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Induced Dimension Reduction Method for Solving Linear Matrix Equations

Journal Article (2016)
Author(s)

Reinaldo Astudillo Rengifo (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Martin van Gijzen (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Research Group
Numerical Analysis
DOI related publication
https://doi.org/10.1016/j.procs.2016.05.313 Final published version
More Info
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Publication Year
2016
Language
English
Research Group
Numerical Analysis
Volume number
80
Pages (from-to)
222-232
Downloads counter
210
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Institutional Repository
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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.