demo_interpol.m

Interpolation routines in matlab

%
%  Demonstrate Polynomial Interpolation
%


disp(' Compute interpolation polynomial using three methods')
disp(' and compare the results ')


disp(' Hit return to continue')
pause

% generate problem data

% example interpolate the error function at x
x  = (1:0.2:3)';
f  = erf(x);
% plot the error function and the interpolating polynomials at z
z  = (0:0.01:4)';
fz = erf(z); 

% Number of points at which to interpolate
n  = size(x(:),1);
% Number of points at which to evaluate interpolation pol.
m  = size(z,1);




% Polynomial interpolation using monomials
V = ones(n,n);
for i = 2:n
  V(:,i) = V(:,i-1).*x;
end
a = V\f;


% Evaluate the interpolation polynomial using Horner's scheme
M = a(n)*ones(m,1);
for i = n-1:-1:1
   M =  a(i)*ones(m,1) + M.*z;
end


% Polynomial interpolation using Lagrange polynomials
L = ones(m,n);
for i = 1:n
    for j = 1:n
        if( i ~= j )
            L(:,i) = L(:,i).*(z-x(j))./(x(i)-x(j));
        end
    end
end


% Polynomial interpolation using Newton polynomials
a = NewtIntPol( x, f, 0 );
N = NewtIntPolEval( x, a, z );
   
   
   
   
subplot(2,2,1)
semilogy( z, fz );
xlabel('x') 
subplot(2,2,2)
semilogy( z, abs(fz-M) ); 
xlabel('x') 
subplot(2,2,3)
semilogy( z, abs(fz-L*f) );
xlabel('x') 
subplot(2,2,4)
semilogy( z, abs(fz-N) );
xlabel('x')
print -deps erf-interp1.eps  
pause
subplot(1,1,1)
semilogy( z, abs(N-M), '--');  hold on
semilogy( z, abs(N-L*f), '-'); hold off
legend('Newton-Monomial','Newton-Lagrange')
title('Error between interpolation polynomials')
xlabel('x')
print -deps erf-interp2.eps