Approximation of functions of large matrices with Kronecker structure

Abstract

We consider the numerical approximation of $$f(\mathcal{A})b$$f(A)b where $$b\in {\mathbb {R}}^{N}$$bźRN and $$\mathcal A$$A is the sum of Kronecker products, that is $$\mathcal{A}=M_2 \otimes I + I \otimes M_1\in {\mathbb {R}}^{N\times N}$$A=M2źI+IźM1źRN×N. Here f is a regular function such that $$f(\mathcal{A})$$f(A) is well defined. We derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations, with special emphasis on the exponential and on completely monotonic functions, for which the new procedure becomes particularly advantageous. Our findings are illustrated by numerical experiments with typical functions used in applications.