Induced Dimension Reduction Method for Solving Linear Matrix Equations

Journal article (2016)

Authors

R.A. Astudillo Rengifo Numerical Analysis - EEMCS
M.B. van Gijzen Numerical Analysis - EEMCS
Research Group
Numerical Analysis (EEMCS) (TU Delft)
Copyright: © 2016 R.A. Astudillo Rengifo, M.B. van Gijzen
DOI
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https://doi.org/10.1016/j.procs.2016.05.313
Krylov subspace methods Preconditioner Matrix linear equations Dimension Reduction method Numerical software
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Published Date

2016

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Numerical Analysis

Abstract

This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization allows us to develop IDR(s) algorithms to approximate the solution of linear matrix equations. The IDR(s) method presented here has two main advantages; firstly, it does not require the computation of inverses of any matrix, and secondly, it allows incorporation of preconditioners. Additionally, we present a simple preconditioner to solve the Sylvester equation based on a fixed point iteration. Several numerical examples illustrate the performance of IDR(s) for solving linear matrix equations. We also present the software implementation.