Computing the Action of the Generating Function of Bernoulli Polynomials on a Matrix with An Application to Non-local Boundary Value Problems

TL;DR

This paper introduces a novel acceleration scheme for efficiently computing the action of the generating function of Bernoulli polynomials on large sparse matrices, with applications to non-local boundary value problems.

Contribution

It integrates Fourier-based methods into Krylov-Lanczos techniques and develops a new acceleration scheme for improved computational efficiency.

Findings

The proposed methods effectively compute the action of Bernoulli polynomial generating functions.

Numerical results demonstrate the efficiency and accuracy of the new algorithms.

The approach accelerates convergence in solving non-local boundary value problems.

Abstract

This paper deals with efficient numerical methods for computing the action of the generating function of Bernoulli polynomials, say $q(\tau,w)$, on a typically large sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of $q(\tau,w)$ have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.