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<code> % A matlab program that constructs a matrix via SVD factors and then compares those to the computed SVD m = 50; % dimension of matrices S = 100; % number of tests for s = 1:S % construct A from random SVD factors [U, tmp] = qr(randn(m,m)); [V, tmp] = qr(randn(m,m)); S = diag(sort(rand(m,1).^6, 'descend')); A = U*S*V'; % compute SVD of A [U2,S2,V2] = svd(A); % change signs of cols of U2 if they don't match U, ditto V for k=1:m uk = U(:,k); if uk'*U2(:,k) < 0 U2(:,k) = -1*U2(:,k); V2(:,k) = -1*V2(:,k); end end kappa = cond(A); loglog(kappa, norm(U-U2), 'r.') hold on loglog(kappa, norm(V-V2), 'b.') loglog(kappa, norm(S-S2), 'k.') loglog(kappa, norm(A-U2*S2*V2'), 'g.'); end legend('norm(U-U2)','norm(V-V2)','norm(S-S2)', 'norm(A-U2*S2*V2)''', ... 'Location', 'NorthWest'); legend boxoff xlabel('cond(A)') axis([1 10^20 10^-16 1]) set(gcf, 'color', [1 1 1]) set(gcf,'inverthardcopy','off') title('SVD errors, A has singvals (0<random<1).^6') </code>
