Polynomial Preconditioners for Conjugate Gradient Calculations

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.