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

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By Walter Gautschi

Purdue University

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

Version 2.0 - published on 05 Oct 2016 doi:10.4231/R7RN35T0 - cite this Archived on 06 Nov 2016

Licensed under Attribution 3.0 Unported

Description

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.

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This is an expanded version of the dataset that contains Matlab and eps files used in the paper Algorithm 957: Evaluation of the repeated integrals of the coerror function by half-range Gauss-Hermite quadrature, ACM Transactions on Mathematical Software 42 (2016), 9:1-9:10.

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