Mathematica圆周率的计算.doc

圆周率的计算实验目的通过使用数学方法，结合数学软件的使用掌握计算圆周率的一些方法 更好的理解圆周率的数学意义 熟练地使用数学软件实验原理不同的半径、不同的大小的圆，圆周率的值总是一个比值比较稳定的数值1、祖冲之的圆周率N22/7, 10 3.142857143N355/113, 10 3.1415929202、无理数的最佳分数逼近a = Pi; e0 = 1; list = ;op = Floorq * a + 0.01; e = Absp/ q a* q;Ife e0, AppenTolist, p/ q; 0 = , q, 1, 100 000;list3,22/7,333/106,355/113,103993/33102,10434/33215,20341/66317,31269/99532a = Pi; e0 = 1; list = ;op = Floorq * a + 0.01; e = Absp - q * a;Printp=, p, ;q=, q, ;e=, e / N;Ife e0, AppenTolist, p/q; 0 = , q, 1, 10;listp=3;q=1;e=0.141593p=6;q=2;e=0.2315p=9;q=3;e=0.42477p=12;q=4;e=0.566371p=15;q=5;e=0.707963p=1;q=6;e=0.49556p=22;q=7;e=0.005142p=25;q=;e=0.132741p=2;q=9;e=0.274334p=31;q=10;e=0.4159273,22/73、乐音的频率比k = 2.0(1/ 12）;music = 1.0, k2, k4, k5, k7, k9, k11, k121., 1.12246, 1.25992, 1.3344, 1.4931, 1.6179, 1.775, 2.freq1 = 1, 2(2/12, 2(4/12), 2(5/12L, 2(7/12), 2(9/12), 2(11/ 12), 2freq2 = 1, 9 / , 5 / 4, 4 / 3, 3 / 2, 5 / 3, 17 / 9, 21, 21/6, 21/3, 25/12, 27/12, 23/4, 211/12, 21,9/,5/4,4/3,3/2,5/3,17/9, 2freq = freq2;m = 512;PlaySin2 Pi * m * t * freq2, t, 0, 0., PlayRange-0, 1SounSampleSounFunctionFunctionPlayTime3,Blockt = 0. + 0.000125 PlayTime3, HSin2p mt freq2 - 0.52., 6400, 000m = 512;freq1 = 1, 2(2 / 12), 2(4 / 12), 2(5 / 12), 2(7 / 12), 2(9 / 12), 2(11 / 12), 2;freq2 = 1, 9 / , 5 / 4, 4 / 3, 3 / 2, 5 / 3, 17 / 9, 2;freq = freq2;Playmusicsong_ := DoPlaySin2Pi* m* t* freqsongi,t, 0, 0.8, PlayRange-0, 1, i, 1, Lengthsong;music = 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;PlaymusicmusicPlaysongsong_ :=Dox = songi, 1;w = Whichx 0, freq-x / 2, x 10, freqx - 10 * 2, True, freqx;y = songi, 2;PlaySin2 Pi* m* t* w, t, 0, 0.4 y, PlayRange-0, 1, i, 1, Lengthsongsong2 = -3, 4, -5, 3, -6, 1, 1, 3, 2, 1, -6, 1,1, 1, -5, 2, 5, 3, 11, 1, 6, 1, 5, 1, 3, 1, 5, 1, 2, 8,2, 3, 3, 1, -7, 2, -6, 2, -5, 3, -6, 1, 1, 2, 2, 2,-3, 2, 1, 2, -6, 1, -5, 1, -6, 1, 1, 1, -5, 8,3, 3, 5, 1, -7, 2, 2, 2, -6, 1, 1, 1, -5, 3, -3, 1,-3, 1, -5, 1, -3, 2, -5, 1, -6, 1, -7, 1, 2, 1, -6, 4,1, 2, 1, 1, 2, 1, 5, 2, 5, 1, 3, 1, 2, 2, 3, 1, 2, 1, 1, 2,-6, 1, -5, 1, -3, 4, 1, 4, -6, 1, 1, 1, -6, 1, -5, 1,-3, 1, -5, 1, -6, 1, 1, 1, -5, 