On the evaluation of the incomplete gamma function

Abstract:

The accurate evaluation of the ${\chi ^2}$ distribution for high degrees of freedom by the usual methods is very difficult (even with a digital computer) because the series to be evaluated would become unbearably long. Also, when a series becomes long, more precision in the numbers used is required in order to offset the effects of round-off errors. On a computer this would mean the use of multiple precision. Accurate tables can be, and have been, prepared by use of the Cornish-Fisher approximation. Comparison of the table values with the values obtained by the method in the writer’s paper show that these tables have an accuracy of about six significant figures. For practical purposes there seems to be no lack of ${\chi ^2}$ tables for high degrees of freedom. The method in the writer’s paper is still useful in checking on the accuracy of tables computed by approximate methods or in producing tables with more significant figures. With single precision it can produce tables of seven figure accuracy at a speed far better than could be by the usual accurate methods. Some unique and useful tables can be produced using this method.