339 64.000000000000000 1.380372270845669

Trapezoidal Rule: clear; close; format long N=2;err=1;x0=0;x1=1;s1=0;r=1.1;

g=@(x) exp(x).*log(x); f=@(x) r^2*exp(x.^r).*x.^(r-1).*log(x); disp('Step approximation') while err>=10^(-5)/2 h=(x1-x0)/N; s=0*h/2; for i=1:N-1 s=s+h*f(x0+h*i); end s=s+f(x1)*h/2; err=abs(s-s1)/(2^2-1); s1=s; disp([N, s]) N=N*2; end r=1.1, the result is shown as follows: Step

approximation

2.000000000000000 -0.623854591742896 4.000000000000000 -0.940995768035230 8.000000000000000 -1.112592600798488 16.000000000000000 -1.207102951940195 32.000000000000000 -1.258874395487081 64.000000000000000 -1.286856326016077 1.0e+002 * 1.280000000000000 -0.013017605045802 1.0e+002 * 2.560000000000000 -0.013095928079541 1.0e+002 * 5.120000000000000 -0.013136609482607 1.0e+003 * 1.024000000000000 -0.001315753063183

1.0e+003 * 2.048000000000000 -0.001316819945001 1.0e+003 * 4.096000000000000 -0.001317360119595 1.0e+003 * 8.192000000000000 -0.001317631941802 1.0e+004 * 1.638400000000000 -0.000131776800227 1.0e+004 * 3.276800000000000 -0.000131783579334 1.0e+004 * 6.553600000000000 -0.000131786943298 1.0e+005 * 1.310720000000000 -0.000013178860660 1.0e+005 * 2.621440000000000 -0.000013178942640

r=1.5, the result is shown as follows: Step

approximation

2.000000000000000 -0.785252399689505 4.000000000000000 -1.102752633918305 8.000000000000000 -1.230956958968851 16.000000000000000 -1.282945971385789 32.000000000000000 -1.303961853848449 64.000000000000000 -1.312392276358369

1.0e+002 * 1.280000000000000 -0.013157431784960 1.0e+002 * 2.560000000000000 -0.013170629138336 1.0e+002 * 5.120000000000000 -0.013175782571961 1.0e+003 * 1.024000000000000 -0.001317777944723 1.0e+003 * 2.048000000000000 -0.001317854789939 1.0e+003 * 4.096000000000000 -0.001317884182432 1.0e+003 * 8.192000000000000 -0.001317895364322

r=2, The result is shown as follows: Step

approximation

2.000000000000000 -0.890018597344417 4.000000000000000 -1.192608285867874 8.000000000000000 -1.282681369490746 16.000000000000000 -1.308174900682751 32.000000000000000 -1.315243410237071 64.000000000000000 -1.317180972500353 1.0e+002 * 1.280000000000000 -0.013177077490157

1.0e+002 * 2.560000000000000 -0.013178500249506 1.0e+002 * 5.120000000000000 -0.013178882384224 1.0e+003 * 1.024000000000000 -0.001317898452849 r=3, the result is shown as follows: Step

approximation

2.000000000000000 -0.883618487324026 4.000000000000000 -1.195009443541396 8.000000000000000 -1.286108348460041 16.000000000000000 -1.309909614973187 32.000000000000000 -1.315905252904677 64.000000000000000 -1.317403523621169 1.0e+002 * 1.280000000000000 -0.013177775970825 1.0e+002 * 2.560000000000000 -0.013178710274392 1.0e+002 * 5.120000000000000 -0.013178943723826 1.0e+003 * 1.024000000000000 -0.001317900206935 r=10, the result is shown as follows: Step

approximation

2.000000000000000 -0.067756290305606

4.000000000000000 -0.605305883381251 8.000000000000000 -1.043582590843233 16.000000000000000 -1.235910892754504 32.000000000000000 -1.296223780878548 64.000000000000000 -1.312400159150013 1.0e+002 * 1.280000000000000 -0.013165213457023 1.0e+002 * 2.560000000000000 -0.013175566157121 1.0e+002 * 5.120000000000000 -0.013178157465841 1.0e+003 * 1.024000000000000 -0.001317880548928 1.0e+003 * 2.048000000000000 -0.001317896750741 1.0e+003 * 4.096000000000000 -0.001317900801271

Simpson Rule: close; clear; N=2;r=1.1;x0=0;x1=1; S1=0;err=1; g=@(x) exp(x).*log(x); f=@(x) r^2*exp(x.^r).*x.^(r-1).*log(x); disp('Step approximation') while err>=(1e-5)/2 h=(x1-x0)/2/N; S=0; for i=1:2*N if (floor(i/2)-i/2==0) a=2; else a=4;

end S=S+a*f(x0+i*h); end S=S-f(x1); S=h/3*S; err=abs(S1-S)/15; S1=S; disp([N, S]) N=2*N; end r=1.1 Step

approximation

2.000000000000000 -1.046709493466008 4.000000000000000 -1.169791545052907 8.000000000000000 -1.238606402320764 16.000000000000000 -1.276131543336043 32.000000000000000 -1.296183636192408 64.000000000000000 -1.306728564101631 1.0e+002 * 1.280000000000000 -0.013122035757454 1.0e+002 * 2.560000000000000 -0.013150169950296 1.0e+002 * 5.120000000000000 -0.013164504348236 1.0e+003 * 1.024000000000000 -0.001317175572273 1.0e+003 * 2.048000000000000 -0.001317540177793 1.0e+003 *

