Partial Fraction Decomposition of Matrices and Parallel Computing
Year: 2019
Author: Frédéric Hecht, Sidi-Mahmoud Kaber
Journal of Mathematical Study, Vol. 52 (2019), Iss. 3 : pp. 244–257
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 Ami 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−1i (or more generally in term of matrices A−ki). Hence the solution of the linear system with matrix A is a linear combination of the solutions of linear systems with matrices Ai (or Aki). These systems are then solved simultaneously on different processors.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v52n3.19.02
Journal of Mathematical Study, Vol. 52 (2019), Iss. 3 : pp. 244–257
Published online: 2019-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 14
Keywords: Partial differential equations parabolic equation finite element methods finite difference methods parallel computing.
Author Details
Frédéric Hecht Email
Sidi-Mahmoud Kaber Email