Nonlinear Extrapolation Estimates of π

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Wen-qing Xu, Sha-sha Wang, Da-chuan Xu
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引用次数: 0

Abstract

The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in ℝ2 and it is well-known that by using linear combinations of these basic estimates, modern extrapolation techniques can greatly speed up the approximation process. Similarly, when n vertices are randomly selected on the circle, the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to π almost surely as n → ∞, and by further applying extrapolation processes, faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons. In this paper, we further develop nonlinear extrapolation methods for approximating π through certain nonlinear functions of the semiperimeter and area of such polygons. We focus on two types of extrapolation estimates of the forms \({{\cal X}_n} = {\cal S}_n^\alpha {\cal A}_n^\beta \) and \({{\cal Y}_n}(p) = {(\alpha {\cal S}_n^p + \beta {\cal A}_n^p)^{1/p}}\) where α + β = 1, p ≠ 0, and \({{\cal S}_n}\) and \({{\cal A}_n}\) respectively represents the semiperimeter and area of a random n-gon inscribed in the unit circle in ℝ2, and \({{\cal X}_n}\) may be viewed as the limit of \({{\cal Y}_n}(p)\) when p → 0. By deriving probabilistic asymptotic expansions with carefully controlled error estimates for \({{\cal X}_n}\) and \({{\cal Y}_n}(p)\), we show that the choice α = 4/3, β= −1/3 minimizes the approximation error in both cases, and their distributions are also asymptotically normal.

π的非线性外推法估计值
众所周知,通过使用这些基本估计值的线性组合,现代外推法可以大大加快近似过程。同样,当在圆上随机选择 n 个顶点时,已知相应的随机内切多边形和外切多边形的半径和面积在 n → ∞ 时几乎肯定收敛于 π,进一步应用外推法,通过对这些随机多边形的半径和面积进行类似的线性组合,也可以获得更快的收敛速度。在本文中,我们将进一步开发非线性外推法,通过此类多边形的半径和面积的某些非线性函数来逼近 π。我们将重点放在两种形式的外推估计上:\({{\cal X}_n} = {\cal S}_n^\alpha {\cal A}_n^\beta \)和\({{\cal Y}_n}(p) = {(\alpha {\cal S}_n^p + \beta {\cal A}_n^p)^{1/p}}\) 其中 α + β = 1, p ≠ 0、和 \({{\cal S}_n}\) 和 \({{\cal A}_n}\) 分别表示在ℝ2中嵌入单位圆的随机 n 个坤的半径和面积,并且 \({{\cal X}_n}\) 可以看作是 p → 0 时 \({{\cal Y}_n}(p)\) 的极限。通过对 \({{\cal X}_n}\) 和 \({{\cal Y}_n}(p)\)的误差估计进行仔细控制,推导出概率渐近展开,我们证明选择 α=4/3, β= -1/3 可以使两种情况下的近似误差最小化,并且它们的分布也是渐近正态的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
70
审稿时长
3.0 months
期刊介绍: Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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