Approximate Calculation of Sums II: Gaussian Type Quadrature
Approximate Calculation of Sums II: Gaussian Type Quadrature
复制标题
和的近似计算 II：高斯型求积
DOI：
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发表时间：
2016
影响因子：
2.9
通讯作者：
Vanessa G. Paschoa
中科院分区：
文献类型：
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作者：
I. Area;D. Dimitrov;E. Godoy;Vanessa G. Paschoa
本文是最近一篇文章的延续[SIAM J. Numer.分析：52（2014），pp. 1867--1886]，在那里我们提出了一种近似计算形式$\sum_{j=1}^{N} f（j）$的和的算法方法。该方法是基于高斯型求积公式的总和，它允许计算的总和与一个非常大的数目的条款$N$减少到总和与一个更小的数目的被加数$n$。本文证明了Weierstrass-Dochev-Durand-Kerner迭代数值方法在显式给定初始条件下收敛于求积公式的节点。比较了几种计算高斯求积公式离散模拟节点的方法。由于，出于实际目的，任何近似的总和应该只使用的值的被加数$f（{j}）$，我们实现了一个简单而有效的程序，另外近似的评估在节点的本地自然样条。提供了明确的数值例子。莫洛夫...
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...