Computing integrals with an exponential weight on the real axis in floating point arithmetic

Abstract

The aim of this work is to propose a fast and reliable algorithm for computing integrals of the type∫−∞∞f(x)e−x2−1x2dx, where f(x) is a sufficiently smooth function, in floating point arithmetic. The algorithm is based on a product integration rule, whose rate of convergence depends only on the regularity of f, since the coefficients of the rule are “exactly” computed by means of suitable recurrence relations here derived. We prove stability and convergence in the space of locally continuous functions on R equipped with weighted uniform norm. By extensive numerical tests, the accuracy of the proposed product rule is compared with that of the Gauss–Hermite quadrature formula w.r.t. the function f(x)e−1x2. The numerical results confirm the effectiveness of the method, supporting the proven theoretical estimates.