On Approximating Univariate NP-Hard Integrals

Abstract: Approximating a definite integral of product of cosines to within an accuracy of n binary digits where the integrand depends on input integers x[k] given in binary radix, is equivalent to counting the number of equal-sum partitions of the integers and is thus a #P problem. Similarly, integrating this function from zero to infinity and deciding whether the result is either zero or infinity is an NP-Complete problem. Efficient numerical integration methods such as the double exponential formula and the sinc approximation have been around since the mid 70's. Noting the hardness of approximating the integral we argue that the proven rates of convergence of such methods cannot possibly be correct since they give rise to an anomalous result as P=#P.