The Lanczos process constructs a sequence of orthonormal vectors vm spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary Lanczos process, starting at an arbitrary Lanczos vector vm and how to use this secondary process to efficiently obtain computable error estimates and error bounds...

No part of the Journal may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from Summary Shifted linear systems are of the form (A − σ k I)x k = b, (1) where A ∈ C N ×N , b ∈ C N and {σ k } Nσ k=1 ∈ C is a sequence of numbers, called shifts. In order to so...

The numerical solution of a time-dependent PDE generally involves the solution of a stiff system of ODEs arising from spatial discretization of the PDE. There are many methods in the literature for solving such systems, such as exponential propagation iterative (EPI) methods, that rely on Krylov projection to compute matrix function-vector products. Unfortunately, as spatial resolution increase...

A multilinear approach based on Grassmann representatives and matrix compounds is presented for the identification of reducing pairs of subspaces that are common to two or more matrices. Similar methods are employed to characterize the deflating pairs of subspaces for a regular matrix pencil A+ sB, namely, pairs of subspaces (L,M) such that AL ⊆ M and BL ⊆ M.

Algebraic solvers based on preconditioned Krylov subspace methods are among the most powerful tools for large scale numerical computations in applied mathematics, sciences, technology, as well as in emerging applications in social sciences. The study of mathematical properties of Krylov subspace methods, in both the cases of exact and inexact computations, is a very active area of research and ...

The Automated Multilevel Substructing method (AMLS ) was recently presented as an alternative to well-established methods for computing eigenvalues of large matrices in the context of structural engineering. This technique is based on exploiting a high level of dimensional reduction via domain decomposition and projection methods. This paper takes a purely algebraic look at the method and expla...

There are classes of linear problems for which the matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. In recent years different authors have investigated the use of, what is called, relaxation strategies for various Krylov subspace methods. These relaxation strategies aim to minimize the amount of work tha...

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that employing subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded H2 or H∞ errors due to the mismatch of the polynomial parts of the full and reducedorder transfer functions. ...

Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, many structural properties of the reduced matrices these are not fully understood. We advance this analysis by deriving bounds on entries rational and their functions, that ensure an a-priori decay as we move away from main diagonal. As opposed to...

We consider the following constrained Rayleigh quotient optimization problem (CRQopt) $$ \min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\, x^{T}x=1\,\mbox{and}\,C^{T}x=b, where $A$ is an $n\times n$ real symmetric matrix and $C$ m$ matrix. Usually, $m\ll n$. The also known as eigenvalue in literature because it becomes if linear constraint $C^{T}x=b$ removed. start by equivalently tra...