三次樣條差值擬合車門曲線分析

1、數(shù)學(xué)實驗（三次樣條）- 1 -數(shù)學(xué)實驗（三次樣條插值）實驗 1：某汽車制造商用三次樣條插值設(shè)計車門的曲線，其中一段的數(shù)據(jù)如下：xi012345678910yi0.00.79 1.532.192.713.033.272.893.06 3.193.29yi0.80.2用三次樣條插值求 S10 ( x) ，用軟件繪制 S10 ( x) 的圖像，即車門的曲線。1、計算 S10 (x) ：程序：/ splineaaaa.cpp :定義控制臺應(yīng)用程序的入口點。/* 三次樣條差值 */*第一步，利用差商，代替導(dǎo)數(shù)，求差商；/*第二步，利用追趕法求解三對角方程組，得到Mi;/*第三步，將求得值帶入三次樣條函數(shù)

2、，求得S(x);#include #include #include #defineN 10doublex N + 1, fxN + 1, hN, HN, fN, a N + 1, bN + 1, cN+ 1, MN + 1, beta N +1, yN + 1,sN; /定義變量數(shù)組；doubletiaojian1, tiaojian2;/邊界條件；求插商；void chashang()inti;for(i = 0; i =N - 1; i+)hi = xi + 1 - xi;for(i = 0; i =N - 1; i+)fi = (fxi + 1 - fxi) / (xi + 1 - x

3、i);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);- 2 -for (i = 1; i =N - 1; i+)Hi = pow(hi, 2);/ 求 hi的平方；利用追趕法解三對角方程組前期初始值；a N = 1;/aN相當(dāng)于 N;b0 = 1;/ 0;c0 = 6.*(f0 - tiaojian1) / h0;/ 相當(dāng)于 d0;c N = 6.*(tiaojian2 - fN - 1) / hN - 1;/ 相當(dāng)于 dN

4、;追趕法計算彎矩向量 Mi; void Zhuigan()inti;beta0 = b0 / 2.;y0 = c0 / 2.;for (i = 1; i =N - 1; i+)betai = bi / (2. - ai * betai - 1);/ 計算 i;for (i = 1; i = 0; i-)/ 逆序求解計算MiMi = yi - betai * Mi + 1;輸出三次樣條差值函數(shù) S1(x)S10(x) void Prints()inti;for (i = 0; iN; i+)printf(s%d=%lf*(%lf-x)3+%lf*(x-%lf)3+%lf*(%lf-x)+%lf*

5、(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);主函數(shù)調(diào)用；void main()inti;x0= 0; fx0 = 0.0;x1= 1; fx1 = 0.79;x2= 2; fx2 = 1.53;x3= 3; fx3 = 2.19;- 3 -x4 = 4; fx4 = 2.71;x5 = 5; fx5 = 3.03;x6 = 6; fx6 = 3.27;x7 = 7; fx7 = 2.89

6、;x8 = 8; fx8 = 3.06;x9 = 9; fx9 = 3.19;x10 = 10; fx10 = 3.29;tiaojian1 = 0.8;tiaojian2 = 0.2;chashang();printf( 追趕法計算所得 Mi:n);Zhuigan();for (i = 0; i =N; i+)printf( %fn, Mi);printf( 三次插值多項式 si=n);Prints();結(jié)果：追趕法計算所得Mi:-0.6038191.1476380.6032690.6392880.3795800.362392-0.149149-0.1857950.2623290.03647

7、70.281761三次插值多項式si=- 4 -2、繪制車門曲線 S10 ( x) ：matlab 程序）：clcclearsymsxsymss ;symssj ;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;- 5 -x=x1,x2,x3,x4,x5,x6,x7,x8,x9,x10;S=s1,s2,s3,s4,s5,s6,s7,s8,s9,s10;QX=plot(x,S,r)set(QX,LineWidth,1.85);holdonplot(x11,s11,k*)holdonWQ=plot(x11,sj,b );hh=ylab