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 Eν(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 Eν(x) in arbitrary-precision arithmetic is discussed.