Ax=b
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(Difference between revisions)
(New page: =Seven ways Matlab can solve thin-matrix linear equality= When <math>b\notin\mathcal{R}(A)</math> <pre> %test backslash timing clc %clear all; close all; fclose all; slow execution by o...) |
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| Line 6: | Line 6: | ||
%test backslash timing | %test backslash timing | ||
clc %clear all; close all; fclose all; slow execution by order of magnitude | clc %clear all; close all; fclose all; slow execution by order of magnitude | ||
| - | A = randn( | + | A = randn(1e6,52); |
% spA = sparse(A); | % spA = sparse(A); | ||
xact = randn(52,1); | xact = randn(52,1); | ||
| Line 14: | Line 14: | ||
opt1.LT = true; opt2.UT = true; | opt1.LT = true; opt2.UT = true; | ||
| + | %form A'*A | ||
[m n] = size(A); | [m n] = size(A); | ||
AA = zeros(n,n); | AA = zeros(n,n); | ||
| Line 104: | Line 105: | ||
disp(['error = ' num2str(norm(x7a - xact)/norm(xact))]) | disp(['error = ' num2str(norm(x7a - xact)/norm(xact))]) | ||
end | end | ||
| - | |||
</pre> | </pre> | ||
| + | |||
| + | [[http://www.convexoptimization.com/wikimization/index.php/Accumulator_Error_Feedback|<tt>csum()</tt> routine]] | ||
| + | (with presorting) increases precision by orders of magnitude. | ||
Revision as of 15:29, 17 December 2017
Seven ways Matlab can solve thin-matrix linear equality
When
%test backslash timing
clc %clear all; close all; fclose all; slow execution by order of magnitude
A = randn(1e6,52);
% spA = sparse(A);
xact = randn(52,1);
b = A*xact;
opt.SYM = true; opt.POSDEF = true;
opt1.LT = true; opt2.UT = true;
%form A'*A
[m n] = size(A);
AA = zeros(n,n);
bb = zeros(n,1);
for i=1:n
for j=1:i
AA(i,j) = csum(A(:,i).*A(:,j));
AA(j,i) = AA(i,j);
end
bb(i) = csum(A(:,i).*b);
end
spAA = sparse(AA);
for i=1:4
if i > 1, disp(' ');disp(' '); end
% disp(' backslash on A')
% tic
% x1 = A\b;
% toc
% disp(['error = ' num2str(norm(x1 - xact)/norm(xact))])
disp(' backslash on A''A')
tic
x1a = AA\bb;
toc
disp(['error = ' num2str(norm(x1a - xact)/norm(xact))])
% disp(' pinv on A')
% tic
% x2 = pinv(A)*b;
% toc
% disp(['error = ' num2str(norm(x2 - xact)/norm(xact))])
disp(' pinv on A''A')
tic
x2a = pinv(AA)*bb;
toc
disp(['error = ' num2str(norm(x2a - xact)/norm(xact))])
% disp(' QR on A')
% tic
% [c,R] = qr(spA,b,0);
% x3 = R\c;
% toc
% disp(['error = ' num2str(norm(x3 - xact)/norm(xact))])
disp(' QR on A''A')
tic
[c,R] = qr(spAA,bb,0);
x3a = R\c;
toc
disp(['error = ' num2str(norm(x3a - xact)/norm(xact))])
disp(' U\(U''\A''*b) on U=chol(A''A)')
tic;
U = chol(AA);
x4 = U\(U'\bb);
toc
disp(['error = ' num2str(norm(x4 - xact)/norm(xact))])
disp(' linsolve on chol(A''A)') %winner on Dec.14 2017
tic
U = chol(AA);
x5 = linsolve(U, linsolve(U',bb,opt1), opt2);
toc
disp(['error = ' num2str(norm(x5 - xact)/norm(xact))])
% disp(' linsolve on A')
% tic
% x6 = linsolve(A,b);
% toc
% disp(['error = ' num2str(norm(x6 - xact)/norm(xact))])
disp(' linsolve on A''A')
tic
x6a = linsolve(AA,bb,opt);
toc
disp(['error = ' num2str(norm(x6a - xact)/norm(xact))])
% disp(' lscov on A')
% tic
% x7 = lscov(A,b);
% toc
% disp(['error = ' num2str(norm(x7 - xact)/norm(xact))])
disp(' lscov on A''A')
tic
x7a = lscov(AA,bb);
toc
disp(['error = ' num2str(norm(x7a - xact)/norm(xact))])
end
[csum() routine] (with presorting) increases precision by orders of magnitude.