实验样条插值法

1、东南大学数学实验学号22009316 成绩实验内容：样条插值法 实验目的三次样条插值函数的求解及应用二 实验内容与要求已知某平原地区的一条公路经过如下坐标点，请用不同的插值方法绘出这条公路（不考虑公路的宽度）。对于表中给出的数据，编程计算三次样条插值函数估计的公路长度。X(m)Y(m)X(m)Y(m)X(m)Y(m)X(m)Y(m)0802021603202004203343064212182340184380328504723020036018836033470422482083722003403468048268212382202320356906627121039024031436012

2、080280200416246280392148120290196430280240390170121300188478296200400180138312186440308思路：由于道路曲折，出现了一个 x 值对应 2 个 y 值的情况，所以将道路分为 2 个函数进行拟合。命令clear% 全部采样点X=0,30,50,70,80,90,120,148,170,180,202,212,230,248,268,271,280,290,300,312,320,340,360, 372,382,390,416,430,478,440,420,380,360,340,320,314,280,240,

3、200;Y=80,64,47,42,48,66,80,120,121,138,160,182,200,208,212,210,200,196,188,186,200,184,188, 200,202,240,246,280,296,308,334,328,334,346,356,360,392,390,400;% 第一段n1=29; X1=0,30,50,70,80,90,120,148,170,180,202,212,230,248,268,271,280,290,300,312,320,340,360,372,382,390,416,430,478; Y1=80,64,47,42,48,6

4、6,80,120,121,138,160,182,200,208,212,210,200,196,188,186,200,184,188,200,202,240,246,280,296;Z1=0:1:478;m1=length(Z1);g1=-8/15,1/3;lmd1(1)=1; mu1(n1)=1; for i=1:n1-1h1(i)=X1(i+1)-X1(i);endd1(1)=6*(Y1(2)-Y1(1)/h1(1)-g1(1)/h1(1);d1(n1)=6*(gfor i=2:n1-11(2)-(Y1(n1)-Y1(n1-1)/-1)/-1);lmd1(i)=h1(i)/(h1(i-

5、1)+h1(i); mu1(i)=1- lmd1(i);d1(i)=6*(Y1(i+1)-Y1(i)/h1(i)-(Y1(i)-Y1(i-1)/h1(i-1)/(h1(i-1)+h1(i);end A1(1,1)=2;A1(1,2)=1;A1(n1,n1-1)=1;A1(n1,n1)=2;for i=2:n1-1 A1(i,i-1)=mu1(i); A1(i,i)=2;A1(i,i+1)=lmd1(i); end M1=inv(A1)*d1; for k=1:m1for i=1:n1-1if Z1(k)=X1(i)&Z1(k)=X2(i)&Z2(k)=X2(i+1)S2(k)=M2(i)*(X2(i+1)-Z2(k)3/(6*h2(i)+M2(i+1)*(Z2(k)-X2(i)3/(6*h2(i)+(Y2(i)-M2(i)*h2(i)2/6)*(X2(i+1)-Z2(k)/h2(i)+(Y2(i+1)-M2(i+1)*h2(i)2/6)*(Z2(k)-X2(i)/h2(i);break end end endplot(Z1,S1,Z2,S2,X,Y,o) % 绘图% 估算公路长度L=0;for t=1:477L=L+(Z1(t)-Z1(t+1)2+(S1(t)-S1(t+1)2)0.5;endfor t=1:277L=L+(Z2(t)