On the Error Computation for Polynomial Based Iteration Methods

Bernd Fischer; Gene H. Golub

doi:10.1007/978-1-4613-9353-5_5

On the Error Computation for Polynomial Based Iteration Methods

Bernd Fischer, Gene H. Golub

Institute of Mathematics

Abstract

In this note we investigate the Chebyshev iteration and the conjugate gradient method applied to the system of linear equations Ax = f where A is a symmetric, positive definite matrix. For both methods we present algorithms which approximate during the iteration process the kth error $k= ||x − xk||A. The algorithms are based on the theory of modified moments and Gaussian quadrature. The proposed schemes are also applicable for other polynomial iteration schemes. Several examples, illustrating the performance of the described methods, are presented.

Original language	English
Title of host publication	Recent Advances in Iterative Methods
Editors	Gene Golub, Mitchell Luskin, Anne Greenbaum
Number of pages	9
Volume	60
Place of Publication	New York, NY
Publisher	Springer New York LLC
Publication date	1994
Pages	59-67
ISBN (Print)	978-1-4613-9355-9
ISBN (Electronic)	978-1-4613-9353-5
DOIs	https://doi.org/10.1007/978-1-4613-9353-5_5
Publication status	Published - 1994

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title = "On the Error Computation for Polynomial Based Iteration Methods",

abstract = "In this note we investigate the Chebyshev iteration and the conjugate gradient method applied to the system of linear equations Ax = f where A is a symmetric, positive definite matrix. For both methods we present algorithms which approximate during the iteration process the kth error $k= ||x − xk||A. The algorithms are based on the theory of modified moments and Gaussian quadrature. The proposed schemes are also applicable for other polynomial iteration schemes. Several examples, illustrating the performance of the described methods, are presented.",

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On the Error Computation for Polynomial Based Iteration Methods. / Fischer, Bernd; Golub, Gene H.
Recent Advances in Iterative Methods. ed. / Gene Golub; Mitchell Luskin; Anne Greenbaum. Vol. 60 New York, NY: Springer New York LLC, 1994. p. 59-67 ( The IMA Volumes in Mathematics and its Applications; Vol. 60).

Research output: Chapters in Books/Reports/Conference Proceedings › Conference contribution › peer-review

TY - GEN

T1 - On the Error Computation for Polynomial Based Iteration Methods

AU - Fischer, Bernd

AU - Golub, Gene H.

PY - 1994

Y1 - 1994

N2 - In this note we investigate the Chebyshev iteration and the conjugate gradient method applied to the system of linear equations Ax = f where A is a symmetric, positive definite matrix. For both methods we present algorithms which approximate during the iteration process the kth error $k= ||x − xk||A. The algorithms are based on the theory of modified moments and Gaussian quadrature. The proposed schemes are also applicable for other polynomial iteration schemes. Several examples, illustrating the performance of the described methods, are presented.

AB - In this note we investigate the Chebyshev iteration and the conjugate gradient method applied to the system of linear equations Ax = f where A is a symmetric, positive definite matrix. For both methods we present algorithms which approximate during the iteration process the kth error $k= ||x − xk||A. The algorithms are based on the theory of modified moments and Gaussian quadrature. The proposed schemes are also applicable for other polynomial iteration schemes. Several examples, illustrating the performance of the described methods, are presented.

U2 - 10.1007/978-1-4613-9353-5_5

DO - 10.1007/978-1-4613-9353-5_5

M3 - Conference contribution

SN - 978-1-4613-9355-9

VL - 60

T3 - The IMA Volumes in Mathematics and its Applications

SP - 59

EP - 67

BT - Recent Advances in Iterative Methods

A2 - Golub, Gene

A2 - Luskin, Mitchell

A2 - Greenbaum, Anne

PB - Springer New York LLC

CY - New York, NY

ER -

On the Error Computation for Polynomial Based Iteration Methods

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