Iterative Solvers in MATLAB

Hello Experts,

I am trying to Solve Ax=B in MATLAB, where A is square matrix of size ~500,000 and B is the vector of same size. I have solved similar equations in COMSOL with iterative solvers like Conjugate Gradient and Algebraic Multigrid Preconditioners.

I have tried many iterative solvers like cgs, bicgs, bicgstab, minres without any preconditioners in MATLAB. But none of them would converge (1e-6). I did not try any preconditioners because I never understood the way to include them in my solver.

Does anyone know how could I implement CG with Algebraic Multigrid Preconditioner in MATLAB? Any other inputs are also welcomed :)

Sanket

[Pranav Gupta]

Hello Sanket,

It has been a while, but I am curious about your success with the AMG preconditioner. I have sparse non-Hermitian (complex symmetric) matrices (100000x100000) with density ~0.5% and ILU preconditioners, etc. are not working. I was wondering if you have suggestions for me, because AMG needs a Hermitian matrix, which I don't have.

Answer 1

[Laura Proctor]

Try the pcg function in MATLAB.

From the documentation:

The five-point finite difference approximation to Laplace's equation on a square, two-dimensional domain provides an example. The following statements use the preconditioned conjugate gradient method preconditioner M = R'*R, where R is the incomplete Cholesky factor of A.

A = delsq(numgrid('S',50));
b = ones(size(A,1),1);
tol = 1.e-3;
maxit = 10;
R = cholinc(A,tol);
[x,flag,err,iter,res] = pcg(A,b,tol,maxit,R',R);

Only four iterations are required to achieve the prescribed accuracy.

[Sanket]

Matlab is running the cholinc command for now. I will get back to you after it is executed

Answer 2

[Sanket]

I used the following command, [matlab_solution,flag,relres,iter,res]=pcg(stiffness,rhs,[],200000,R',R);

After 1669 iterations, the solver stopped giving flag 3 with relres=14.0242.

The solution obtained is way incorrect. Do you know what could solve it?

One of the conditions in pcg is square matrix must be symmetric and positive definite. How would I verify that?

Also, I tried cond and condest function but I get out of memory errors.

Sanket