A note on Krylov subspace methods for singular systems

Krylov subspace methods for solving linear systems

Inexact Krylov Subspace Methods for Linear Systems

There is a class of linear problems for which the computation of the matrix-vector product is very expensive since a time consuming approximation method is necessary to compute it with some prescribed relative precision. In this paper we investigate the effect of an approximately computed matrix-vector product on the convergence and accuracy of several Krylov subspace solvers. The obtained insi...

A New Approach for Solving Singular Systems in Topology Optimization Using Krylov Subspace Methods

In topology optimization, the design parameter of the element that does not give any contribution to the objective function vanishes. This causes the stiffness matrix to become singular. To avoid the breakdown caused by this singularity, the previous studies employ some additional procedures. These additional procedures, however, have some problems. On the other hand, convergences of Krylov sub...

On Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems

The Biorthogonal Lanczos and the Biconjugate Gradients methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems. Sonneveld [19] obtained the Conjugate Gradient Squared by squaring the matrix polynomials of the Biconjugate Gra­ dients method. Here we square the Biorthogonal Lanczos, the Biconjugate Residual and the Biconjugate Orth...

Subspace Correction Methods for Singular Systems

We investigate the application of minimal residual and orthogonal residualsubspace correction methods to singular linear systems Ax = b. Special em-phasis is put on the special case of Krylov subspace methods. If A has index 1(i.e., if all Jordan blocks associated with the eigenvalue λ = 0 of A are 1× 1)the behaviour of these iterative methods is well understood (see, e.g., ...