Partial Fraction Decomposition of Matrices and Parallel Computing

Abstract

We are interested in the design of parallel numerical schemes for linear systems. We give an effective solution to this problem in the following case: the matrix A of the linear system is the product of p nonsingular matrices Am i with specific shape: Ai = I−hiX for a fixed matrix X and real numbers hi. Although having a special form, these matrices Ai arise frequently in the discretization of evolutionary Partial Differential Equations. For example, one step of the implicit Euler scheme for the evolution equation u′=Xu reads (I−hX)un+1 =un. Iterating m times such a scheme leads to a linear system Aun+m = un. The idea is to express A−1 as a linear combination of elementary matrices A−1 i (or more generally in term of matrices A −k i ). Hence the solution of the linear system with matrix A is a linear combination of the solutions of linear systems with matrices Ai (or Ak i ). These systems are then solved simultaneously on different processors. AMS subject classifications: 65M60, 65Y05, 35K45, 74S05, 74S20