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

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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,

### Accepted Answer

Alan Stevens
on 10 Sep 2021

### More Answers (1)

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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