Adaptive polynomial preconditioning for hermitian indefinite linear systems

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.

[1]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[2]  E. Cheney.  Introduction to approximation theory , 1966 .

[3]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[4]  Gene H. Golub,et al.  Richardson''s non-stationary matrix iterative procedure. , 1972 .

[5]  Åke Björck,et al.  Numerical Methods , 1995, Handbook of Marine Craft Hydrodynamics and Motion Control.

[6]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[7]  O. Axelsson.  A class of iterative methods for finite element equations , 1976 .

[8]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[9]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[10]  Kang C. Jea,et al.  Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods , 1980 .

[11]  J. Grcar.  Analyses of the lanczos algorithm and of the approximation problem in richardson's method , 1981 .

[12]  John R. Rice,et al.  Extremal Polynomials with Application to Richardson Iteration for Indefinite Linear Systems , 1982 .

[13]  C. Micchelli,et al.  Polynomial Preconditioners for Conjugate Gradient Calculations , 1983 .

[14]  T. Manteuffel,et al.  Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method , 1984 .

[15]  T. Jordan.  CONJUGATE GRADIENT PRECONDITIONERS FOR VECTOR AND PARALLEL PROCESSORS , 1984 .

[16]  L. Adams.  m-Step Preconditioned Conjugate Gradient Methods , 1985 .

[17]  Y. Saad,et al.  Practical Use of Polynomial Preconditionings for the Conjugate Gradient Method , 1985 .

[18]  Owe Axelsson,et al.  A survey of preconditioned iterative methods for linear systems of algebraic equations , 1985 .

[19]  P. Brown,et al.  Matrix-free methods for stiff systems of ODE's , 1986 .

[20]  Anthony T. Chronopoulos.  A class of parallel iterative methods implemented on multiprocessors , 1987 .

[21]  P. Saylor,et al.  Leapfrog variants of iterative methods for linear algebraic equations , 1988 .

[22]  S. Ashby.  Polynomial Preconditioning for Conjugate Gradient Methods , 1988 .

[23]  C.-C. Jay Kuo,et al.  Parallel elliptic preconditioners: Fourier analysis and performance on the connection machine , 1989 .

[24]  S. Ashby,et al.  A numerical investigation of the conjugate gradient method as applied to three‐dimensional groundwater flow problems in randomly heterogeneous porous media , 1989 .

[25]  G. Golub,et al.  Estimates of Eigenvalues for Iterative Methods , 1989 .

[26]  T. Manteuffel,et al.  A taxonomy for conjugate gradient methods , 1990 .