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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} }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.

*Journal of Mathematical Study*.

*52*(3). 244-257. doi:10.4208/jms.v52n3.19.02