On the similarities between the quasi-Newton inverse least squares method and GMRes

Rob Haelterman; Joris Degroote; Dirk Van Heule; Jan Vierendeels

doi:10.1137/090750354

On the similarities between the quasi-Newton inverse least squares method and GMRes

Rob Haelterman, Joris Degroote, Dirk Van Heule, Jan Vierendeels

University of Ghent

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

Abstract

We show how one of the best-known Krylov subspace methods, the generalized minimal residual method (GMRes), can be interpreted as a quasi-Newton method and how the quasi-Newton inverse least squares method (QN-ILS) relates to Krylov subspace methods in general and to GMRes in particular when applied to linear systems. We also show that we can modify QN-ILS in order to make it analytically equivalent to GMRes, without the need for extra matrix-vector products.

Original language	English
Pages (from-to)	4660-4679
Number of pages	20
Journal	SIAM Journal on Numerical Analysis
Volume	47
Issue number	6
DOIs	https://doi.org/10.1137/090750354
Publication status	Published - 2009

Keywords

Generalized minimal residual method
Iterative method
Least squares
Linear algebra
Quasi-Newton method
Rank-one update
Secant method

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10.1137/090750354

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abstract = "We show how one of the best-known Krylov subspace methods, the generalized minimal residual method (GMRes), can be interpreted as a quasi-Newton method and how the quasi-Newton inverse least squares method (QN-ILS) relates to Krylov subspace methods in general and to GMRes in particular when applied to linear systems. We also show that we can modify QN-ILS in order to make it analytically equivalent to GMRes, without the need for extra matrix-vector products.",

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AU - Degroote, Joris

AU - Van Heule, Dirk

AU - Vierendeels, Jan

PY - 2009

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AB - We show how one of the best-known Krylov subspace methods, the generalized minimal residual method (GMRes), can be interpreted as a quasi-Newton method and how the quasi-Newton inverse least squares method (QN-ILS) relates to Krylov subspace methods in general and to GMRes in particular when applied to linear systems. We also show that we can modify QN-ILS in order to make it analytically equivalent to GMRes, without the need for extra matrix-vector products.

KW - Generalized minimal residual method

KW - Iterative method

KW - Least squares

KW - Linear algebra

KW - Quasi-Newton method

KW - Rank-one update

KW - Secant method

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JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 6

ER -

On the similarities between the quasi-Newton inverse least squares method and GMRes

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Keywords

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