矩阵与数值分析(7)

plot(X,f) P=

x*(11153755408641612500000*x^9 19582893765599948750000*x^7 6457490747185549971250*x^5 452380176499959382000*x^3

+ + -

-

55768777028538456250000*x^8 256281087074564729875000*x^6 1150585140486343857745625*x^4 2327269296084898261641325*x^2

+ + + +

4715634276489610818*x - 1414847701376318005254584)

画图程序： X=0:0.1:1; x1=0:0.0001:1; f1=sin(x1*pi); syms x x0=0:0.001:1; f=sin(pi*X);

O=Nweton_xhazhi(X,f); x=x0; Y=eval(O); plot(x0,Y,'r'); hold on plot(x1,f1);

1.210.80.60.40.20-0.200.10.20.30.40.50.60.70.80.91

由上图可知：红色图形是利用插值多项式画出的，蓝色图像是利用原函数画出的，将两者对比我们可以得出结论：该插值多项式具有收敛性

11：对函数f(x)?1,x?[?5,5]，取不同的节点数n,用等距节点作Lagrange1?x2插值，观察Runge现象。

Lagrange插值通用程序：

%X为节点，y为节点对应的函数值，x0为插值节点 function Lagrange(X,y,x0) syms x; [~,n]=size(X); sum=0; for i=1:n

sum=sum+y(i)*L(X,i); end x=x0; p=eval(sum); plot(x0,p,'r-') hold on

plot(X,y,'b-');

function l=L(X,k) syms x; sum1=1; sum2=1; [~,n]=size(X); if (k==1) for i=k+1:n

sum1=sum1*(x-X(i)); sum2=sum2*(X(k)-X(i)); l=sum1/sum2; end else

for i=1:k-1

sum1=sum1*(x-X(i)); sum2=sum2*(X(k)-X(i)); end for i=k+1:n

sum1=sum1*(x-X(i)); sum2=sum2*(X(k)-X(i)); end

l=sum1/sum2; end

取不同的节点数n,用等距节点作Lagrange插值，观察Runge现象:

当n=10;

X=-5:0.1: 5;f=1./(1+X.^2); x0=-5:5:5;Lagrange(X,f,x0) x1=-5:0.0001:5; y1=1./(1+x1.^2); plot(x1,y1)

1.41.210.80.60.40.20-5-4-3-2-1012345

当n=20;

X=-5:0.1:5; f=1./(1+X.^2); x0=-5:0.5:5; Lagrange(X,f,x0) x1=-5:0.0001:5; y1=1./(1+x1.^2); plot(x1,y1)

1.41.210.80.60.40.20-5-4-3-2-1012345

当n=50;

X=-5:0.1:5; f=1./(1+X.^2); x0=-5:0.2:5; Lagrange(X,f,x0) x1=-5:0.0001:5; y1=1./(1+x1.^2); plot(x1,y1)

1.41.210.80.60.40.20-5-4-3-2-1012345

n=100; X=-5:0.1:5; f=1./(1+X.^2); x0=-5:0.1:5;