Approximate Calculation of Sums II: Gaussian Type Quadrature

Abstract

The present paper is a continuation of a recent article [SIAM J. Numer. Anal., 52 (2014), pp. 1867--1886], where we proposed an algorithmic approach for approximate calculation of sums of the form $\sum_{j=1}^{N} f(j)$. The method is based on a Gaussian type quadrature formula for sums, which allows the calculation of sums with a very large number of terms $N$ to be reduced to sums with a much smaller number of summands $n$. In this paper we prove that the Weierstrass--Dochev--Durand--Kerner iterative numerical method, with explicitly given initial conditions, converges to the nodes of the quadrature formula. Several methods for computing the nodes of the discrete analogue of the Gaussian quadrature formula are compared. Since, for practical purposes, any approximation of a sum should use only the values of the summands $f({j})$, we implement a simple but efficient procedure to additionally approximate the evaluations at the nodes by local natural splines. Explicit numerical examples are provided. Moreove...