Inaccuracy in solving simultaneous equations using matrix
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So i was solving a system of n linear equations. My coefficient matrix is a tridiagonal one.
However as i was decreasing the values inside matrix or increasing n the error was increasing rapidly.
clear
n=1000;
B=(1:n);
B=B';
A=full(gallery('tridiag',n,0.341,0.232,0.741));
x=A\B;
c=A*x-B;
error=0;
for i=1:n
error=error+abs(c(i,1));
end
error
%error = 2.174626266011847e+155
Here the system is in the form Ax=B
ideally c should contain only zero.
Can anyone a suggest a method so that i can decrease the net error.
NOTE: I also tried the Thomas Algorithm even that gave an error of similar order .
3 个评论
Walter Roberson
2020-6-3
>> rcond(A)
ans =
5.96371748872238e-170
Much too small for reliable numeric solution. rank(A) says 999 rather than 1000.
Switching to symbolic toolbox is able to get full rank and exact solution.
Susmit Kumar Mishra
2020-6-3
Walter Roberson
2020-6-3
编辑：Walter Roberson
2020-6-3
n = 1000;
B = (1:n).';
A = sym( full(gallery('tridiag',n,0.341,0.232,0.741)) ); %11 seconds
x = A\B; %not fast!! 43 seconds
c = A*x-B;
error = sum(abs(c));
disp(error)
采纳的回答
Bjorn Gustavsson
2020-6-3
If you run this code-snippet:
N = round(logspace(1,3,7));
for i1 = 1:numel(N),
n = N(i1);
B=(1:n)';
A=full(gallery('tridiag',n,0.341,0.232,0.741));
[U,S,V] = svd(A);
ph(i1) = plot(diag(S),'.-','linewidth',2,'markersize',15,'color',rand(1,3));
title(n)
drawnow
end
% Pause
set(gca,'xscale','log')
% Pause
set(gca,'yscale','log')
You will see that the smallest eigenvalue is way smaller than the next smallest. Such matrices A are illconditioned and are a bit more problematic to solve. For more information and better tools to handle such problems have a look at regtools.
HTH
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