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.
Cite this work
Researchers should cite this work as follows:
- Walter Gautschi (2016). INERFC: Evaluation of the Repeated Integrals of the Coerror Function by Half-Range Gauss-Hermite Quadrature. (Version 2.0). Purdue University Research Repository. doi:10.4231/R7RN35T0
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.