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Adaptive polynomial preconditioning for hermitian indefinite linear systems

  • Preconditioned Conjugate Gradient Methods
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Abstract

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(λ), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.

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Authors and Affiliations

  1. Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, P.O. Box 808, 94550, Livermore, CA, USA

    Steven F. Ashby

  2. Computational Mathematics Group, University of Colorado at Denver, 1200, Larimer Street, 80204, Denver, CO, USA

    Thomas A. Manteuffel

  3. Department of Computer Science, University of Illinois at Urbana-Champaign, 1304 West Springfield Avenue, 61801, Urbana, IL, USA

    Paul E. Saylor

Authors
  1. Steven F. Ashby
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  2. Thomas A. Manteuffel
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  3. Paul E. Saylor
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Additional information

This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Dept. of Energy, by Lawrence Livermore National Laboratory under contract W-7405-ENG-48.

This research was supported in part by the Dept. of Energy and the National Science Foundation under grant DMS 8704169.

This research was supported in part by U.S. Dept. of Energy grant DEFG02-87ER25026 and by National Science Foundation grant DMS 8703226.

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Ashby, S.F., Manteuffel, T.A. & Saylor, P.E. Adaptive polynomial preconditioning for hermitian indefinite linear systems. BIT 29, 583–609 (1989). https://doi.org/10.1007/BF01932736

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  • DOI: https://doi.org/10.1007/BF01932736

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