An Extended-Rational Arnoldi Method for Large Matrix Exponential Evaluations

Abstract

The numerical computation of a matrix function such as $\exp {(-tA)V}$, where A is an $n\times n$ large and sparse matrix, V is an $n \times p$ block with $p\ll n$, and $t>0$ arises in various applications including network analysis, the solution of time-dependent partial differential equations (PDE’s) and others. In this work, we propose the use of the global extended-rational Arnoldi method for computing approximations of such functions. The derived method projects the initial problem onto the global extended-rational Krylov subspace $\mathcal {RK}^{e}_m(A,V)=\text {span}\{\prod \limits \nolimits _{i=1}^m(A+s_iI_n)^{-1}V,\ldots ,(A+s_1I_n)^{-1}V,V$ $,AV, \ldots ,A^{m-1}V\}$ of a low dimension. An adaptive procedure of getting the shifts $\{s_1,\ldots ,s_m\}$ during the algorithmic process is given and analyzed. Applications to the solution of time-dependent PDE’s and to network analysis are presented. Numerical examples are presented to show the performance of the global extended-rational Arnoldi process.