Project 4 Solutions

Project 4 Solutions - Project 4 Group name: Unknowns Group...

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Project 4 Group name: Unknowns Group members: Annon, Sofia, Hristian, Divya 1. Matlab code: function x=mgslsq(A,b) [Q,R]=mgs([A b]); R1=R(1:size(R,1)-1,1:size(R,2)-1); c=R(1:size(R,1)-1,size(R,2)); x=backsub(R1,c); a) >> A=hilb(6);A=A(:,1:4);b=A*ones(4,1); >> x=mgslsq(A,b) x = 0 0 0 0 >> norm(A*x-b) ans = 2.8578 >> x=A\b x = 1.0000 1.0000 1.0000 1.0000 >> norm(A*x-b) ans = 1.0991e-015 We can see that the second method is much better, it gives a much smaller residual. b) >> A=ones(6)+eps*eye(6);A=A(:,1:4);b=A*ones(4,1);
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>> x=mgslsq(A,b) x = 0 0 0 0 >> norm(A*x-b) ans = 9.7980 >> x=A\b Warning: Rank deficient, rank = 1, tol = 3.2634e-015. x = 4.0000 0 0 0 >> norm(A*x-b) ans = 1.9860e-015 The second method gives a smaller residual. --------------------------------------------------------------------------------------------------------------------------------------------------- ------------- 2. Matlab code: function [Q,R]=givensqr(A) [m,n]=size(A); R=A; Q=eye(m,m); for i=1:n for j=i+1:m r= norm([R(i,i);R(j,i)]); c=R(i,i)/r; s=R(j,i)/r; temp=R(i,:); R(i,:)=c*R(i,:)+s*R(j,:); R(j,:)=-s*temp+c*R(j,:); temp=Q(i,:);
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Q(i,:)=c*Q(i,:)+s*Q(j,:); Q(j,:)=-s*temp+c*Q(j,:); end end Q=Q'; (1) >> A=rand(6) A = 0.1509 0.4966 0.3420 0.8385 0.6946 0.1730 0.6979 0.8998 0.2897 0.5681 0.6213 0.9797 0.3784 0.8216 0.3412 0.3704 0.7948 0.2714 0.8600 0.6449 0.5341 0.7027 0.9568 0.2523 0.8537 0.8180 0.7271 0.5466 0.5226 0.8757 0.5936 0.6602 0.3093 0.4449 0.8801 0.7373 >> [Q,R]=givensqr(A) Q = 0.0959 0.5321 0.5057 0.6559 -0.0866 0.1194 0.4437 0.2382 -0.6160 0.1561 -0.5787 -0.0868 0.2406 0.6604 0.0011 -0.4495 0.3920 -0.3877 0.5468 -0.4446 0.0642 0.3670 0.3416 -0.4978 0.5428 -0.1583 0.5366 -0.4534 -0.3748 0.2153 0.3774 0.0355 -0.2697 0.0565 0.4966 0.7306 R = 1.5728 1.6903 1.0469 1.2704 1.6726 1.4082 0.0000 0.6284 0.1347 0.4430 0.5656 0.2801 0.0000 -0.0000 0.3359 0.2929 0.0738 -0.2285 0.0000 0.0000 0 0.5074 0.3593 -0.1183 0.0000 0.0000 0 0.0000 0.4600 -0.3515 0.0000 -0.0000 0 -0.0000 0 0.4319 >> Q*R-A ans = 1.0e-015 * -0.0555 0 0.0555 0.4441 0.1110 -0.1943 -0.1110 -0.2220 -0.1665 -0.1110 -0.6661 0.1110 -0.1665 -0.3331 -0.1110 -0.3331 -0.4441 -0.3331 -0.1110 -0.1110 0 -0.2220 -0.3331 -0.1110
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-0.1110 0.1110 0.1110 0.1110 0 -0.1110 -0.1110 0 0 0 -0.1110 0 >> Q'*Q ans = 1.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 1.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 1.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 1.0000 0 -0.0000 -0.0000 0.0000 0.0000 0 1.0000 We can see that Q*R is in fact equal to A. We know that Q is orthogonal
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Project 4 Solutions - Project 4 Group name: Unknowns Group...

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