Ax=b

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(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...)
Current revision (20:46, 20 December 2017) (edit) (undo)
 
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=Seven ways Matlab can solve thin-matrix linear equality=
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==Seven ways Matlab can optimally solve normal equations given thin matrix <math>A</math>==
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For <math>b\notin\mathcal{R}(A)\,,\,</math> find best fit <math>x</math> by least squares such that <math>A_{}x\approx b\,</math>:
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When <math>b\notin\mathcal{R}(A)</math>
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<pre>
<pre>
%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
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A = randn(600*96000,52);
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m=1e6; n=52;
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% spA = sparse(A);
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A = randn(m,n);
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xact = randn(52,1);
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xact = randn(n,1);
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b = A*xact;
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b = A*xact;
opt.SYM = true; opt.POSDEF = true;
opt.SYM = true; opt.POSDEF = true;
opt1.LT = true; opt2.UT = true;
opt1.LT = true; opt2.UT = true;
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[m n] = size(A);
 
AA = zeros(n,n);
AA = zeros(n,n);
bb = zeros(n,1);
bb = zeros(n,1);
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%form A'A for accuracy, precision, and symmetry
for i=1:n
for i=1:n
for j=1:i
for j=1:i
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AA(j,i) = AA(i,j);
AA(j,i) = AA(i,j);
end
end
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bb(i) = csum(A(:,i).*b);
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bb(i) = csum(A(:,i).*b);
end
end
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% AA = A'*A;
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% AA = (AA + AA')/2;
spAA = sparse(AA);
spAA = sparse(AA);
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for i=1:4
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% spA = sparse(A);
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for i=1:3 %Manually disable unnecessary OS processes. Runs faster on subsequent loops.
if i > 1, disp(' ');disp(' '); end
if i > 1, disp(' ');disp(' '); end
% disp(' backslash on A')
% disp(' backslash on A')
% tic
% tic
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% x1 = A\b;
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% x1a = A\b;
% toc
% toc
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% disp(['error = ' num2str(norm(x1 - xact)/norm(xact))])
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% disp(['error = ' num2str(norm(x1a - xact)/norm(xact))])
disp(' backslash on A''A')
disp(' backslash on A''A')
tic
tic
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x1a = AA\bb;
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x1 = AA\bb;
toc
toc
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disp(['error = ' num2str(norm(x1a - xact)/norm(xact))])
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disp(['error = ' num2str(norm(x1 - xact)/norm(xact))])
% disp(' pinv on A')
% disp(' pinv on A')
% tic
% tic
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% x2 = pinv(A)*b;
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% x2a = pinv(A)*b;
% toc
% toc
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% disp(['error = ' num2str(norm(x2 - xact)/norm(xact))])
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% disp(['error = ' num2str(norm(x2a - xact)/norm(xact))])
disp(' pinv on A''A')
disp(' pinv on A''A')
tic
tic
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x2a = pinv(AA)*bb;
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x2 = pinv(AA)*bb;
toc
toc
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disp(['error = ' num2str(norm(x2a - xact)/norm(xact))])
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disp(['error = ' num2str(norm(x2 - xact)/norm(xact))])
% disp(' QR on A')
% disp(' QR on A')
% tic
% tic
% [c,R] = qr(spA,b,0);
% [c,R] = qr(spA,b,0);
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% x3 = R\c;
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% x3a = R\c;
% toc
% toc
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% disp(['error = ' num2str(norm(x3 - xact)/norm(xact))])
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% disp(['error = ' num2str(norm(x3a - xact)/norm(xact))])
disp(' QR on A''A')
disp(' QR on A''A')
tic
tic
[c,R] = qr(spAA,bb,0);
[c,R] = qr(spAA,bb,0);
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x3a = R\c;
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x3 = R\c;
toc
toc
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disp(['error = ' num2str(norm(x3a - xact)/norm(xact))])
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disp(['error = ' num2str(norm(x3 - xact)/norm(xact))])
disp(' U\(U''\A''*b) on U=chol(A''A)')
disp(' U\(U''\A''*b) on U=chol(A''A)')
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% disp(' linsolve on A')
% disp(' linsolve on A')
% tic
% tic
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% x6 = linsolve(A,b);
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% x6a = linsolve(A,b);
% toc
% toc
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% disp(['error = ' num2str(norm(x6 - xact)/norm(xact))])
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% disp(['error = ' num2str(norm(x6a - xact)/norm(xact))])
disp(' linsolve on A''A')
disp(' linsolve on A''A')
tic
tic
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x6a = linsolve(AA,bb,opt);
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x6 = linsolve(AA,bb,opt);
toc
toc
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disp(['error = ' num2str(norm(x6a - xact)/norm(xact))])
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disp(['error = ' num2str(norm(x6 - xact)/norm(xact))])
% disp(' lscov on A')
% disp(' lscov on A')
% tic
% tic
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% x7 = lscov(A,b);
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% x7a = lscov(A,b);
% toc
% toc
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% disp(['error = ' num2str(norm(x7 - xact)/norm(xact))])
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% disp(['error = ' num2str(norm(x7a - xact)/norm(xact))])
disp(' lscov on A''A')
disp(' lscov on A''A')
tic
tic
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x7a = lscov(AA,bb);
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x7 = lscov(AA,bb);
toc
toc
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disp(['error = ' num2str(norm(x7a - xact)/norm(xact))])
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disp(['error = ' num2str(norm(x7 - xact)/norm(xact))])
end
end
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</pre>
</pre>
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[[Accumulator_Error_Feedback | <tt>csum()</tt> routine]]
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(with presorting) can increase precision by orders of magnitude.

