Numerical Analysis

Midterm Problem Set 2 Solutions

II.

> with(CurveFitting);

> xlist:=[0.0,0.04,0.14,0.21,1.22,1.5,1.93];

> ylist:=[0.0,.849,1.894,1.986,1.895,1.5,.694];

> IP:=PolynomialInterpolation(xlist,ylist,x);

free spline:

> FSp:=Spline(xlist,ylist,x);

> plot({FSp,IP},x=0..2);

clamped spline: use forward difference at and a backward difference at

> df0:=(ylist[2]-ylist[1])/(xlist[2]-xlist[1]);

> dfn:=(ylist[7]-ylist[6])/(xlist[7]-xlist[6]);

> read "/home/fac/little/public_html/Num01/ClampedSpline.map";

> CSp:=clampedspline(xlist,ylist,df0,dfn,x);

> plot({CSp,IP},x=0..2);

IV.

> f:=x->sqrt(1+cos(x)^2);

> plot(abs(diff(f(x),x$2)),x=0..48);

> plot(abs(diff(f(x),x$4)),x=0..48);

Error bound for composite Trapezoidal Rule:

> fsolve(1/12*48*h^2=.0001,h);

> nvaltrap:=200*48;

Error bound for composite Simpson's Rule:

> hvalsimp:=fsolve(7/180*48*h^4=.0001,h);

> nvalsimp:=48/hvalsimp[2];

> with(student):

> approx:=evalf(simpson(f(x),x=0..48,562));

> exact:=evalf(Int(f(x),x=0..48));

> simperror:=abs(exact-approx);

>