The Leja method revisited: backward error analysis for the matrix exponential

TL;DR

This paper provides a backward error analysis for the Leja polynomial interpolation method, enabling precise parameter selection for computing matrix exponentials efficiently and reliably.

Contribution

It introduces a backward error analysis framework for the Leja method, improving parameter choice and implementation for matrix exponential computations.

Findings

Backward error analysis guides parameter selection.

Enhanced efficiency and reliability in matrix exponential computation.

Numerical examples demonstrate practical performance improvements.

Abstract

The Leja method is a polynomial interpolation procedure that can be used to compute matrix functions. In particular, computing the action of the matrix exponential on a given vector is a typical application. This quantity is required, e.g., in exponential integrators. The Leja method essentially depends on three parameters: the scaling parameter, the location of the interpolation points, and the degree of interpolation. We present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy. Additional aspects that are required for an efficient and reliable implementation are discussed. Numerical examples that illustrate the performance of our Matlab code are included.