实验报告课程名称：数学实验实验名称：π的近似计算实验目的、要求：1.了解圆周率π的计算历程。

2.了解计算π的割圆术、韦达公式、级数法、拉马努金公式、迭代法。

3.学习、掌握MATLAB 软件有关的命令。

实验仪器：安装有MA TLAB 软件的计算机实验步骤：一、 实验内容1．内容π是人们经常使用的数学常数，对π的研究已经持续了2500多年，今天，这种探索还在继续中。

1.割圆术。

2.韦达（VieTa ）公式。

3.利用级数计算π。

4.拉马努金（Ranmaunujan ）公式。

5.迭代方法。

6.π的两百位近似值。

计算π的近似值：2. 原理1、 刘徽的迭代公式1106.2 6.2 6.2 6.224, 3.2,1n n n n n x x s x x ++=--==2、利用韦达（VieTa ）公式22222222222...2222π++++++= 3、莱布尼茨级数 n 1(1)=421nn π∞=-+∑4、级数加速后的公式2121n 0n 011(1)1(1)116arctan 4arctan 164523921521239k k k k k k π∞∞++==--=-=⋅-⋅++∑∑5、拉马努金公式4n 0122(4)!110326396=9801396n n n π∞=+⋅∑（n!）二、实验结果练习1 用刘徽的迭代公式11 6.206.2 6.2 6.224, 3.2,1n n n n x x s x x ++=--==计算π的近似值。

相应的MA TLAB 代码为>>clear;>>x=1;>>for i=1:30>>x=vpa (sqrt(2-sqrt(4-x^2)),15)%计算精度为15位有效数字>>S=vpa(3*2^i*x,10)>>end计算可得x =.517638********* S =3.105828541x =.261052384440103 S =3.132628613 …练习题 1.1106.2 6.2 6.2 6.224, 3.2,1n n n n n x x s x x ++=--==,计算π的近似值,迭代50次，有效数字取为100位。

解：x=.5176380902050415899751101278525311499834060668945312500000000000000000000000000000000000000000000000S =3.105828541x =.26105238444010321659775747351901 S =3.132628613x =.13080625846028615048946927650964 S =3.139350203x =.65438165643552292570302790136363e-1 S =3.141031951x =.32723463252973567509131365534310e-1 S =3.141452472x =.16362279207874260682204029836769e-1 S =3.141557608x =.81812080524695802451715035172989e-2 S =3.141583892x =.40906125823281907567941283992211e-2 S =3.141590463x =.20453073606766093463703416909438e-2 S =3.141592106x =.10226538140273951344639001691572e-2 S =3.141592517x =.51132692372483469411943625174689e-3 S =3.141592619x =.25566346395130951352151834359580e-3 S =3.141592645x =.12783173223676627836851115115739e-3 S =3.141592651x =.63915866151022079410225244625580e-4 S =3.141592653x =.31957933079590907234139446466181e-4 S =3.141592653x =.15978966540305437058221873847277e-4 S =3.141592654x =.79894832702164664592540752808922e-5 S =3.141592654x =.39947416351162017209013439939351e-5 S =3.141592654x =.19973708175590969218580033534076e-5 S =3.141592654x =.99868540877967296859318388180810e-6 S =3.141592654x =.49934270438985204773114636404083e-6 S =3.141592654x =.24967135219492796927298869033863e-6 S =3.141592654x =.12483567609746422765596560845085e-6 S =3.141592654x =.62417838048732144468262246567435e-7 S =3.141592654x =.31208919024366076239396409864371e-7 S =3.141592654x =.15604459512183038119698204932186e-7 S =3.141592654x =.78022297560915174577429878338312e-8 S =3.141592654x =.39011148780457619330837231814377e-8 S =3.141592654x =.19505574390228809665418615907189e-8 S =3.141592654x =.97527871951145330011984785335838e-9 S =3.141592654x =.48763935975575228375775804183490e-9 S =3.141592654x =.24381967987797867667021545728558e-9 S =3.141592654x =.12190983993919440791778038578154e-9 S =3.141592654x =.60954919969597203958890192890768e-10 S =3.141592654x =.30477459984388462814097367475721e-10 S =3.141592654x =.15238729993014509737727583788009e-10 S =3.141592654x =.76193649997883681907846134144596e-11 S =3.141592655x = .38096825064564107321692503847144e-11 S =3.141592660x =.19048412532282053660846251923572e-11 S =3.141592660x =.95242065286300884583045335519372e-12 S =3.141592747x =.47621035268040950056566904654351e-12 S =3.141592920x =.23810522883800767133458981166171e-12 S =3.141593613x =.11905250942336326914690870705313e-12 S =3.141590842x =.59526464702684973075452865921534e-13 S =3.141601925x =.29764072302022114258805759148830e-13 S =3.141690584x =.14882876066137216880187288510066e-13 S =3.141867896x =.74431176263713581056781550557580e-14 S =3.142577041x =.37282703764614497175334507121310e-14 S =3.148244452x =.18708286933869706927918743661583e-14 S =3.159548777x =.94868329805051379959966806332982e-15 S =3.2043673112. 利用利用韦达公式，构造出一类算法来计算π的近似值，并进行实际计算，评价算法效果。

解：i =1 pai =2.828427124746190097607 error =.313165528843603140856i =2 pai =3.061467458920718232305 error =.80125194669075006158e-1i =3 pai =3.121445152258052404243 error =.20147501331740834220e-1i =4 pai =3.136548490545939249172 error =.5044163043853989291e-2i =5 pai =3.140331156954752859018 error =.1261496635040379445e-2i =6 pai =3.141277250932772720971 error =.315402657020517492e-3i =7 pai =3.141513801144301048398 error =.78852445492190065e-4i =8 pai =3.141572940367091461665 error =.19713222701776798e-4i =9 pai =3.141587725277159716372 error =.4928312633522091e-5i =10 pai =3.141591421511200086144 error =.1232078593152319e-5i =11 pai =3.141592345570117831013 error =.308019675407450e-6i =12 pai =3.141592576584872625610 error =.77004920612853e-7i =13 pai =3.141592634338562821979 error =.19251230416484e-7i =14 pai =3.141592648776985437415 error =.4812807801048e-8i =15 pai =3.141592652386591008208 error =.1203202230255e-8i =16 pai =3.141592653288992575574 error =.300800662889e-9i =17 pai =3.141592653514592793026 error =.75200445437e-10i =18 pai =3.141592653570993021778 error =.188********e-10i =19 pai =3.141592653585093078979 error =.4700159484e-11i =20 pai =3.141592653588617918871 error =.1175319592e-11i =21 pai =3.141592653589499303264 error =.293935199e-12i =22 pai =3.141592653589719736536 error =.73501927e-13i =23 pai =3.141592653589774844854 error =.18393609e-13i =24 pai =3.141592653589788447537 error =.4790926e-14i =25 pai =3.141592653589788447537 error =.4790926e-14实验心得了解圆周率π的计算历程，并了解计算π的割圆术、韦达公式、级数法、拉马努金公式、迭代法。