Jens-Peter M. Zemke

Technische Universität Hamburg | TUHH · Institute of Mathematics

We give two generalizations of the induced dimension reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi-shift quasi-minimal residual IDR variant. These variants are based on a generalized Hessenberg decomposition. We present a new, more stable way to compute basis vectors in IDR. Numerical examples are pre...

The induced dimension reduction (IDR) technique developed by Sonneveld and van Gijzen [1] is a powerful concept resulting in a variety of transpose-free Krylov subspace methods based on short-term recurrences. We present the main differences between and similarities of IDR methods and classical Krylov subspace methods; our tool of trade is the so-c...

We generalize an augmented rounding error result that was proven for the symmetric Lanczos process in [C.C. Paige, An augmented stability result for the Lanczos Hermitian matrix tridiagonalization process, SIAM J. Matrix Anal. Appl. 31 (2010) 2347–2359], to the two-sided (usually unsymmetric) Lanczos process for tridiagonalizing a square matrix. We...

The Induced Dimension Reduction (IDR) method, which has been introduced as a transpose-free Krylov space method for solving nonsymmetric linear systems, can also be used to determine approximate eigenvalues of a matrix or operator. The IDR residual polynomials are the products of a residual polynomial constructed by successively appending linear sm...

We introduce the framework of “abstract perturbed Krylov methods”. This is a new and unifying point of view on Krylov subspace methods based solely on the matrix equation and the assumption that the matrix Ck is unreduced Hessenberg. We give polynomial expressions relating the Ritz vectors, quasi-orthogonal residual iterates and quasi-minimal resid...

We present the framework of “abstract perturbed Krylov methods”, a new, unified point of view on different types of Krylov subspace methods. We give a brief informal sketch of polynomial representations of QOR approximations to solutions of linear systems and eigenvectors. The results are applicable to exact arithmetic, finite precision computation...

Explicit relations between eigenvalues, eigenmatrix entries and matrix elements of unreduced Hessenberg matrices are derived. The main result is based on the Taylor expansion of the adjugate of zI-H on the one hand and inherent properties of Hessenberg matrix structure on the other hand. This result is utilized to construct computable relations bet...

We show under very general assumptions that error bounds for an individual eigenvector of a matrix can be computed if and only if the geometric multiplicity of the corresponding eigenvalue is one. Basically, this is true if not computing exactly like in computer algebra methods. We first show, under general assumptions, that nontrivial error bounds...

In this paper we examine the behaviour of finite precision Krylov methods for the algebraic eigenproblem. Our approach presupposes only some basic knowledge in linear algebra and may serve as a basis for the examination of a wide variety of Krylov methods. The analysis carried out does not yield exact bounds on the accuracy of Ritz values or the sp...

This thesis is concerned with the error analysis of the most common Krylov subspace methods for the solution of the algebraic eigenvalue problem and the solution of linear systems. It contains a new form of error analysis for these methods. The main concern lies in the unification and extension of well-known error analysis approaches. Die Dissertat...