Particle Swarm Optimization to solve matrix inversion

Hi everyone,

I am Huda, and I am working with matrix inversion. I am really new with PSO and I would like to know is it possible to use PSO for a large size matrix inversion (eg., 16x16, or 64x64)? From a linear equation, Ax=b, we can find x by several techniques such as Gauss-Jordan elimination, and we know that AA^(-1) =I. Thus, I wonder if PSO can solve this problem by computing A^(-1) iteratively.

If yes, can you share with me any material that can guide me to start my work. But, If matrix inversion with PSO will give bad result, can you advise any technique that suitable to use?

Thank you for your response.

[John D'Errico]

Use of a partical swarm optimizer to solve for the inverse of even a moderately large matrix is a flat out incredibly poor idea.

Since there are quite accurate means to solve whatever problem it is that you really want to solve using linear algebra, use those methods.

Is the reason you wish to do this because the matrix is singular? If so, there is nothing PSO can do for you. So is there a valid reason why you think there is some better way to do linear algebra, than to just use linear algebra?

[John D'Errico]

Please don't use an answer to make acomment. Moved from an answer:

"My goal is to find the inversion of a nonsingular matrix. Why I am interested to apply PSO for matrix inversion, is because I found that PSO is kind of an iterative method to get the best solution. I do use a few iterative techniques to invert a nonsingular matrix in order to reduce the computational complexity. I wonder, if I can further reduce the complexity if I can apply PSO to invert the matrix."

[John D'Errico]

Again, no. Just because PSO can be used for optimizations, does not mean it would be good to solve a linear algebra problem. That is something that has been implemented in a very computationally efficient way. Use of PSO for that would in fact be not a reduction in computation complexity, but a MASSIVE increase in that complexity. Just because some tool CAN theoretically be used to solve some problem does not mean it would be even remotely a good tool for that purpose. PSO would be an obscenely bad way to solve a linear algebra problem, compared to the existing tools we have available.

[Nurulhuda Ismail]

Ok then, thank you for your comment.

[Nurulhuda Ismail]

Instead of PSO, is there any other optimizer that suitable for matrix inversion? Your idea is really appreciated.Thank you.

Answer 1

[Walter Roberson]

N = 8;
rng(12345);
M = randi([-50 50], N, N);
obj = @(v) sum(sum((M * reshape(v,N,N) - eye(N)).^2));

N2 = N.^2;
LB = []; UB = []; 
options = optimoptions(@particleswarm, 'Display', 'final', 'MaxIterations', 500*N2, 'PlotFcn', @pswplotbestf );

tic
[bestinv, fval] = particleswarm(obj, N2, LB, UB, options);
time_needed_ps = toc;

Minvps = reshape(bestinv, N, N);

tic
Minv = inv(M);
time_needed_numeric = toc;

residue = sum((Minvps(:) - Minv(:)).^2);
fprintf('\nResidue between particle swarm inverse and numeric inverse is: %.6g\n', residue);
fprintf('\nDifference between particle swarm inverse and numeric inverse is: \n\n')
disp(Minvps - Minv)

fprintf('\ntime required for particle swarm partial inverse: %.6gs\n', time_needed_ps);
fprintf('time required for numeric inverse:                %.6gs\n', time_needed_numeric);

Approximate time multiplication for the particle swarm approach: 620000 times more expensive. The calculated inverse will not necessarily be especially close -- it is not bad for 5 x 5, but by 8 x 8 the residue is still about a million.

[Walter Roberson]

For N = 16, the above code takes about 1668 seconds on my system before it runs out of iterations with a residue above 2E6, and clocks in close to 70000 times slower than a simple inv()

[Nurulhuda Ismail]

Thank you. I got an error while I run in MATLAB which is because I dont install the Global Optimizaion Toolbox. I will re-run the codes once I install this toolbox.

Regarding to a matrix inversion, instead of PSO, do you know any other optimizer that I can be used to do matrix inversion?

[Walter Roberson]

On my system inv() of a 64x64 array takes less than 1e-4 seconds. Any optimization routine would have to finish faster than that to be worth bothering with.