Fractional decomposition of matrices and parallel computing
Résumé
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 A m i with specific shape: Ai = I −hiX for a fixed matrix X and real numbers hi. Although having the special form, these matrices Ai arise frequently in the discretization of evolutionary Partial Differential Equations. The idea is to express A −1 as a linear combination of elementary 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 A k i. These systems are solved simultaneously on different processors.
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