4.096000000000000 -0.001317722549204 1.0e+003 * 8.192000000000000 -0.001317813355760 1.0e+004 * 1.638400000000000 -0.000131785839036 r=1.5 Step

approximation

2.000000000000000 -1.208586045327905 4.000000000000000 -1.273691733985700 8.000000000000000 -1.300275642191435 16.000000000000000 -1.310967148002669 32.000000000000000 -1.315202417195009 64.000000000000000 -1.316860145875236 1.0e+002 * 1.280000000000000 -0.013175028256128 1.0e+002 * 2.560000000000000 -0.013177500383170 1.0e+002 * 5.120000000000000 -0.013178445072319 1.0e+003 * 1.024000000000000 -0.001317880405011 r=2 Step

approximation

2.000000000000000 -1.293471515375693 4.000000000000000 -1.312705730698371

8.000000000000000 -1.316672744413419 16.000000000000000 -1.317599580088510 32.000000000000000 -1.317826826588113 64.000000000000000 -1.317883341187503 r=3 Step

approximation

2.000000000000000 -1.298806428947186 4.000000000000000 -1.316474650099589 8.000000000000000 -1.317843370477569 16.000000000000000 -1.317903798881840 r=10 Step

approximation

2.000000000000000 -0.784489081073132 4.000000000000000 -1.189674826663894 8.000000000000000 -1.300020326724927 16.000000000000000 -1.316328076919896 32.000000000000000 -1.317792285240502 64.000000000000000 -1.317895074553085 1.0e+002 * 1.280000000000000 -0.013179017057153

6. Solution. b) clear; close; format long N=2;err=1; x0=0;x1=2/pi;s1=100; f=@(x) x.^2.*sin(1./x); disp('Step approximation') while err>=10^(-3)/2 h=(x1-x0)/N; s=0*h/2; for i=1:N-1 s=s+h*f(x0+h*i); end s=s+f(x1)*h/2; err=abs(s1-s); s1=s; disp([N, s]) N=N*2;

end Result: Step

approximation

2.000000000000000 0.064503068866399 4.000000000000000 0.063673513579651 8.000000000000000 0.059387564754992 16.000000000000000 0.058902335031313

c) clear; close; format long N=2;err=1; x0=0;x1=1; s1=111; f=@(x) 3*cos(x.^3); disp('Step approximation') while err>=10^(-8)/2 h=(x1-x0)/N; s=f(x0)*h/2; for i=1:N-1 s=s+h*f(x0+h*i); end s=s+f(x1)*h/2; err=abs(s1-s)/3; s1=s; disp([N, s]) N=N*2; end Step

approximation

2.000000000000000 2.643523230245098 4.000000000000000 2.755912152904892 8.000000000000000 2.785264907428469 16.000000000000000 2.792648819817276

32.000000000000000 2.794497055948642 64.000000000000000 2.794959246696237 1.0e+002 * 1.280000000000000 0.027950748024685 1.0e+002 * 2.560000000000000 0.027951036919146 1.0e+002 * 5.120000000000000 0.027951109143075 1.0e+003 * 1.024000000000000 0.002795112719908 1.0e+003 * 2.048000000000000 0.002795113171308 1.0e+003 * 4.096000000000000 0.002795113284158 1.0e+003 * 8.192000000000000 0.002795113312370 1.0e+004 * 1.638400000000000 0.000279511331942 6. Solution. First we plot the graph of f, p_1(x), p_3(x) and p_5(x). close;clear; x0=-1;x1=1; x=x0:0.001:x1; f=@(x) sin(pi/2*x); p5= @(x) pi* x/2 - (pi^3* x.^3)/48 + (pi^5 *x.^5)/3840; p3=@(x) pi* x/2 - (pi^3* x.^3)/48; p1=@(x) pi* x/2; plot(x,p5(x),'--') hold on plot(x,p3(x),'-.') hold on

plot(x,p1(x),':') hold on plot(x,f(x)) legend('n=5','n=3','n=1','analytic') The result is shown on the first graph. And also we plot the error of p_1(x), p_3(x) and p_5(x) with respect to f. close;clear; x0=-1;x1=1; x=x0:0.001:x1; f=@(x) sin(pi/2*x); p5= @(x) pi* x/2 - (pi^3* x.^3)/48 + (pi^5 *x.^5)/3840; p3=@(x) pi* x/2 - (pi^3* x.^3)/48; p1=@(x) pi* x/2; plot(x,abs(f(x)-p5(x)),'--') hold on plot(x,abs(f(x)-p3(x)),'-.') hold on plot(x,abs(f(x)-p1(x))) legend('n=5','n=3','n=1'); The result is shown on the second graph.