A study of defect-based error estimates for the Krylov approximation of φ-functions

Abstract

Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

Exponential Multistep Methods for Stiff Delay Differential Equations

“…It is obvious that efficiently computing the ϕ-functions is critical for the implementation of exponential multistep methods. Several efforts have been made on this line; see [23][24][25][26]. For large-scale problems, it is more efficient to compute the product of a matrix exponential function with a vector directly without the explicit form of the matrix exponential.…”