Accuracy Control for Parallel Evaluation of Matrix Rational Functions

Frequently, one needs to evaluate expressions of the form [p(A)] 1q(A)b, where A 2 RN N , b 2 RN and p; q are polynomialswith degree q degree p. Algorithms based on the partial fraction representation of q=p when evaluating [p(A)] 1q(A)b lend themselves to parallelism but may yield poor accuracy. We discuss how to determine an incomplete partial fraction representation of q=p to achieve accuracy and examine its effects on the parallelism. 1 Statement of the Problem and Discussion In this note we are concerned with the problem of accuracy control in the evaluation of x := [p(A)] 1q(A)b (1) where A 2 RN N , b 2 RN , p and q are polynomials such that: 1. The degree of q is not larger than the degree of p; 2. Polynomial p has a factorization of the formQnj=1(t tj ) with distinct roots tj ; 3. p and q have no common roots. The above conditions will be assumed to hold throughout this paper. When q 1, the standard way of computing (1) on a serial computer has been to solve systems (A tjI)xj = xj 1, for j = 1; : : : ; n, with x0 = b. This evaluation is based on the Stevens Institute of Technology, Department of Pure and Applied Mathematics, Hoboken, NJ 07030. Research supported in part by the Design and Manufacturing Institute of Stevens Institute of Technology. y University of Illinois at Urbana-Champaign, Department of Computer Science and Center for Supercomputing Research and Development, Urbana, IL 61801. Supported in part by the National Science Foundation under grant NSF CCR-9120105. z Kent State University, Department of Mathematics and Computer Science, Kent, OH 44242. Supported in part by NSF grants DMS-9002884 and DMS-9205531.