Volume 52, Issue 3
Partial Fraction Decomposition of Matrices and Parallel Computing

J. Math. Study, 52 (2019), pp. 244-257.

Published online: 2019-09

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• 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.

• Keywords

Partial differential equations, parabolic equation, finite element methods, finite difference methods, parallel computing.

65M60, 65Y05, 35K45, 74S05, 74S20

hecht@ljll.math.upmc.fr (Fredéric Hecht)

kaber@ljll.math.upmc.fr (Sidi-Mahmoud Kaber)

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@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} }
TY - JOUR T1 - Partial Fraction Decomposition of Matrices and Parallel Computing AU - Hecht , Fredéric AU - Kaber , Sidi-Mahmoud JO - Journal of Mathematical Study VL - 3 SP - 244 EP - 257 PY - 2019 DA - 2019/09 SN - 52 DO - http://doi.org/10.4208/jms.v52n3.19.02 UR - https://global-sci.org/intro/article_detail/jms/13297.html KW - Partial differential equations, parabolic equation, finite element methods, finite difference methods, parallel computing. AB -

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.

Fredéric Hecht & Sidi-Mahmoud Kaber. (2019). Partial Fraction Decomposition of Matrices and Parallel Computing. Journal of Mathematical Study. 52 (3). 244-257. doi:10.4208/jms.v52n3.19.02
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