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

The aim of this work is to propose a fast and reliable algorithm for computing integrals of the type 00 1 -00 - x 2 - 1 f ( x ) e x2 dx , 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 confirm the effectiveness of the method, supporting the proven theoretical estimates. (c) 2023 IMACS. Published by Elsevier B.V. All rights reserved. 1 x 2 . The numerical results