% Applying Lagrange and cubic spline interpolation on Ruddy duck example
% See Example 3 on pages 149-152 of Burden and Faires
% Kartik's MATLAB session
% September 17, 2007.
% To run this demo, download this file into your MATLAB directory.
% Start MATLAB and at the MATLAB prompt >> type
% ruddyduck
% You should get a plot as output

% (x,f(x)) values from Table 3.17
x = [0.9 1.3 1.9 2.1 2.6 3.0 3.9 4.4 4.7 5.0 6.0 7.0 8.0 9.2 10.5 11.3 11.6 12.0 12.6 13.0 13.3]';
y = [1.3 1.5 1.85 2.1 2.6 2.7 2.4 2.15 2.05 2.1 2.25 2.3 2.25 1.95 1.4 0.9 0.7 0.6 0.5 0.4 0.25]';

% Generate a grid of points
xgrid = 0.9:0.01:13.3;
lgrid = length(xgrid);

% Estimate the parameters for the natural spline using Kartik's MATLAB code
spar = naturalcubicsplineInt(x,y);

for i=1:lgrid,
   % Find the Lagrange interpolated value at xgrid(i) 
   yLgrid(i) = lagrangeINT(x,y,xgrid(i));
   index = find(x-xgrid(i) > 0);
   if isempty(index) == 0, 
      index = index(1)-1;
   else 
      index = 20; 
   end 
   val = (xgrid(i)-x(index));
   % Find the natural spline interpolated value at xgrdi(i)
   ySgrid(i) = spar(4*(index-1)+1) + spar(4*(index-1)+2)*val + spar(4*(index-1)+3)*(val^2) + spar(4*(index-1)+4)*(val^3);
end

figure(1);
% Plot the values. Compare this plot with Figures 3.12 and 3.13 in Burden
% and Faires
plot(xgrid,[yLgrid(:) ySgrid(:)]);
legend('Lagrange interpolation', 'Cubic spline interpolation');
xlabel('x');
ylabel('f(x)');
title('Approximating the top profile of the Ruddy duck');