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

Abstract This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say $$q(\tau ,A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , 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 $$q(\tau ,w)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.