Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) for given data and then finding area

Hello all

    I am trying to do Piecewise Cubic Hermite Interpolation on the data given below and then I want to get the area covered by the polynomials with x axis. I think, I am misunderstanding the meaning of coefficients returned by pchip command, but not sure. Does anyonw know what could be the problem?

x = [5.8808 6.5137 7.1828 7.8953];

y = [31.2472 33.9977 36.7661 39.3567];

pp = pchip(x,y)

If I see see pp,it gives pp as

   form: 'pp'

    breaks: [5.8808 6.5137 7.1828 7.8953]

     coefs: [3x4 double]

    pieces: 3

     order: 4

       dim: 1

and pp.coefs are

-0.0112 -0.1529 4.4472 31.2472

-0.3613 0.0884 4.2401 33.9977

-0.0422 -0.3028 3.8731 36.7661

I think these are the polynomials representing the three intervals

[5.8808:6.5137],

[6.5137:7.1828],

[7.1828:7.8953]

But when I find the y values corresponding to x values using these polynomials, it gives wrong values.

It gives negative y values for second polynomial . Even third polynomial do not seem to satisfy the points.

I used these commands for obtaining the values

For example:- (for second polynomial)

xs = linspace(6.5137, 7.1828, 200);

y = polyval(pp.coefs(2,:),xs);

plot(xs,y)

I want to find the area under the curve covered by this plot, that's why I am trying to find the polynomial. Is there any other way to do it or if anyone could find the problem in the commands that I am using, please let me know.

Thanks

Bhomik Luthra

Answer 1

[Andrei Bobrov]

x = [5.8808 6.5137 7.1828 7.8953];
y = [31.2472 33.9977 36.7661 39.3567];

pch = pchip(x,y);

out = fnval(fnint(pch),x([2,3]))*[-1;1];    % if you have 'Curve Fitting Toolbox'

or

out = integral(@(x)ppval(pch,x),x(2),x(3)); %

[Bhomik Luthra]

Thanks for your valuable answer, but I still have the same doubt. What exactly are pch.coefs? Could you please tell me what do they represent?

Thanks Bhomik

[Andrei Bobrov]

please read listing of the function ppval (in Command Window: >> open ppval).

[b,c,l,k,dd]=unmkpp(pp);
%{
b = pp.breaks;
c = pp.coefs;
k = pp.order;
%}

xx = linspace(b(2),b(3),200);

sizexx = size(xx);
lx = numel(xx); 
xs = reshape(xx,1,lx);

[~,index] = histc(xs,[-inf,b(2:l),inf]);
xs = xs-b(index);
v = c(index,1);
for i=2:k
   v = xs(:).*v + c(index,i);
end

or

v = ppval(pp,xx);