Approximation of the matrix exponential for matrices with a skinny field of values

Abstract

The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential $$\exp (A)v$$ has to be approximated by $$\left( p_m(s^{-1}A)\right) ^sv=\exp (A+\varDelta A)v$$ . We propose here a rigorous bound for the relative backward error $$\left\Vert \varDelta A\right\Vert _{2}/\left\Vert A\right\Vert _{2}$$ , which is of particular interest for matrices whose field of values is skinny, such as the discretization of the advection–diffusion or the Schrodinger operators. The numerical results confirm the superiority of the new approach with respect to methods based on the classical power series expansion of the backward error for the matrices of our interest, both in terms of computational cost and achieved accuracy.