Question

Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following interpolating polynomials, and use MATLAB to graph both the interpolating polynomials and the data points: a) The piecewise linear Lagrange interpolating polynomialx) b) The piecewise quadratic Lagrange interpolating polynomial q(x) c) Newton's divided difference interpolation pa(x) of degree s 4

Answer 1

P, P) lundtiay limea Aunt 与十(M2) ( 1) -0.25 1-25-025(a-1 二 x-2)に7-1)

40 32

1.25-5 25 = 125-1 ·25 Or -2-1 ple )1 035 2 .ttrom)'가 Ljn シ
Matlab Code

x=[-2 0 1 2 4];
y=[5 1 1.25 1 -7];
plot(x,y)

Output

4 -2 -4 -6 -8 -2 4

(a)Linear Lagrange interpolation

x=linspace(-2,4);
if -2<=x<0
y=6+x/2;
end
if 0<=x<1
y=1+x/8;
end
if 1<=x<2
y=1.5-0.25*x;
end
if 2<=x<4
y=9-4*x;
end
plot(x,y)

Output

20 15 10 -5 -10 -2 4

(b)

x=linspace(-2,4);
if -2<=x<=1
y=(x.^2+2*x)*(1.25/3)+(5/6)*(x.^2-x)-(1/2)*(x.^2+x-2);
end
if 1<=x<=4
y=(-7/3)*(x-1).*(x-2)+(1.25/3)*(x-2).*(x-4)-(1/2)*(x-1).*(x-4);
end
plot(x,y)

Output

-5 -10 -15 20 -25 -30 -2 -1

(c)

x=linspace(-2,4);
y=5-2*(x+2)+0.25*x.*(x+2)-x.*(x-2).*(x+2)*(0.09375)-x.*(x+2).*(x-1).*(x-2)*(0.12326);
plot(x,y)

0 -5 -10 -15 20 -25 L -2 4 -1