Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials
Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials
复制标题
和的近似计算 I：格拉姆多项式零点的界限
DOI：
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发表时间：
2014
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
设$N$为正整数，$x_{j}$为$N$等距点。我们提出了一种近似计算 ${\sum_{j=1}^{N} F(x_{j})}$ 形式的和的算法方法。该方法基于高斯型求和公式， \[ \sum_j=1^N F(x_j) \approx \sum_k=1^n B_n,k F(g_n,k(N)), n ll N, \] 其中 $g_{n,k}(N)$ 是所谓的格拉姆多项式的零点。这允许将具有大量项 $N$ 的和的计算减少为具有少得多的被加数 $n$ 的和的计算。构建此类公式的第一个任务是计算其节点 $g_{n,k}(N)$。在本文中，我们获得了 $g_{n,k}(N)$ 的精确下限和上限。数值实验表明，对零点 $g_{n,k}(N)$ 的估计非常尖锐，并且所提出的计算和的方法是有效的。
Let $N$ be a positive integer and $x_{j}$ be $N$ equidistant points. We propose an algorithmic approach for approximate calculation of sums of the form ${\sum_{j=1}^{N} F(x_{j})}$. The method is based on the Gaussian type quadrature formula for sums, \[ \sum_j=1^N F(x_j) \approx \sum_k=1^n B_n,k F(g_n,k(N)), n ll N, \] where $g_{n,k}(N)$ are the zeros of the so-called Gram polynomials. This allows the calculation of sums with very large number of terms $N$ to be reduced to sums with a much smaller number of summands $n$. The first task in constructing such a formula is to calculate its nodes $g_{n,k}(N)$. In this paper we obtain precise lower and upper bounds for $g_{n,k}(N)$. Numerical experiments show that the estimates for the zeros $g_{n,k}(N)$ are very sharp and that the proposed method for calculation of sums is efficient.