A Rational Krylov Subspace Method for the Computation of the Matrix Exponential Operator

TL;DR

This paper introduces a rational Krylov subspace method utilizing a rational block Lanczos algorithm to efficiently approximate matrix exponential functions for large sparse matrices, with proven error bounds and numerical validation.

Contribution

It develops a novel rational Krylov subspace approach with error estimates for computing matrix exponentials and Cauchy-Stieltjes functions, enhancing computational efficiency.

Findings

The method achieves high accuracy in approximating e^tA B.

Numerical results demonstrate improved computational efficiency.

Error bounds provide theoretical guarantees for convergence.

Abstract

The computation of approximating e^tA B, where A is a large sparse matrix and B is a rectangular matrix, serves as a crucial element in numerous scientific and engineering calculations. A powerful way to consider this problem is to use Krylov subspace methods. The purpose of this work is to approximate the matrix exponential and some Cauchy-Stieltjes functions on a block vectors B of R^n*p using a rational block Lanczos algorithm. We also derive some error estimates and error bound for the convergence of the rational approximation and finally numerical results attest to the computational efficiency of the proposed method.