Section: New Results

Fractional decomposition of matrices and parallel computing

In [40] 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 $A_i^m$ with specific shape: $A_i=I-h_{i}X$ for a fixed matrix $X$ and real numbers $h_i$. Although having the special form, these matrices $A_i$ arise frequently in the discretization of evolutionary Partial Differential Equations. The idea is to express $A^{-1}$ as a linear combination of elementary matrices $A_i^{-k}$. Hence the solution of the linear system with matrix $A$ is a linear combination of the solutions of linear systems with matrices $A_i^k$. These systems are solved simultaneously on different processors.