f=inline('1./(1+12*x.^2)');
%% Example 1: Seven points
%First define the points for the polynomial:
xp=linspace(-1,1,7);
yp=f(xp);
%Find the coefficients using the Vandermonde matrix:
V=vander(xp);
C=V\yp';
xx=linspace(-1,1,200); %Pts for f, P
yy1=f(xx);
yy2=polyval(C,xx);
plot(xp,yp,'r*',xx,yy1,xx,yy2);
legend('Data','f(x)','Polynomial')
title('Interpolation using 7 points, Degree 6 poly')
%% Example 2: 15 Points
%First define the points for the polynomial:
xp=linspace(-1,1,15);
yp=f(xp);
%Find the coefficients using the Vandermonde matrix:
V=vander(xp);
C=V\yp';
xx=linspace(-1,1,200); %Pts for f, P
yy1=f(xx);
yy2=polyval(C,xx);

figure(2)
plot(xp,yp,'r*',xx,yy1,xx,yy2);
title('Interpolation using 15 points, Degree 14 Poly')
legend('Data','f(x)','Polynomial')
%% Example 3: 30 Points, degree 9
%First define the points for the polynomial:
xp=linspace(-1,1,30);
yp=f(xp);
%Find the coefficients using vander1.m (written by us)
V=vander1(xp',10);
C=V\yp';
xx=linspace(-1,1,200); %Pts for f, P
yy1=f(xx);
yy2=polyval(C,xx);
figure(3)
plot(xp,yp,'r*',xx,yy1,xx,yy2);
title('Interpolation using 30 points, Degree 10 Poly')
legend('Data','f(x)','Polynomial')