MATLAB: Plotting a smooth curve from points

smooth curves from discrete points with zero slope

I'm trying to plot a smooth line for 6 points in such a way that slope at each is zero. I have successfully achieved it for middle points using 'pchip' but I also want it for extreme points i.e. x=0 and x=15 which I am unable to do while using 'pchip'.
Here is my code
clear
clc
x = [0;3;6;9;12;15];
y = [0;1.9190;-3.2287;3.5133;-2.6825;1];
xi = linspace(min(x), max(x), 150);                     % Evenly-Spaced Interpolation Vector
yi = interp1(x, y, xi, 'pchip');
figure
plot(x, y, 'b')
hold on
plot(xi, yi, '-r')
hold off
grid
xlabel('X')
ylabel('Y')
legend('Original Data', 'Interpolation', 'Location', 'NE')

Best Answer

Direct analytic method using piecewise cublic polynomial. The curve is first-order differentiable, but not second order differentiable (as with sublic spline or result with my first answer using BSFK) 
x = [0;3;6;9;12;15];
y = [0;1.9190;-3.2287;3.5133;-2.6825;1];
[xx,is] = sort(x(:));
yy = y(is);
yy = yy(:);
dx = diff(xx);
dy = diff(yy);
y0 = yy(1:end-1);
n = numel(xx)-1;
coefs = [-2*dy./(dx.^3), 3*dy./(dx.^2), 0*dy, y0];
pp = struct('form', 'pp',...
    'breaks', xx(:)',...
    'coefs', coefs,...
    'pieces', n, ...
    'order', 4,...
    'dim', 1);
figure
xi = linspace(min(x),max(x));
yi = ppval(pp,xi);
plot(x,y,'b-o',xi,yi,'r');
xlim([min(x),max(x)])
grid on
Propably this is how John generates his curve
x = [0 1 1.5 2 4];
y = x;
Related Question