flintlib
Integration of functions that don't have closed-form expressions
I was wondering if it was possible to perform rigorous integration using Arb with integrands which do not have a closed-form expression (i.e. that are obtained numerically for only a discrete set of points, in my case represented in C as two arrays (x,f(x)) of type double), and if so, how?
Looking over the documentation, I've tried using polynomial interpolation of the integrand for this task, however my particular function doesn't appear to be well approximated by a polynomial, no matter how high a degree I choose.
On the other hand, if I interpolate the function using double precision using, say, GSL, and in the integration function I use acb_set_d to directly set the result to the interpolated value, the integration algorithm doesn't appear to probe the domain of integration at all.
Is this feature implemented in Arb? If it's not, would you consider adding it in some future release?
If the integrand is only known from a discrete set of points, then it is impossible to perform rigorous integration. You need some regularity condition for error bounds.
acb_calc_integrate allows integrating any kind of interpolating function that can be evaluated at arbitrary balls, but you do need to evaluate it in a way that preserves enclosures (and respects the analyticity conditions).
