@Article{JMS-52-244, author = {Hecht , Fredéric and Kaber , Sidi-Mahmoud }, title = {Partial Fraction Decomposition of Matrices and Parallel Computing}, journal = {Journal of Mathematical Study}, year = {2019}, volume = {52}, number = {3}, pages = {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 \$A_i^m\$ with specific shape: \$A_i= I -{h_i}X\$ for a fixed matrix \$X\$ and real numbers \$h_i\$. Although having a  special form, these matrices \$A_i\$ 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)u^{n+1}=u^n\$. Iterating \$m\$ times such a scheme leads to a linear system \$Au^{n+m}=u^n\$. The idea is to express \$A^{-1}\$ as a linear combination of elementary  matrices \$A_i^{-1}\$ (or more generally in term of 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\$ (or \$A_i^k\$). These systems are then solved simultaneously on different processors.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v52n3.19.02}, url = {http://global-sci.org/intro/article_detail/jms/13297.html} }