Open AccessInduced Dimension Reduction Method for Solving Linear Matrix Equations
Author(s) -
Reinaldo Astudillo,
Martin B. van Gijzen
Publication year - 2016
Publication title -
procedia computer science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.334
H-Index - 76
ISSN - 1877-0509
DOI - 10.1016/j.procs.2016.05.313
Subject(s) - preconditioner , coefficient matrix , computer science , linear equation , dimension (graph theory) , matrix (chemical analysis) , reduction (mathematics) , mathematics , system of linear equations , linear system , dimensionality reduction , generalization , mathematical optimization , iterative method , algorithm , eigenvalues and eigenvectors , mathematical analysis , artificial intelligence , pure mathematics , geometry , quantum mechanics , physics , materials science , composite material
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
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