(1.6)-2023數(shù)值實驗-練習(xí)5

練習(xí)S=0;fori=1:100

S=S+i;endu=1;fori=1:10ifi~=3

u=u*i;endend第四章：多項式插值練習(xí)一、Lagrange插值多項式functionL=lar_fun(X,Y,x)n=length(X);L=0;fori=1:n

u=1.0;

forj=1:nifj~=iu=u.*(x-X(j))./(X(i)-X(j));endendL=L+u*Y(i);endclear,clcX=[1,2,3,4,5];Y=[1,4,7,8,6];x=2.5;L=lar_fun(X,Y,x)u=1:0.01:5;v=lar_fun(X,Y,u);plot(u,v,X,Y,'ro')主程序functionA=D_fun(X,Y)n=length(X);A=zeros(n);A(:,1)=X;A(:,2)=Y;forj=3:n+1fori=j-1:nA(i,j)=(A(i,j-1)-A(i-1,j-1))/(X(i)-X(i-(j-2)));endend1、差商表二、Newton插值多項式functionf=N_fun(X,Y)n=length(X);symsx;A=D_fun(X,Y);co=diag(A,1);f=co(1);fori=2:nu=1.0;fork=1:i-1u=u*(x-X(k));endf=f+u*co(i);endf=expand(f);二、Newton插值多項式2、Newton插值多項式函數(shù)子程序clear,clcX=[1,2,3,4,5];Y=[1,3,4,2,1];f=N_fun(X,Y)g=polyfit(X,Y,4)二、Newton插值多項式3、Newton插值多項式主程序例如：已知：X=[1,2,3,4,5];Y=[1,4,7,8,6];求：

三次多項式插值多項式三、插值MATLAB命令：polyfit（含擬合）clear,clcX=[1,2,3,4,5];Y=[1,4,7,8,6];p4=polyfit(X,Y,4)xx=1:0.01:5;y4=polyval(p4,xx);plot(xx,y4,X,Y,'r*')legend('4thInter','datapoint')clear,clcfplot(@(x)1./(1+25.*x.*x),[-1.,1.])x=-1:0.2:1;y=1./(1+25*x.^2);holdonplot(x,y,'r*')p=polyfit(x,y,10);xx=-1:0.01:1;yy=polyval(p,xx);plot(xx,yy)四、Runge現(xiàn)象functionm=s1(x,y,y0,yn)n=length(x);A=diag(2*ones(1,n));d=zeros(1,n);h=diff(x);mu=zeros(1,n-1);mu(1)=1;lambda=zeros(1,n-1);lambda(n-1)=1;fori=2:n-1mu(i)=h(i-1)/(h(i)+h(i-1));endlambda=1-mu;A=A+diag(mu,-1)+diag(lambda,1);fori=2:n-1d(i)=6/(h(i-1)+h(i))*((y(i+1)-y(i))/h(i)-(y(i)-y(i-1))/h(i-1));endd(1)=6/h(1)*((y(2)-y(1))/h(1)-y0);d(n)=6/h(n-1)*(yn-(y(n)-y(n-1))/h(n-1));m=A\d';五、第一類三次樣條插值多項式的M系數(shù)clear,clcx=[0,1,2,3];y=[0,0.5,2,1.5];y0=0.2;yn=-1；n=length(x);m=s1(x,y,y0,yn);

h=diff(x);symsxxS=zeros(n-1,4);fori=1:n-1a=(y(i+1)-y(i))/h(i)-h(i)*(m(i+1)-m(i))/6;

b=(y(i)/h(i)-h(i)*m(i)/6)*x(i+1)+(-y(i+1)/h(i)+h(i)*m(i+1)/6)*x(i);s=expand(m(i)*(x(i+1)-xx)^3/6/h(i)+m(i+1)*(xx-x(i))^3/6/h(i)+a*xx+b);S(i,:)=sym2poly(s);endS三次樣條插值多項式主程序plot(x,y,'r*'),holdonxx1=0:0.01:1;xx2=1:0.01:2;xx3=2:0.01:3;yy1=polyval(S(1,:),xx1);