# Questions tagged [krylov-method]

Referring to Krylov Subspaces and the methods of solutions to linear systems of equations which exploit these spaces.

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### Does incomplete LU preconditioning improve the asymptotic scaling of Krylov subspace methods?

It is well known that unpreconditioned Krylov subspace methods applied to the finite-difference-discretised Poisson equation with $n$ grid points per direction require $O(n \, |\log(\varepsilon)|)$ ...
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### Does LAPACK offer routines for Krylov sub-space based solvers and nonlinear solvers?

I have skimmed through the LAPACK user guide, but I could not find if LAPACK offers routines for Krylov Subspace based methods (such as CG or BiCGSTAB etc) and Newton method based nonlinear solvers. ...
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### What are the major differences between GMRES and FOM?

I am reading Professor Saad's "Iterative Methods for Sparse Linear Systems" (2nd edition). The basic algorithm for FOM is given on page 166 and the basic algorithm for GMRES is given on page 172. ...
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### How to find a good preconditioner to the system $(A^T A + \lambda I) x = A^T b$?

The system in the title has a damper factor $\lambda > 0$ and the matrix $A$ is sparse and rectangular, with a structure I can exploit to solve matrix vector products very fast. My current solver, ...
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I am reading Chapter 5 of Saad's iterative methods book, and I don't understand section 5.2.1 about the two propositions of optimality results. In the statements of the propositions, what does it mean ...
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### Is reduced stiffness matrix positive definite too?

The stiffness matrix $K$ in a finite element analysis is a symmetric positive definite matrix. When we introduce essential boundary conditions, we remove rows and columns associated with prescribed ...
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### Convergence/stagnation of BiCGStab(l)

I am solving 3D time-harmonic Maxwell FDFD problems (which result in huge sparse linear systems) using BiCGStab(l). I have tried out a bunch of different methods and for my specific use case, it seems ...
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### Are there any libraries out there that implement block Krylov subspace methods?

Question Are there libraries out there that implement block Krylov subspace methods? (I was not able to find any from a simple Google search.) Background Right now, I am working with a code that ...
My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow): \begin{pmatrix}A & B^T \\ B & C\end{pmatrix} where $C = 0$ is the typical case (I have even ...