Matlab integration of numerical data

I have numerical data for function f, calculated for a vector x (not a defined function of x). Can I create a functional form out of this so that I can use something like f(y) in further calculations and integrals ?

x = 0:1:10; F is calculated for eac x. Example : Want to integrate f(x+c) between some limits, (not 0,10) and for any constant c.

Answer 1

[Star Strider]

The cumtrapz funciton will likely do what you want, although I am not certain what result you want.

Answer 2

[John D'Errico]

In order for it to be truly general, you will need to interpolate the "function". For example, we might do this:

x = 0:10;
y = sin(x); % some function

Now, we wish to integrate y, but only based on the values we have generated. Remember that integrating beyond the linits of our data will be bad idea, because then we are forced to extrapolate the function. And THAT is a bad idea, unless our extrapolant is one chosen carefully. A spline will be a terribly poor tool to extrapolate.

f = spline(x,y);
fint = fnint(f); % this lives in the curve fitting toolbox.

Now, assume we wish the integral of y, between x and x + c. In this example, I'll use x=0.1223, and c=4.75.

fintxc = @(x,c) fnval(fint,x+c) - fnval(fint,x);
fintxc(0.1223,4.75)

ans = 0.8435

How well did we do? Remember, this can be no more than an approximation. We can use integral on the original function to do the work, although I could be more intelligent, since I know the integral of the sine function. I'll take the lazy way here.

integral(@sin,0.1223,0.1223 + 4.75)

ans = 0.8333

Which given the coarseness of the original sample, is not at all bad.

[Sib RV]

Ok, thanks. Some interpolation need to be done, which is inaccurate and not a good idea.

Why I did not try the obvious integration: F is not some well defined function of x. It is the result of some calculations done in several steps.

[John D'Errico]

It depends on how nasty is the function, and how densely sampled. Interpolation can be very well behaved. No, you probably don't want to do linear interpolation, but you could, and it would provide an integratino estimate that is essentially equivalent to a trapezoidal rule.

That is why a spline is a good choice, since it is effectively a higher orer interpolant. Or you might want to use a shape preserving interpolant, like pchip. Finally, you could even use a tool like my SLM toolbox, which can produce spline interpolants that will be well-behaved.

[Sib RV]

Thank you.