Abstract: An algorithm for computing the incomplete gamma function ?*(a, z) for real values of the parameter a and negative real values of the argument z is presented. The algorithm combines the use of series expansions, Poincaré-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy ~10-13 in the parameter region (a, z) ? [- 500, 500] × [- 500, 0) can be obtained when computing the function ?*(a, z) with the Fortran 90 module IncgamNEG implementing the algorithm.

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