Nonlinear Extrapolation Estimates of π  

作　　者：Wen-qing Xu Sha-sha Wang Da-chuan Xu 

机构地区：[1]Beijing Institute for Scientific and Engineering Computing,Beijing University of Technology,Beijing,100124,China [2]Department of Mathematics and Statistics,California State University,Long Beach,California,90840,USA [3]Department of Mathematics and Physics,Shijiazhuang Tiedao University,Shijiazhuang,050043,China

出　　处：《Acta Mathematicae Applicatae Sinica》2024年第1期91-108,共18页应用数学学报（英文版）

基　　金：supported in part by the National Natural Science Foundation of China (No.12131003)。

摘　　要：The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R~2 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 χ_n=S_n~αA_n~β and Y_n(p)=(αS_n~p+βA_n~p)~(1/p) where α+β=1,p≠0,and Sn and An respectively represents the semiperimeter and area of a random n-gon inscribed in the unit circle in R~2,and Xn may be viewed as the limit of Y_n(p) when p→0.By deriving probabilistic asymptotic expansions with carefully controlled error estimates for Xn and 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.

关 键 词：Archimedean polygons random polygons approximations ofπ extrapolation processes asymptotic convergence 

分 类 号：O212.1[理学—概率论与数理统计]