The Lanczos and conjugate gradient algorithms : from theory to finite precision computations / Gérard Meurant.

Format:

Book

Author/Creator:

Meurant, Gérard A.

Contributor:

Society for Industrial and Applied Mathematics.

Series:

Software, environments, tools ; 19.

Software, environments, tools

Software, environments, tools ; SE19

Language:

English

Subjects (All):

Conjugate gradient methods.

Algorithms--Methodology.

Algorithms.

Physical Description:

1 online resource (xv, 365 p. ) ill. ;

Place of Publication:

Philadelphia, Pa. : Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 2006.

Language Note:

English

System Details:

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader.

Summary:

The Lanczos and conjugate gradient (CG) algorithms are fascinating numerical algorithms. This book presents the most comprehensive discussion to date of the use of these methods for computing eigenvalues and solving linear systems in both exact and floating point arithmetic. The author synthesizes the research done over the past 30 years, describing and explaining the "average" behavior of these methods and providing new insight into their properties in finite precision. Many examples are given that show significant results obtained by researchers in the field. The author emphasizes how both algorithms can be used efficiently in finite precision arithmetic, regardless of the growth of rounding errors that occurs. He details the mathematical properties of both algorithms and demonstrates how the CG algorithm is derived from the Lanczos algorithm. Loss of orthogonality involved with using the Lanczos algorithm, ways to improve the maximum attainable accuracy of CG computations, and what modifications need to be made when the CG method is used with a preconditioner are addressed.

Contents:

Preface

Chapter 1. The Lanczos algorithm in exact arithmetic

Chapter 2. The CG algorithm in exact arithmetic

Chapter 3. A historical perspective on the Lanczos algorithm in finite precision

Chapter 4. The Lanczos algorithm in finite precision

Chapter 5. The CG algorithm in finite precision

Chapter 6. The maximum attainable accuracy

Chapter 7. Estimates of norms of the error in finite precision

Chapter 8. The preconditioned CG algorithm

Chapter 9. Miscellaneous - Appendix.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references.

Description based on title page of print version.

ISBN:

0-89871-814-7

Publisher Number:

SE19 siam

SE19 SIAM