Polynomial Preconditioners for Conjugate Gradient Calculations

• Published in: SIAM journal on numerical analysis Vol. 20; no. 2; pp. 362 - 376
• Main Authors: Johnson, Olin G., Micchelli, Charles A., Paul, George
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.04.1983
• Subjects: Algebraic conjugates; Algorithms; Approximation; Eigenvalues; Least squares; Matrices; Polynomials; Preconditioning; Recurrence relations; Weighting functions
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0720025

Dubois, Greenbaum and Rodrigue proposed using a truncated Neumann series as an approximation to the inverse of a matrix A for the purpose of preconditioning conjugate gradient iterative approximations to Ax = b. If we assume that A has been symmetrically scaled to have unit diagonal and is thus of the form (I - G), then the Neumann series is a power series in G with unit coefficients. The incomplete inverse was thought of as a replacement of the incomplete Cholesky decomposition suggested by Meijerink and van der Vorst in the family of methods ICCG (n). The motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors. We here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in G. We then show how to select the parameters to minimize the condition number of the product of the polynomial and (I - G). Theoretically the resulting algorithm is the best of the class involving polynomial preconditioners. We also show that polynomial preconditioners which minimize the mean square error with respect to a large class of weight functions are positive definite. We give recurrence relations for the computation of both classes of polynomial preconditioners.