An Extended-Rational Arnoldi Method for Large Matrix Exponential Evaluations
Résumé
The numerical computation of a matrix function such as exp (−t A)V , where A is an n × n
large and sparse matrix, V is an n × p block with p 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 RKe
m (A, V ) =
span{∏m
i=1(A + si I n )−1 V , . . . , (A + s1 I n )−1 V , V , AV , . . . , A m−1 V } of a low dimension.
An adaptive procedure of getting the shifts {s1, . . . , sm } 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