Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems

• Published in: Advances in computational mathematics Vol. 51; no. 2
• Main Authors: Aceto, Lidia, Gemignani, Luca
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.04.2025
• Subjects: Algorithms; Boundary value problems; Computation; Fourier series; Mathematics; Numerical methods; Polynomials; Sparse matrices
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-025-10231-1

This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say $$q(\tau ,A)$$ q ( τ , A ) , on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of $$q(\tau ,w)$$ q ( τ , 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.