Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes

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Department of Electrical and Computer Engineering; Department of Physics


A numerical evaluator for the confluent hypergeometric function for complexarguments with large magnitudes using a direct summation of Kummer's series is described. Extended precision subroutines using large integer arrays to accumulate a single numerator and denominator are ultimately used in a single division to arrive at the final result. The accuracy has been verified through a variety of tests and they show the evaluator to be consistently accurate to thirteen significant figures, and on rare occasion accurate to only nine for magnitudes of the arguments ranging into the thousands in any quadrant in the complex plane. Because the evaluator automatically determines the number of significant figures of machine precision, and because it is written in FORTRAN 77, tests on various computers have shown the evaluator to provide consistently accurate results, making the evaluator very portable. The principal drawback is that, for certain arguments, the evaluator is slow, however, the evaluator remains valuable as a benchmark even in such cases.

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Journal of Computational and Applied Mathematics