s06 - Ma 449 Numerical Applied Mathematics Model Solutions...

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Unformatted text preview: Ma 449: Numerical Applied Mathematics Model Solutions to Homework Assignment 6 Prof. Wickerhauser Exercise 2(b) of Section 5.1, p.259. MATLAB CODE: <>x=[-6 -2 0 2 6]; y=[-5.3 -3.5 -1.7 0.2 4]; f=[-6 -2.84 -1.26 0.32 3.48]; <>one=[1 1 1 1 1]; n=5; <>xmean=one*x'/n XMEA = 0. <>ymean=one*y'/n YMEA = -1.2600 <>x-xmean*one ANS = -6. -2. 0. 2. 6. <>sxy = (x-xmean*one)*(y-ymean*one)' SXY = 63.2000 <>sx2 = (x-xmean*one)*(x-xmean*one)' SX2 = 80. <>a = sxy/sx2 A = .7900 <>b=ymean-a*xmean B = -1.2600 <>l=a*x+b*one L = -6.0000 -2.8400 -1.2600 .3200 3.4800 <>l-f ANS = 1.0E-15 * .0000 .0000 -.2220 -.3886 -.8882 <>l-y ANS = -.7000 .6600 .4400 .1200 -.5200 <>e2=(l-y)*(l-y)' E2 = .5299 Exercise 5 of Section 5.1, p.260. Let SX2, SX1, SY1, and SXY be the sums of x^2, x, y, and x*y, respectively, from Equation Eq. 10 of p.255. Then Eq. 10 can be written in matrix form as: / SX2 SX1 \ /A\ = /SXY\ \ SX1 N / \B/ \SY1/ This 2x2 system can be solved explicitly: /A\ = -1 / N -SX1 \ /SXY\ , \B/ D \ -SX1 SX2 / \SY1/ where D = SX2*N-SX1*SX1. Multiplying this out gives A = ( N*SXY - SX1*SY1)/D; B = (SX2*SY1 - SX1*SXY)/D Exercise 6 of Section 5.1, p.260. Let xm = SX1/N be the mean value of x(1),...,x(N). Then note that [x(k)-xm]*[x(k)-xm] = x(k)*x(k) - 2*xm*x(k) + xm*xm, so summing this over k=1,2,...,N gives Sum [x(k)-xm]*[x(k)-xm] = SX2 - 2*xm*SX1 + N*xm*xm k = SX2 - 2*SX1*SX1/N + SX1*SX1/N = SX2 - SX1*SX1/N = D/N This shows that D = N*Sum [x(k)-xm]*[x(k)-xm], k which is nonzero if x(1),...,x(N) contains at least 2 distinct points, since then [x(k)-xm] must be nonzero for some k. Algorithm 4(a,b,c) of Section 5.1, p.262. MATLAB CODE: OUTPUT: % generate the function: n=50; x=[1:n] * 0.1; y=x+cos(sqrt([1:n])); %(a) compute least-squares line y = ax+b: sx1=sum(x); sy1=sum(y); sx2=sum(x.*x); sxy=sum(x.*y); d=sx2*n-sx1*sx1; a=( n*sxy - sx1*sy1)/d A = 1.4186 b=(sx2*sy1 - sx1*sxy)/d B = -.8808 l=a*x+b*ones(x); %(b) compute the RMS error: e2=norm(l-y)/sqrt(n) E2 = .4032 %(c) plot the error: plot(l-y) * ** ** * * * * * * * * * * * * * * ** * * * * * ** ** ** * ** * * * ** ** * ** * * * * Exercise 2(c) of Section 5.2, p.275. MATLAB CODE: OUTPUT: x=[-2 -1 0 1 2]; y=[10 1 0 2 9]; n=5; % compute least-squares parabols y = ax^2+bx+c: sx1=sum(x); sx2=sum(x.*x); sx3=sum(x.*x.*x); sx4=sum(x.*x.*x.*x); sy1=sum(y); sxy=sum(x.*y); sxxy=sum(x.*x.*y); mx=[sx4 sx3 sx2 ; sx3 sx2 sx1 ; sx2 sx1 n]; rh=[sxxy sxy sy1]' ; coef=mx\rh; a=coef(1), A = 2.5000 b=coef(2), B = -.1000 c=coef(3), C = -.6000 p=a*x.*x+b*x+c*ones(x), P = 9.6000 2.0000 -.6000 1.8000 9.2000 y, Y = 10. 1. 0. 2. 9....
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This note was uploaded on 05/16/2010 for the course MATH 449 taught by Professor Wickerhauser during the Fall '09 term at Washington University in St. Louis.

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s06 - Ma 449 Numerical Applied Mathematics Model Solutions...

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