matlab實驗九 π的近似計算

1、實驗九n的近似計算實驗目的】了解圓周率n的計算歷程。了解計算n的割圓術、韋達公式、級數法、拉馬努金公式、迭 代法。學習、掌握MATLAB軟件有關的命令?！緦嶒瀮热荨坷庙f達（VieTa）公式2 V2 J2 + V2 J2 + /2 +y 2 + J2 + J2 +n 2222計算n的近似值。【實驗準備】割圓術韋達（VieTa）公式利用級數計算n拉馬努金（Ranmaunujan ）公式迭代方法n的兩百位近似值【實驗重點】圓周率的計算歷程【實驗難點】1. 圓周率的各種計算公式【實驗方法與步驟】練習1用劉徽的迭代公式兀.=：2_, s. = 32n., x = 1 62+i62n 62+16 2 0

2、計算n的近似值。相應的 MATLAB 代碼為clear;x=1;for i=1:30 x=vpa (sqrt(2-sqrt(4-x2)，15)%計算精度為 15 位有效數字S=vpa(3*2Ai*x,10)end計算可得x = .517638090205042 S = 3.105828541x = .261052384440103 S = 3.132628613x = .130806258460286 S= 3.139350203x = .654381656435522e-1S = 3.141031951x = .327234632529735e-1S = 3.141452472x = .16

3、3622792078742e-1S = 3.141557608x = .818120805246955e-2S = 3.141583892x = .409061258232818e-2S = 3.141590463x = .204530736067660e-2S = 3.141592106x = .102265381402739e-2S = 3.141592517x = .511326923724832e-3S = 3.141592619x = .255663463951308e-3S = 3.141592645x = .127831732236766e-3S = 3.141592651x =

4、 .639158661510219e-4S = 3.141592653x = .319579330795908e-4S = 3.141592653x = .159789665403054e-4S = 3.141592654x = .798948327021645e-5S = 3.141592654x = .399474163511619e-5S = 3.141592654x = .199737081755909e-5S = 3.141592654x = .998685408779670e-6S = 3.141592654x = .499342704389851e-6S = 3.14159265

5、4x = .249671352194927e-6S = 3.141592654x = .124835676097464e-6S = 3.141592654x = .624178380487320e-7S = 3.141592654x = .312089190243660e-7S = 3.141592654x = .156044595121830e-7S = 3.141592654x = .780222975609150e-8S = 3.141592654x = .390111487804575e-8S = 3.141592654x = .195055743902288e-8S = 3.1415

6、92654x = .975278719511453e-9S = 3.141592654練習 2 用韋達公式2 _ V2 J2 + 巧 J2 + 722+ J2 + 72 + 近n 2222計算n的近似值。相應的 MATLAB 代碼為clear;x=1;for i=l:20;ix=vpa(x*cos(pi/2A(i+l),30);%計算精度為 30 位有效數字pai=vpa(2/x,22)error=vpa(pai-2/x,22)end計算可得i=1 pa i = 2.828427124746190097603error = -.3774484188e-21i=2 pa i = 3.061467

7、458920718232298error = -.2223605969e-21i=3 pa i = 3.121445152258052404216error = -.579142540e-22i=4 pa i = 3.136548490545939249125error = .2408548301e-21i=5 pa i = 3.140331156954752858963error = -.3651343370e-21i=6 pa i = 3.141277250932772720892error = -.1464196722e-21i=7 pa i = 3.141513801144301048

8、318error = .4482597912e-21i=8 pa i = 3.141572940367091461583error = .596014783e-22i=9 pa i = 3.141587725277159716287error = -.1189029953e-21i=10 pa i = 3.141591421511200086037error = .3229610904e-21i=11 pa i = 3.141592345570117830926error = .4967305463e-21i=12 pa i = 3.141592576584872625535error = -

9、.4344678192e-21i=13 pa i = 3.141592634338562821891error = .4862357044e-21i=14 pa i = 3.141592648776985437337error = .2757983366e-21i=15 pa i = 3.141592652386591008149error = .956629086e-22i=16 pa i = 3.141592653288992575505error = .4169549797e-21i=17 pa i = 3.141592653514592792966error = -.2783657489e-21i=18 pa i = 3.141592653570993021726error = -.1396314172e-21i=19 pa i = 3.141592653585093078916error = -.1682315366e-21i=20 pa i = 3.14159265358861791