# speed up integrating the same function over many overlapping intervals?

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Jeff Miller on 10 Sep 2021
Answered: Paul on 12 Sep 2021
I need to compute many different integrals of the same function fn, essentially like this:
Xint = size(X);
for i=1:numel(X)
Xint(i) = integral(fn,0,X(i)); % fn is continuous
end
The problem is that X is large and computation of fn is slow, so this simple loop takes quite a long time. My question is, what is the best way to speed up this loop?
It does seem like considerable speed-up should be possible, because each time through the loop the integral computation evaluates fn at many almost-the-same points from 0 to X(i)--the range of the X(i)'s is about 400-2000. But what's the best approach? E.g. maybe just order the X's and compute the integrals in pieces, 0 to min(X), then min(X) to 2nd-smallest-X, etc (but isn't there a danger that the errors of approximation will accumulate)? Or maybe precompute fn at a bunch of X's and use trapz (but how to choose those to-be-precomputed X's to ensure the desired level of accuracy)? Or precompute fn, spline it, and then let integral work with the spline_of_fn?
Given all the work that has been done on numerical methods of integration, it seems like someone must have studied how to do this most effectively, but I can't find anything on it. So, I'd appreciate any pointers or tips.
Thanks,
Jeff Miller on 11 Sep 2021
@Paul My reason for the assumption is not very good--it's just based on a vague understanding of how these integrals get computed to a convergence criterion, leaving a little residual error in each integral. Often I'd expect the piecewise errors to more or less cancel each other out when the pieces are summed, but I doubt that would be guaranteed for all functions.
It would be illuminating to experiment with full-range versus piece-wise integrals for some functions where the correct value can be computed analytically.

Alan Stevens on 10 Sep 2021
How about using cumtrapz to do the cumulative integral from 0 to max(X) just once, then select the sections specified by X(i).
Jeff Miller on 12 Sep 2021
Thanks for your thoughts. Just a small followup comment:
The adaptive step size methods are no doubt very helpful in computing a single integral (say 0 to X1), but it seems all the computations involved in step size adaptation have to be redone from scratch when the integration range changes (say 0 to X2). My X vector can easily contain several hundred values, so computation of all the different integrals could in principle be speeded up by somehow saving some or all of the computations relevant to step size adaptation.
For comparison, I approximated a single integral using cumtrapz with a naive equal spacing of x values. Roughly, I needed so many x values (to match integral()'s precision) that the cumtrapz method took 2-3 times as long as integral(). But of course the cumtrapz method wins when the number of integrations increases, because integral() has to redo all the function calculations and cumtrapz 'remembers' them from the earlier computations. For repeated integrations, it seems like it should be possible in principle to have the best of both worlds--adaptive choice of step sizes but also some retention of the function values computed along the way. Not that I would have any idea how to code such a thing...

Paul on 12 Sep 2021
In your question you said that fn is continuous. If it's sufficiently smooth, ideally with continuous first derivative, you can use an ode solver to get the value of the integral at the points you specify in the tspan vector. For example:
fn = @(x) (sin(x));
odefun = @(t,x) (fn(t));
X = linspace(0,1,10)*2*pi;
[~,intvalues] = ode23(odefun,X,0); % note, X(1) = 0, using ode23 jsut to illustrate
plot(X,intvalues,X,-cos(X)+1,'o'),grid % compare to truth solution
Of course you'll have to consider all the other options for the ode solver.
I have absolutely no idea how accurate or fast this solution is compared to any other solutions discussed in this thread, but at least it will give you the values of the integral at the specified values in X in one go.

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