Is dynamic interpolation of input of function for 'quad' integration possible?

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Daniel
Daniel 2012-1-30
My task is to integrate a function given limited data, e.g.
max_energy = 1000;
quad(@(x) a.*x.*exp(-b.*x),0,max_energy)
where a, b are vectors, functions of energy. My problem is that it seems quad uses Simpson's rule iteratively until some tolerance is met, so the array size of a, b would need to change dynamically to match. Is it possible to interpolate between the data points I have to match quad? Must I do it in advance or use a different integration method?

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Andrew Newell
Andrew Newell 2012-1-30
Given that your data are irregularly spaced, you may have to use interpolation after all. If your points are (xdata,ydata), you could integrate as below:
pp = interp1(xdata,ydata,'pchip','pp');
f = @(x) ppval(pp,x);
max_energy = 1000;
quad(f,0,max_energy)
Just don't assume the integral is as accurate as the default tolerance.
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Daniel
Daniel 2012-1-31
The method you've provided works; my 'default' interp1 use does not. Does it use linear interpolation if you don't specify a method? Do you know what problem the method I tried has that the 'pchip' method doesn't have?
Andrew Newell
Andrew Newell 2012-1-31
The default method is linear. I don't know why you're getting the NaN, but if you use interp in the form yi = interp1(x,Y,xi), you have to keep recalculating the interpolating polynomials.

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Andrew Newell
Andrew Newell 2012-1-30
Any increase in accuracy based on interpolating a and b would probably be illusory. If your points are regularly spaced, you could use trapz:
y = a.*x.*exp(-b.*x);
yInt = trapz(x,y);
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Bård Skaflestad
Bård Skaflestad 2012-1-30
@Andrew
Thanks a lot -- I'd missed that part of the documentation. I obviously need to pay more attention...

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Bård Skaflestad
Bård Skaflestad 2012-1-30
All of MATLAB's quadrature methods require an integrand that can be evaluated at vector inputs and return an equally sized vector result.
Do you mean to say that your a and b in some way depend on x? If so, you may have to implement a traditional function (i.e., .m) file that evaluates both a and b along with the resulting integrand.
For instance,
function y = integrand(x)
a = some_function(x);
b = some_other_function(x);
y = a .* x .* exp(-b .* x);
end
Does this help at all?
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Daniel
Daniel 2012-1-30
It does help, thank you, but I don't think I have an expression for a(x), b(x) -- yes, their values depend on the variable of integration -- that instead their values are experimentally determined, hence the need for interpolation. Is curve fitting the way to go, then, based on the existing data for a, b? But the data is seemingly logarithmic -- will fitting to polynomials work for something whose behavior changes significantly depending on the range of x? (and I have never done curve fitting before, so I would need to learn; would you recommend resources?)

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