Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

Tony F. Chan; Lisette De Pillis; Henk Van Der Vorst

doi:10.1023/a:1011637511962

Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

Tony F. Chan^*, Lisette De Pillis, Henk Van Der Vorst

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix-vector products per iteration without accessing A^T. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix-vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.

Original language	English (US)
Pages (from-to)	51-66
Number of pages	16
Journal	Numerical Algorithms
Volume	17
Issue number	1-2
DOIs	https://doi.org/10.1023/a:1011637511962
State	Published - 1998
Externally published	Yes

Keywords

Bi-conjugate gradients algorithm
Krylov subspace methods
Lanczos algorithm
Nonsymmetric linear systems
Quasi-minimal residual algorithm

ASJC Scopus subject areas

Applied Mathematics

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10.1023/a:1011637511962

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@article{0293173343d54a609ac8be9b42ddf65b,

title = "Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems",

abstract = "We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix-vector products per iteration without accessing AT. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix-vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.",

keywords = "Bi-conjugate gradients algorithm, Krylov subspace methods, Lanczos algorithm, Nonsymmetric linear systems, Quasi-minimal residual algorithm",

author = "Chan, {Tony F.} and {De Pillis}, Lisette and {Van Der Vorst}, Henk",

year = "1998",

doi = "10.1023/a:1011637511962",

language = "English (US)",

volume = "17",

pages = "51--66",

journal = "Numerical Algorithms",

issn = "1017-1398",

publisher = "Baltzer Science Publishers, Baarn/Kluwer Academic Publishers",

number = "1-2",

}

TY - JOUR

T1 - Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

AU - Chan, Tony F.

AU - De Pillis, Lisette

AU - Van Der Vorst, Henk

PY - 1998

Y1 - 1998

N2 - We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix-vector products per iteration without accessing AT. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix-vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.

AB - We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix-vector products per iteration without accessing AT. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix-vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.

KW - Bi-conjugate gradients algorithm

KW - Krylov subspace methods

KW - Lanczos algorithm

KW - Nonsymmetric linear systems

KW - Quasi-minimal residual algorithm

UR - http://www.scopus.com/inward/record.url?scp=0032382996&partnerID=8YFLogxK

U2 - 10.1023/a:1011637511962

DO - 10.1023/a:1011637511962

M3 - Article

AN - SCOPUS:0032382996

SN - 1017-1398

VL - 17

SP - 51

EP - 66

JO - Numerical Algorithms

JF - Numerical Algorithms

IS - 1-2

ER -

Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

Abstract

Keywords

ASJC Scopus subject areas

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