A Moment-Matching Arnoldi Iteration for Linear Combinations of $\varphi$ Functions

The action of the matrix exponential and related $\varphi$ functions on vectors plays an important role in the application of exponential integrators to ordinary differential equations. For the efficient evaluation of linear combinations of such actions we consider a new Krylov subspace algorithm. By employing Cauchy's integral formula an error representation of the numerical approximation is given. This is used to derive a priori error bounds that describe well the convergence behavior of the algorithm. Further, an efficient a posteriori estimate is constructed. Numerical experiments illustrating the convergence behavior are given in MATLAB.