For a given real n n matrix A and initial vectors v 1 and w 1 , we examine the sensitivity of the tridiagonal matrix T and the biorthogonal sets of vectors of the Lanczos reduction to small changes in A, v 1 and w 1. We also consider the sensitivity of the developing Krylov subspaces.

Reliable estimates for the condition number of a large, sparse, real matrix A are important in many applications. To get an approximation for the condition number κ(A), an approximation for the smallest singular value is needed. Standard Krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value. Therefore, we study extended Krylov subspaces which tu...

Several Krylov-type procedures are introduced that generalize matrix Krylov methods for tensor computations. They are denoted minimal Krylov recursion, maximal Krylov recursion, and contracted tensor product Krylov recursion. It is proved that, for a given tensor A with multilinear rank-(p, q, r), the minimal Krylov recursion extracts the correct subspaces associated to the tensor in p+ q+r num...

It will be shown that extended Krylov subspaces –under some assumptions– can be retrieved without any explicit inversion or system solves involved. Instead we do the necessary computations of A−1v in an implicit way using the information from an enlarged standard Krylov subspace. It is well-known that both for classical and extended Krylov spaces, direct unitary similarity transformations exist...

This paper presents an efficient algorithm to solve total variation (TV) regularizations of images contaminated by a both blur and noise. The unconstrained structure of the problem suggests that one can solve a constrained optimization problem by transforming the original unconstrained minimization problem to an equivalent constrained minimization one. An augmented Lagrangian method is develope...

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. G...

We consider a two-directional Krylov subspace Kk(A[j], b[j]), where besides the dimensionality k of the subspace increases, the matrix A[j] and vector b[j] which induce the subspace may also augment. Specifically, we consider the case where the matrix A[j] and the vector b[j] are augmented by block triangular bordering. We present a two-directional Arnoldi process to efficiently generate a sequ...

In this paper we consider the numerical solution of projected generalized continuous-time Lyapunov equations with low-rank right-hand sides. The interest in this problem stems from stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. Two projection methods are proposed for calculating low-rank approximate solutions. One is based ...

We present a general framework for a number of techniques based on projection methods onàugmented Krylov subspaces'. These methods include the deeated GM-RES algorithm, an inner-outer FGMRES iteration algorithm, and the class of block Krylov methods. Augmented Krylov subspace methods often show a signiicant improvement in convergence rate when compared with their standard counterparts using the...