INERFC: Evaluation of the Repeated Integrals of the Coerror Function by Half-Range Gauss-Hermite Quadrature

<p>The integrals in the title, functions of the real variable x and integer parameter n, are of considerable interest in physics and chemistry, notably in problems involving heat and mass transfer. They are traditionally evaluated by the three-term recurrence relation that they satisfy. This involves, even if done carefully, controlled loss of accuracy. On the other hand, a whole sequence of n+2 integrals is produced, as may be required in some applications. Here, we propose a method based on quadrature that, involving the summation of a finite sum of positive terms, is perfectly stable and allows the computation of just one of these integrals. The quadrature entails non-classical Gaussian integration and the half-range Hermite polynomials orthogonal with respect to the weight function exp(-t^2) on the half-infinite interval from zero to infinity. An important issue is the determination of a natural domain in the (n,x)-plane in which to evaluate the function.</p>