Algorithm 969

Abstract

An algorithm for computing the incomplete gamma function γ * (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 γ * (a, z) with the Fortran 90 module IncgamNEG implementing the algorithm.

FDTD: Solving 1+1D delay PDE in parallel

“…We note that combining (a) and (b) gives the standard approach to real-valued z [36], and that for very large |z| the computation may not converge [44], but the convergence range is large enough for our purposes. The reason is that the time T max ∆ is where the second light cone intersects with the box boundary.…”

“…Fortunately, this problem is recently tackled in Ref. [44], and we incorporate their findings into our implementation, which works 16 6) in Ref. [44]:…”

Efficient Computation of Generalized Bessel Function Distributions

“…In such cases Gautschi 1979 [21] proposed that it may be preferable to use asymptotic methods, e.g. the uniform asymptotic expansions of Temme 1977Temme -1979 (later publication of Temme 1985 [26], A.R DiDonato and A.H. Morris, Jr. 1986 [25] includes the computation of the incompete gamma function inverses from Temme's uniform asymptotic expansions and Temme 1996 [26] includes negative values, and the complete refined algorithm in Gil, Ruiz-Antolín, Segura, and Temme 2017 [33]). Jones and Thron 1986 [23] Based on the work on Greengard and Rokhlin 2017-2018 [34] Numerical experiments show that in this regime, for all > 0, < 10,000 evaluation of ( , ) using…”