On Inexact Preconditioners for Nonsymmetric Matrices

Zhong Zhi Bai; Michael K. Ng

doi:10.1137/040604091

On Inexact Preconditioners for Nonsymmetric Matrices

Zhong Zhi Bai^*
, Michael K. Ng

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

114 Citations (Scopus)

Abstract

Inexact versions of the block-triangular preconditioned for nonsymmetric matrices of block two-by-two structures in [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput, 21 (2000), pp. 1969-1972] and [I. C. F. Ipsen, SIAM J. Sci. Comput, 23 (2001), pp. 1050-1051] are presented, and the two preconditioners for symmetric block two-by-two matrices in [C. Durazzi and V. Ruggiero, Numer. Linear Algebra Appl., 10 (2003), pp. 673-688] are extended to general nonsymmetric matrices. Moreover, we precisely describe the spectral properties of the preconditioned matrices and the finite-step termination properties of the preconditioned Krylov subspace iteration methods with an optimal or Galerkin property, with respect to these preconditioners. Several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners.

Original language	English
Pages (from-to)	1710-1724
Number of pages	15
Journal	SIAM Journal on Scientific Computing
Volume	26
Issue number	5
DOIs	https://doi.org/10.1137/040604091
Publication status	Published - Jan 2005

User-Defined Keywords

Block two-by-two matrices
Minimal polynomial
Preconditioners

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10.1137/040604091

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title = "On Inexact Preconditioners for Nonsymmetric Matrices",

abstract = "Inexact versions of the block-triangular preconditioned for nonsymmetric matrices of block two-by-two structures in [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput, 21 (2000), pp. 1969-1972] and [I. C. F. Ipsen, SIAM J. Sci. Comput, 23 (2001), pp. 1050-1051] are presented, and the two preconditioners for symmetric block two-by-two matrices in [C. Durazzi and V. Ruggiero, Numer. Linear Algebra Appl., 10 (2003), pp. 673-688] are extended to general nonsymmetric matrices. Moreover, we precisely describe the spectral properties of the preconditioned matrices and the finite-step termination properties of the preconditioned Krylov subspace iteration methods with an optimal or Galerkin property, with respect to these preconditioners. Several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners.",

keywords = "Block two-by-two matrices, Minimal polynomial, Preconditioners",

author = "Bai, \{Zhong Zhi\} and Ng, \{Michael K.\}",

note = "Funding Information: This author{\textquoteright}s research was subsidized by The Special Funds for Major State Basic Research Projects G1999032803. This author{\textquoteright}s research was supported in part by RGC grants 7130/02P, 7046/03P, and 7035/04P. Publisher copyright: {\textcopyright} 2005 Society for Industrial and Applied Mathematics",

year = "2005",

month = jan,

doi = "10.1137/040604091",

language = "English",

volume = "26",

pages = "1710--1724",

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AU - Bai, Zhong Zhi

AU - Ng, Michael K.

N1 - Funding Information: This author’s research was subsidized by The Special Funds for Major State Basic Research Projects G1999032803. This author’s research was supported in part by RGC grants 7130/02P, 7046/03P, and 7035/04P. Publisher copyright: © 2005 Society for Industrial and Applied Mathematics

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N2 - Inexact versions of the block-triangular preconditioned for nonsymmetric matrices of block two-by-two structures in [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput, 21 (2000), pp. 1969-1972] and [I. C. F. Ipsen, SIAM J. Sci. Comput, 23 (2001), pp. 1050-1051] are presented, and the two preconditioners for symmetric block two-by-two matrices in [C. Durazzi and V. Ruggiero, Numer. Linear Algebra Appl., 10 (2003), pp. 673-688] are extended to general nonsymmetric matrices. Moreover, we precisely describe the spectral properties of the preconditioned matrices and the finite-step termination properties of the preconditioned Krylov subspace iteration methods with an optimal or Galerkin property, with respect to these preconditioners. Several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners.

AB - Inexact versions of the block-triangular preconditioned for nonsymmetric matrices of block two-by-two structures in [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput, 21 (2000), pp. 1969-1972] and [I. C. F. Ipsen, SIAM J. Sci. Comput, 23 (2001), pp. 1050-1051] are presented, and the two preconditioners for symmetric block two-by-two matrices in [C. Durazzi and V. Ruggiero, Numer. Linear Algebra Appl., 10 (2003), pp. 673-688] are extended to general nonsymmetric matrices. Moreover, we precisely describe the spectral properties of the preconditioned matrices and the finite-step termination properties of the preconditioned Krylov subspace iteration methods with an optimal or Galerkin property, with respect to these preconditioners. Several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners.

KW - Block two-by-two matrices

KW - Minimal polynomial

KW - Preconditioners

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U2 - 10.1137/040604091

DO - 10.1137/040604091

M3 - Journal article

AN - SCOPUS:25444458272

SN - 1064-8275

VL - 26

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EP - 1724

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 5

ER -

On Inexact Preconditioners for Nonsymmetric Matrices

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