New rigorous numerical integration in Arb
The next version of Arb will introduce the function acb_calc_integrate for computing definite integrals $\int_a^b f(x) dx$ with rigorous error bounds. I already discussed the possibility to use Arb for numerical integration four years ago, but the old toy implementation (which used Taylor series) was both inefficient and hard to use. The new integration code is far more efficient, more general, and easier to use. Combining industrial-grade™ adaptive subdivision with degree-adaptive Gauss-Legendre (GL) quadrature, it achieves near-optimal convergence for holomorphic integrands while being able to cope with a wide range of pathological functions. Minimal extra information is required from the user apart from black-box evaluation of $f$ plus the ability to test for holomorphicity in order to use GL quadrature; in particular, function derivatives and a priori locations of discontinuities are not required. The main limitation is that only proper integrals are supported directly. The integration code supports both absolute and relative accuracy goals and aborts gracefully (and configurably) if convergence is too slow.
In this blog post, I will discuss the algorithm and present several examples of integrals of varying difficulty. The integration code is available in the git master of Arb, with some documentation, and a demo program implementing most of the integrals in this blog post is available as examples/integrals.c.
