matlab---三次样条插值

4444多项式插值与函数最佳逼近多项式插值与函数最佳逼近多项式插值与函数最佳逼近多项式插值与函数最佳逼近37373737（上机题）3333次样条插值函数：（1）编制求第一型3次样条插值函数的通用程序；（2）已知汽车门曲线型值点的数据如下：i012345xi012345yi2.513.304.044.705.225.54i678910xi678910yi5.785.405.575.705.80端点条件为8.0'0=y，2.0'10=y，用所编程序求车门的3次样条插值函数S（x），并打印出9,,1,0),5.0(⋯=+iiS。用matlab编写通用程序为：function[Sx]=Threch(X,Y,dy0,dyn)%X为输入变量x的数值%Y为函数值y的数值%dy0为左端一阶导数值%dyn为右端一阶导数值%Sx为输出的函数表达式n=length(X)-1;d=zeros(n+1,1);h=zeros(1,n-1);f1=zeros(1,n-1);f2=zeros(1,n-2);fori=1:n%求函数的一阶差商h(i)=X(i+1)-X(i);f1(i)=(Y(i+1)-Y(i))/h(i);endfori=2:n%求函数的二阶差商f2(i)=(f1(i)-f1(i-1))/(X(i+1)-X(i-1));d(i)=6*f2(i);endd(1)=6*(f1(1)-dy0)/h(1);d(n+1)=6*(dyn-f1(n-1))/h(n-1);%赋初值A=zeros(n+1,n+1);B=zeros(1,n-1);C=zeros(1,n-1);fori=1:n-1B(i)=h(i)/(h(i)+h(i+1));C(i)=1-B(i);endA(1,2)=1;A(n+1,n)=1;fori=1:n+1A(i,i)=2;endfori=2;nA(i,i-1)=B(i-1);A(i,i+1)=C(i-1);endM=A\d;symsx;fori=1:nSx(i)=collect(Y(i)+(f1(i)-(M(i)/3+M(i+1)/6)*h(i))*(x-X(i))...+M(i)/2*(x-X(i))^2+(M(i+1)-M(i))/(6*h(i))*(x-X(i))^3);digits(4);Sx(i)=vpa(Sx(i));endfori=1:ndisp('S(x)=');fprintf('%s(%d,%d)\n',char(Sx(i)),X(i),X(i+1));endS=zeros(1,n);fori=1:nx=X(i)+0.5;S(i)=Y(i)+(f1(i)-(M(i)/3+M(i+1)/6)*h(i))*(x-X(i))...+M(i)/2*(x-X(i))^2+(M(i+1)-M(i))/(6*h(i))*(x-X(i))^3;enddisp('S(i+0.5)');disp('iX(i+0.5)S(i+0.5)');fori=1:nfprintf('%d%.4f%.4f\n',i,X(i)+0.5,S(i));endEnd在运行窗口输入：X=[012345678910];Y=[2.513.304.044.705.225.545.785.405.575.705.80];Threch(X,Y,0.8,0.2)运行结果如下：S(x)=-0.005714*x^3-0.004286*x^2+0.8*x+2.51(0,1)S(x)=-0.01286*x^3+0.01714*x^2+0.7786*x+2.517(1,2)S(x)=-0.015*x^3+0.03*x^2+0.795*x+2.45(2,3)S(x)=-0.015*x^3+0.03*x^2+0.865*x+2.24(3,4)S(x)=0.03*x^3-0.51*x^2+3.08*x-0.86(4,5)S(x)=-0.135*x^3+1.965*x^2-9.09*x+18.74(5,6)S(x)=0.2925*x^3-5.73*x^2+36.96*x-72.9(6,7)S(x)=-0.1475*x^3+3.51*x^2-27.55*x+76.87(7,8)S(x)=0.0025*x^3-0.09*x^2+1.118*x+1.11(8,9)S(x)=0.04625*x^3-1.271*x^2+11.72*x-30.53(9,10)S(i+0.5)iX(i+0.5)S(i+0.5)10.50002.908221.50003.680232.50004.390643.50004.991954.50005.406365.50005.725676.50005.596687.50005.437298.50005.6416109.50005.7383