Fast and accurate algorithm for the generalized exponential integral Eν(x) for positive real order

We describe an algorithm for the numerical evaluation of the generalized exponential integral E ν ( x ) for positive values of ν and x . A detailed description of the numerical methods used in the algorithm is provided, including error bounds. Different approaches from earlier algorithms are also su...

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Bibliographic Details
Published inNumerical algorithms Vol. 77; no. 2; pp. 603 - 630
Main Author Navas-Palencia, Guillermo
Format Journal Article
LanguageEnglish
Published New York Springer US 01.02.2018
Springer Nature B.V
Subjects
Algebra
Algorithms
Computer Science
Integrals
Numeric Computing
Numerical Analysis
Numerical methods
Open source software
Original Paper
Robustness (mathematics)
Software packages
Theory of Computation
Asymptotic expansion
Generalized exponential integral
Numerical evaluation of special functions
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ISSN1017-1398
1572-9265
DOI10.1007/s11075-017-0331-z

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Summary:We describe an algorithm for the numerical evaluation of the generalized exponential integral E ν ( x ) for positive values of ν and x . A detailed description of the numerical methods used in the algorithm is provided, including error bounds. Different approaches from earlier algorithms are also summarised. The performance and accuracy of the resulting algorithm is analysed and compared with open-source software packages. This analysis shows that our implementation is competitive and more robust than other state-of-the-art codes. Finally, a brief study of the implementation of E ν ( x ) in arbitrary-precision arithmetic is discussed.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-017-0331-z