4;Playsongsong2song1 = 3, 1, 5, 1, 6, 1, 6, 0.5, 5, 0.5, 6, 1, 3, 1,2, 2, 3, 1, 5, 1, 6, 1, 6, 0.5, 5, 0.5, 6, 1, 3, 3,3, 1, 5, 1, 6, 1, 6, 0.5, 5, 0.5, 6, 1, 3, 1, 2, 2,5, 1, 3, 1, 2, 0.5, 3, 0.5, 2, 0.5, 1, 0.5, 2, 1, -6, 3,-6, 1, 2, 2, 5, 1, 3, 3, 2, 0.5, 1, 0.5, -6, 4,5, 1, 3, 1, 2, 0.5, 3, 0.5, 2, 0.5, 1, 0.5, 2, 1, -6, 3,5, 1, 3, 1, 2, 0.5, 3, 0.5, 2, 0.5, 1, 0.5, 2, 1, 6, 3;Playsongsong14、单位圆的面积等于fx_ := ;fig = Plotfx, 0, x, 0, 1, AspectRatio- 1n = 10; fig1 = ; fig2 = ;DoAppendTofig1, Line1 / n* i, 0, 1/ n*i, f1/ n*i,AppendTofig2, Line1 / n* (i 1), f1 / n*i, 1 / n*i, f1 / n*i,i, 1, n;Showfig, Graphicsfig1, Graphicsfig2fig3 = ; fig4 = ;DoAppendTofig3, Line1 / n*i,0, 1/ n*i,f1/ n*（i 1）,AppendTofig4, Line1 / n*(i 1), f1 / n * (i 1), 1/ n*i, f1/ n*(i 1),i, 1, n; Showfig, Graphicsfig3, Graphicsfig4os1 = N4 * Sumfk / 10m / 10m, k, 1, 10m;s2 = N4 * SumfHk - 1L / 10m / 10m, k, 1, 10m;Prints1, s2, Hs1 + s2L / 2, m, 1, 4fig5 = ;DoAppendTofig5, Line1 / n * Hi - 1L, f1 / n * Hi - 1L, 1 / n * i, f1 / n * i,i, 1, n; Showfig, Graphicsfig1, Graphicsfig5om = 10t;s3 = N4 * (f0 + f1) / 2 / m + Sumfk / m / m, k, 1, m - 1L, 20;s4 = N4 * (f0 + f1) / m + 2 * Sumfk / m / m, k, 1, m - 1 +4 * Sumf(k + 1 / 2) / m / m, k, 0, m - 1L / 6, 20; Prints3, s4, t, 3, 43.141555466911027637, 3.141575191227769063.1415914776113222011, 3.1415924912201991625、级数展开法Tx_, n_ := Sum（-1）（k 1） * x(2 k 1) / (2 k 1), k, 1, n;N4*T1, 10 000, 20NPi, 203.1414926535900432353.1415926535979323462643332795024220.N4 * (4 T1 / 5, 100 - T1 / 239, 40), 150NPi, 1503.14159265358979323853.141592653589793238462643383279502884197169399375105820974944592307816406286208998.628034825342117067982148086513282306647093844609550582231725113323563.141592653589793238462643383279502884197169399375105820974944592307816406286208998.628034825342117067982148086513282306647093844609550582231725359408136、蒙特卡罗法（随机模拟法）fig = Plot , x, 0, 1, AspectRatio-1, PlotStyle- RGBColor1, 0, 0;fig0 = GraphicsLine0, 1, 1, 1, 1, 0;Showfig, fig0n = 1000; fig1 = ; temp = 0;Dox = Random; y = Random;AppendTofig1, Pointx, y;Ifx2+ y2=1, temp +, , i, 1, n;Showfig, fig0, Graphicsfig1Ntemp * 4 / nn = 10 000; p = ;om = 0; o