Jacobi Method in calculating solution, number of iterations, and relative residue

Question

Zhukun Wang 2021-3-16

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/774542-jacobi-method-in-calculating-solution-number-of-iterations-and-relative-residue

编辑： John D'Errico 2021-3-16

N=50; % 50;

A=gallery('poisson',N);

n=size(A,1);

xs=ones(n,1);

b=A*xs(:);

MaxIter=10*n;

TOL=1.e-4;

t=cputime;

[x,info,relres]=myJacobi(A,b,MaxIter,TOL,n); % Jacobi example

mycput = cputime - t;

relerr = norm(xs-x)/norm(xs);

fprintf('size(A,1) = %d, relerr = %8.5e\n',n, relerr);

fprintf('time = %8.5e\n', mycput);

function [x,info,relres] = myJacobi(A,b,MaxIter,TOL,n)

%Jacobi Method

x=zeros(n,1);

k=1;

relres = 1;

info=0;

while k<=MaxIter && relres>TOL

for i=1:n

x(i)=(b(i)-A(i,[1:i-1,i+1:n])*x([1:i-1,i+1:n]))/A(i,i);

end

k=k+1;

info=info+1;

relres = norm(A*x-b)/norm(b);

end

disp(k)

end

This is my code of Jacobi method for calculating a certain system. My issue is that I think the program runs very slow and how would we solve this issue. Note that I want to calculate three things which are "x","info" and "relres". "x" is the solution, "relres" is the reltive residue and "info" records total number of iterations when the algorithm doesn't fail. Could someone help with this? Thanks a lot.

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Answer 1

John D'Errico 2021-3-16

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https://ww2.mathworks.cn/matlabcentral/answers/774542-jacobi-method-in-calculating-solution-number-of-iterations-and-relative-residue#answer_649352

编辑：John D'Errico 2021-3-16

在 MATLAB Online 中打开

If you think your code runs slowly, an important thing you want to do is look at the profiler tools in MATLAB.

help profile

Does it find a bottleneck in your code? In fact, I don't see anything that would be notably poor in your code. Could you have written it "better"? Arguably so, but your code does not appear poor in my eyes.

Perhaps as importantly is to understand that you DON'T want to use this method in general.

https://en.wikipedia.org/wiki/Jacobi_method

In the global spectrum of methods, it is rarely a good choice of methods. Yes, it is a good method to teach students, because they need to learn to program to solve other problems. In some cases, I've seen similar methods employed even for nonlinear problems that were known to be well-behaved.

But to ask why it runs slowly, perhaps better is to understand WHY and WHEN the Jacobi method converges, when it will fail to converge, and when it will converge rapidly. On what class of problems will it be rapidly convergent? For this, you should do some reading. Or perhaps it is something your instructor would like to explain to you. (Since yours is not a question about MATLAB at all.) After all, it is likely this is one reason you were asked to code this method.

https://en.wikipedia.org/wiki/Spectral_radius