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What are the major differences between GMRES and FOM?

There is one major difference between GMRES over FOM. It is also the reason why I would recommend GMRES over FOM. In exact arithmetic, the residuals obtained by GMRES form a decreasing sequence. You ...
• 1,391
Accepted

When do not use preconditioners for sparse linear system of equations?

In my experience, you always need (or better use) some form of preconditioning. The type and complexity of the precondition would vary depending on the task though. From Y. Saad, Iterative Methods for ...
• 8,652
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• 1,340
Accepted

Preconditioner for dense matrix "with diagonal predominance"

In case it may be of help to others, here's the result of the testing on a 30x30 influence matrix.It was made using non-restarted GMRES. GMRES without preconditioner (red curve) converges in 13 ...
Accepted

• 1,558

Doubt regarding GMRES(m) and preconditioned GMRES

It's just a difference in how the authors decided to write the algorithm. Left-preconditioned GMRES is the same as regular GMRES where $A$ is replaced with $M^{-1}A$ and $b$ is replaced with $M^{-1}b$....
1 vote

relres in gmres MATLAB

The advantage of that definition is that computing relres comes "for free" from the GMRES iteration. You could switch to the other definition without the preconditioner, but then you'd have ...
• 11.3k
1 vote

How GMRES method finds smallest singular value and the corresponding singular vectors of a matrix?

I kinda see what you are asking. There is a relationship between singular values of a matrix $A$ and the matrix $A^HA$: $\sigma_i^2(A)=\lambda_i(A^HA)$. So in theory, you can use inverse power method ...
• 2,476
1 vote

Library to solve dense linear system with GMRES

PETSC might be a good option. Not super user friendly, but its good with lots of options.
• 2,079
1 vote

How does gmres method iteration behave for this non-diagonalizable matrix?

First some comments on why such matrices are hard for solvers. Notice that if $U$ is upper diagonal like ...
• 932
1 vote

Number of GMRES iterations increase when stepping forward in time, using the Newton method

It looks to me like the most helpful thing you can do is increase the number of krylov vectors you are using. This is somewhat different from the typical behavior I've seen where the first step takes ...
• 2,079
1 vote

How to set an initial guess for the iterative solver in Comsol?

I am not too familiar with the details of the iterative solve in Comsol; however, quick googling did reveal any easy way to set up a custom initial guess for an iterative linear solver. (I ready to be ...
• 8,652
1 vote

GMRES : incomplete Krylov-subspace

If $A$ is invertible and the Krylov subspace $v,Av,A^2v,\ldots$ stops to expand after $m$ steps, then GMRES (and other reasonable Krylov methods) converge to the exact solution in at most $m$ steps (...
• 161

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