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On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling
Author(s) -
Freund Roland W.,
Hochbruck Marlis
Publication year - 1994
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680010406
Subject(s) - transpose , krylov subspace , mathematics , coefficient matrix , invertible matrix , linear system , markov chain , algorithm , hermitian matrix , convergence (economics) , matrix (chemical analysis) , iterative method , eigenvalues and eigenvectors , mathematical analysis , pure mathematics , quantum mechanics , physics , statistics , materials science , economics , composite material , economic growth
Recently, Freund and Nachtigal proposed the quasi‐minimal residual algorithm (QMR) for solving general nonsingular non‐Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix—vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose‐free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov‐subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.