Ax=b

From Wikimization

Revision as of 16:28, 17 December 2017 (edit) Ranjelin (Talk | contribs) ← Previous diff		Revision as of 18:24, 17 December 2017 (edit) (undo) Ranjelin (Talk | contribs) Next diff →
Line 1:		Line 1:
-	==Seven ways Matlab can solve thin-matrix linear equality==	+	==Seven ways Matlab can optimally solve normal equations given thin <math>A</math>==
-	For <math>b\notin\mathcal{R}(A)</math>	+	For <math>b\notin\mathcal{R}(A)\,,\,</math> find best fit <math>x</math> such that <math>A_{}x\approx b\,</math>:
	<pre>		<pre>
	%test backslash timing		%test backslash timing

---

Revision as of 18:24, 17 December 2017

Seven ways Matlab can optimally solve normal equations given thin

For find best fit such that :

%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.