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.

翻译：本文研究高效数值方法，用于计算伯努利多项式生成函数（记为$q(\tau,w)$）作用于典型大规模稀疏矩阵的算法。该问题在求解某些非局部边值问题时出现。基于$q(\tau,w)$傅里叶展开的方法已在科学文献中有所探讨。本文的贡献主要体现在两个方面：首先，我们将这些方法置于经典的Krylov-Lanczos（多项式-有理函数）技术框架中，用于加速傅里叶级数收敛。这使得我们能够将在此背景下发展的收敛性结果应用于所研究的函数。其次，我们设计了一种新的加速方案。文中给出的数值结果验证了所提算法的有效性。