An Extended-Rational Arnoldi Method for Large Matrix Exponential Evaluations

The numerical computation of a matrix function such as exp (-t A)V, where A is an n x n large and sparse matrix, V is an n x 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 RKme(A, V) = span{Pi(m)(i=1)(A + s(i) I-n)V-1, ..., (A + s(1) I-n)V-1, V, AV, ..., A(m-1)V} of a low dimension. An adaptive procedure of getting the shifts {s(1), ..., 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.