Formally Verified Approximations of Definite Integrals

Abstract

Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis.This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. Our approach also handles improper integrals, provided that a factor of the integrand belongs to a catalog of identified integrable functions. This work has been integrated to the CoqInterval library.

Plotting in a Formally Verified Way

“…These polynomial approximations were instrumental when devising the integral tactic for guaranteed numerical quadrature [2]. Indeed, once (p, ∆) has been computed, one can easily enclose the integrals of p(x) and of p(x) − f (x) over X, and thus the integral of f (x) = p(x) − (p(x) − f (x)) over X.…”

Numerical Integration in Arbitrary-Precision Ball Arithmetic

“…For example, |x| on R extends to the function √ z 2 of z = x + yi, which equals z in the right plane and −z in the left plane with a branch cut on Re(z) = 0. 7 We provide as library methods such extensions of sgn(x), |x|, x , x , max(x, y), min(x, y), with builtin branch cut detection. Table 3 shows integrals with mid-interval jumps or kinks, including one complex integral crossing a branch cut discontinuity (D 2 ).…”

“…Table 3 shows integrals with mid-interval jumps or kinks, including one complex integral crossing a branch cut discontinuity (D 2 ). The example D 0 , where p(x) changes sign once on [0, 1], is due to Helfgott (see comments in [7]).…”

“…For example, an interface in Sage [11] exists (thanks to Marc Mezzarobba and Vincent Delecroix), which we demonstrate by computing We obtain nearly 100 digits with a rigorous error bound in 0.04 seconds (0.02 s when using C directly). This relatively difficult test integral (f (x) changes sign 950 times) was introduced by Rump [9] who observed that the quad function in Matlab took over a second only to return the erroneous 0.2511 (Rump's interval package Intlab computes 7 digits in about one second; see also [7]).…”