On Krylov Subspace Approximations to the Matrix Exponential Operator

Abstract

Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkin-type Krylov methods for linear equations, namely, the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum, or the pseudospectrum. The convergence to exp$(\tau A)v$ is faster than that of corresponding Krylov methods for the solution of linear equations $(I-\tau A)x=v$, which usually arise in the numerical solution of stiff ordinary differential equations (ODEs). We therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.