Numerical methods for the computation of the confluent and Gauss hypergeometric functions

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The...

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Bibliographic Details
Published in:Numerical algorithms Vol. 74; no. 3; pp. 821 - 866
Main Authors: Pearson, John W., Olver, Sheehan, Porter, Mason A.
Format: Journal Article
Language:English
Published: New York Springer US 01.03.2017
Springer Nature B.V
Subjects:
Algebra
Algorithms
Applied mathematics
Asymptotic methods
Asymptotic series
Computation
Computer Science
Differential equations
Hypergeometric functions
Numeric Computing
Numerical Analysis
Numerical methods
Original Paper
Parameters
Quadratures
Recommender systems
Theory of Computation
Confluent hypergeometric function
33C15
Gauss hypergeometric function
Secondary: 41A58
41A60
Primary: 33C05
Computation of special functions
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-016-0173-0