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

Abstract

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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Acknowledgments

The author acknowledges Javier Segura for reading the preliminary version of the manuscript and providing useful remarks. The author also thanks the anonymous reviewer for indicating errors and suggesting improvements.

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Correspondence to Guillermo Navas-Palencia.

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Navas-Palencia, G. Fast and accurate algorithm for the generalized exponential integral E ν (x) for positive real order. Numer Algor 77, 603–630 (2018). https://doi.org/10.1007/s11075-017-0331-z

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Keywords

  • Generalized exponential integral
  • Numerical evaluation of special functions
  • Asymptotic expansion