Katherine Porter

March 25, 1999

A Polynomial Iterative Algorithm for Nonsymmetric Matrices

This talk will introduce a new method to solve the system Ax = b, where A is a nonsymmetric, sparse matrix whose symmetric part is positive definite. Polynomial approximations to the reciprocal function are applied to A within an iterative algorithm to approximate A−1. The polynomials arise from two sources: Lagrange interpolation of the reciprocal function and optimization, in the least squares sense, of the error between the polynomial of fixed degree and the reciprocal function, in a specified region of the complex plane. Iterative Refinement is the basis of the algorithm, using the polynomials in A in Horner form, to solve the systems. Numerical examples from partial differential equation are used to display the effectiveness of the iterative algorithm and for comparison with current methods such as Generalized Minimal Residual (GMRES), Conjugate Gradient Squared (CGS), and Transpose-Free Quasi-Minimal Residual (TFQMR). Convergence and error theorems are presented for the new algorithm. Other potential uses for the polynomials and the iterative algorithm are also introduced and future research is proposed.