Induced Dimension Reduction Method for Solving Linear Matrix Equations

Reinaldo Astudillo Rengifo, Martin van Gijzen

Research output: Contribution to journalConference articleScientificpeer-review

4 Citations (Scopus)
25 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)222-232
Number of pages11
JournalProcedia Computer Science
Publication statusPublished - 2016


  • Matrix linear equations
  • Krylov subspace methods
  • Dimension Reduction method
  • Preconditioner
  • Numerical software


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