Current revision

Seven ways Matlab can optimally solve normal equations given thin matrix LaTeX: A

For LaTeX: b\notin\mathcal{R}(A)\,,\, find best fit LaTeX: x by least squares such that LaTeX: A_{}x\approx b\,:

%test backslash timing
clc  %clear all; close all; fclose all; slow execution by order of magnitude
m=1e6;  n=52;
A    = randn(m,n);
xact = randn(n,1); 
b    = A*xact;

opt.SYM = true;  opt.POSDEF = true;
opt1.LT = true;  opt2.UT = true;

AA = zeros(n,n);
bb = zeros(n,1);
%form A'A for accuracy, precision, and symmetry
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
% AA = A'*A;
% AA = (AA + AA')/2;
spAA = sparse(AA);
% spA = sparse(A);
for i=1:3  %Manually disable unnecessary OS processes. Runs faster on subsequent loops.
   if i > 1, disp(' ');disp(' '); end

%    disp('   backslash on A')
%    tic
%    x1a = A\b;
%    toc
%    disp(['error = ' num2str(norm(x1a - xact)/norm(xact))])

   disp('   backslash on A''A')
   tic
   x1 = AA\bb;
   toc
   disp(['error = ' num2str(norm(x1 - xact)/norm(xact))])

%    disp('   pinv on A')
%    tic
%    x2a = pinv(A)*b;
%    toc
%    disp(['error = ' num2str(norm(x2a - xact)/norm(xact))])

   disp('   pinv on A''A')
   tic
   x2 = pinv(AA)*bb;
   toc
   disp(['error = ' num2str(norm(x2 - xact)/norm(xact))])

%    disp('   QR on A')
%    tic
%    [c,R] = qr(spA,b,0);  
%    x3a = R\c;
%    toc
%    disp(['error = ' num2str(norm(x3a - xact)/norm(xact))])

   disp('   QR on A''A')
   tic
   [c,R] = qr(spAA,bb,0);  
   x3 = R\c;
   toc
   disp(['error = ' num2str(norm(x3 - 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
%    x6a = linsolve(A,b);
%    toc
%    disp(['error = ' num2str(norm(x6a - xact)/norm(xact))])

   disp('   linsolve on A''A')
   tic
   x6 = linsolve(AA,bb,opt);
   toc
   disp(['error = ' num2str(norm(x6 - xact)/norm(xact))])

%    disp('   lscov on A')
%    tic
%    x7a = lscov(A,b);
%    toc
%    disp(['error = ' num2str(norm(x7a - xact)/norm(xact))])

   disp('   lscov on A''A')
   tic
   x7 = lscov(AA,bb);
   toc
   disp(['error = ' num2str(norm(x7 - xact)/norm(xact))])
end

csum() routine (with presorting) can increase precision by orders of magnitude.

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