三次样条差值拟合车门曲线.doc

数学实验（三次样条） 数学实验（三次样条插值）实验1： 某汽车制造商用三次样条插值设计车门的曲线，其中一段的数据如下：0123456789100.00.791.532.192.713.033.272.893.063.193.290.80.2用三次样条插值求，用软件绘制的图像，即车门的曲线。1、 计算：程序：/ splineaaaa.cpp : 定义控制台应用程序的入口点。/*三次样条差值*/*第一步，利用差商，代替导数，求差商；/*第二步，利用追赶法求解三对角方程组，得到Mi;/*第三步，将求得值带入三次样条函数，求得S(x);#include#include#include#define N 10double xN + 1, fxN + 1, hN, HN, fN, aN + 1, bN + 1, cN + 1, MN + 1, betaN + 1, yN + 1,sN;/定义变量数组；double tiaojian1, tiaojian2; /边界条件；/求插商；void chashang() int i;for (i = 0; i = N - 1; i+)hi = xi + 1 - xi;for (i = 0; i = N - 1; i+)fi = (fxi + 1 - fxi) / (xi + 1 - xi); for (i = 1; i = N - 1; i+)ai = hi - 1 / (hi - 1 + hi);bi = hi / (hi - 1 + hi);ci = 3 * (ai * fi - 1 + bi * fi);for (i = 1; i = N - 1; i+)Hi = pow(hi, 2);/求hi的平方；/利用追赶法解三对角方程组前期初始值；aN = 1;/aN相当于N; b0 = 1;/0;c0 = 6.*(f0 - tiaojian1) / h0;/相当于d0;cN = 6.*(tiaojian2 - fN - 1) / hN - 1;/相当于dN;/追赶法计算弯矩向量Mi;void Zhuigan()int i;beta0 = b0 / 2.;y0 = c0 / 2.;for (i = 1; i = N - 1; i+)betai = bi / (2. - ai * betai - 1); /计算i;for (i = 1; i = 0; i-) /逆序求解计算Mi Mi = yi - betai * Mi + 1;/输出三次样条差值函数S1(x)S10(x)void Prints()int i;for (i = 0; iN; i+)printf(s%d=%lf*(%lf-x)3+%lf*(x-%lf)3+%lf*(%lf-x)+%lf*(x-%lf)n,i+1, Mi / (6.*hi), xi + 1, Mi + 1 / (6.*hi), xi, fxi / hi - Mi * Hi / 6., xi + 1, fxi + 1 / hi - Mi + 1 * Hi / 6., xi);/主函数调用；void main()int i;x0 = 0; fx0 = 0.0;x1 = 1; fx1 = 0.79;x2 = 2; fx2 = 1.53;x3 = 3; fx3 = 2.19;x4 = 4; fx4 = 2.71;x5 = 5; fx5 = 3.03;x6 = 6; fx6 = 3.27;x7 = 7; fx7 = 2.89;x8 = 8; fx8 = 3.06;x9 = 9; fx9 = 3.19;x10 = 10; fx10 = 3.29;tiaojian1 = 0.8;tiaojian2 = 0.2;chashang();printf(追赶法计算所得Mi:n);Zhuigan();for (i = 0; i = N; i+)printf(%fn, Mi);printf(三次插值多项式si=n);Prints();结果：追赶法计算所得Mi:-0.6038191.1476380.6032690.6392880.3795800.362392-0.149149-0.1857950.2623290.0364770.281761三次插值多项式si=s1=-0.100636.*(1.000000-x1).3+0.191273.*x1.3+0.790000.*x1;s2=0.191273.*(2.000000-x2).3+0.100545.*(x2-1.000000).3+0.598727.*(2.000000-x2)+1.429455.*(x2-1.000000);s3=0.100545.*(3.000000-x3).3+0.106548.*(x3-2.000000).3+1.429455.*(3.000000-x3)+2.083452.*(x3-2.000000);s4=0.106548.*(4.000000-x4).3+0.063263.*(x4-3.000000).3+2.083452.*(4.000000-x4)+2.646737.*(x4-3.000000);s5=0.063263.*(5.000000-x5).3+0.060399.*(x5-4.000000).3+2.646737.*(5.000000-x5)+2.969601.*(x5-4.000000);s6=0.060399.*(6.000000-x6).3+-0.024858.*(x6-5.000000).3+2.969601.*(6.000000-x6)+3.294858.*(x6-5.000000);s7=-0.024858.*(7.000000-x7).3+-0.030966.*(x7-6.000000).3+3.294858.*(7.000000-x7)+2.920966.*(x7-6.000000);s8=-0.030966.*(8.000000-x8).3+0.043722.*(x8-7.000000).3+2.920966.*(8.000000-x8)+3.016278.*(x8-7.000000);s9=0.043722.*(9.000000-x9).3+0.006080.*(x9-8.000000).3+3.016278.*(9.000000-x9)+3.183920.*(x9-8.000000);s10=0.006080.*(10.000000-x10).3+0.046960.*(x10-9.000000).3+3.183920.*(10.000000-x10)+3.243040.*(x10-9.000000);2、绘制车门曲线：（matlab程序）：clcclearsyms xsyms s;syms sj;x11=0:1:10;s11=0.0;0.79;1.53;2.19;2.71;3.03;3.27;2.89;3.06;3.19;3.29;pp=csape(x11,s11,complete,0.8,0.2);sj=ppval(pp,x11);x1=0.0001:0.01:1;x2=1:0.01:2;x3=2:0.01:3;x4=3:0.01:4;x5=4:0.01:5;x6=5:0.01:6;x7=6:0.01:7;x8=7:0.01:8;x9=8:0.01:9;x10=9:0.01:10;s1=-0.100636.*(1.000000-x1).3+0.191273.*x1.3+0.790000.*x1;s2=0.191273.*(2.000000-x2).3+0.100545.*(x2-1.000000).3+0.598727.*(2.000000-x2)+1.429455.*(x2-1.000000);s3=0.100545.*(3.000000-x3).3+0.106548.*(x3-2.000000).3+1.429455.*(3.000000-x3)+2.083452.*(x3-2.000000);s4=0.106548.*(4.000000-x4).3+0.063263.*(x4-3.000000).3+2.083452.*(4.000000-x4)+2.646737.*(x4-3.000000);s5=0.063263.*(5.000000-x5).3+0.060399.*(x5-4.000000).3+2.646737.*(5.000000-x5)+2.969601.*(x5-4.000000);s6=0.060399.*(6.000000-x6).3+-0.024858.*(x6-5.000000).3+2.969601.*(6.000000-x6)+3.294858.*(x6-5.000000);s7=-0.024858.*(7.000000-x7).3+-0.030966.*(x7-6.000000).3+3.294858.*(7.000000-x7)+2.920966.*(x7-6.000000);s8=-0.030966.*(8.000000-x8).3+0.043722.*(x8-7.000000).3+2.920966.*(8.000000-x8)+3.016278.*(x8-7.000000);s9=0.043722.*(9.000000-x9).3+0.006080.*(x9-8.000000).3+3.016278.*(9.000000-x9)+3.183920.*(x9-8.000000);s10=0.006080.*(10.000000-x10).3+0.046960.*(x10-9.000000).3+3.183920.*(10.000000-x10)+3.243040.*(x10-9.000000);x=x1,x2,x3,x4,x5,x6,x7,x8,x9,x10;S=s1,s2,s