pFq values different from Mathematica

[Dylan Marques]

Hi

I found some different values between this implementation of the generalized hypergeometric function and from mathematica 12.0 if z < 0:

using HypergeometricFunctions:
pFq([1/2], [1., 3/2], -1000) = 1.5267456019790377e7
pFq([1/2], [1., 3/2], -10000) = -5.471268327327299e65

using Mathematica 12.0
N [HypergeometricPFQ[{1/2}, {1, 3/2}, -1000]] = 0.0152208
N [HypergeometricPFQ[{1/2}, {1, 3/2}, -10000]] = 0.00472887

using positive values of z the functions give the same result:
pFq([1/2], [1., 3/2], 10000) = 1.0224158131796494e83

N[HypergeometricPFQ[{1/2}, {1, 3/2}, 10000]] = 1.02242*10^83

Is there any reason for the results to be different?

[Dream Scatter]

Hypergeometric functions require many different algorithms and techniques to converge properly, any single algorithm or technique usually doesn't converge for all inpute values.

[Mikael Slevinsky]

Looks like if we removed the parameter length check here

HypergeometricFunctions.jl/src/generalized.jl

Lines 15 to 20 in 5866fe5

     elseif abs(z) ≤ ρ || length(α) ≤ length(β) 
         return pFqmaclaurin(α, β, float(z); kwds...) 
     else 
         return pFqweniger(α, β, float(z); kwds...) 
     end 
 end 

then the rational approximation algorithm would do quite a bit better than the Maclaurin series

julia> HypergeometricFunctions.pFqweniger([1/2], [1, 3/2], -1000)
0.015220788412071432

julia> HypergeometricFunctions.pFqweniger([1/2], [1, 3/2], big(-1000))
0.01522078841245579637174821040509930075759919800570185853201355946706413542259123

julia> HypergeometricFunctions.pFqweniger([1/2], [1, 3/2], -10000)
0.004728869998571868

julia> HypergeometricFunctions.pFqweniger([1/2], [1, 3/2], big(-10000))
0.004728870002692929177025108485606062277803713334109646653656286413090573959288139

I don't think Mathematica uses finite-precision floating-point, so it's tough to draw comparisons. This package applies generic algorithms using the data types you supply: no gimmicks or tricks. Conversely, there's no guarantee of correctness.

[James Cook]

It appears that Float64s run out precision for the particular arguments + implementation. Fortunately, we can use ArbNumerics and choose how many digits we want to use:

julia> using HypergeometricFunctions

julia> using ArbNumerics

julia> x = ArbReal(10000, digits=64)
10000.0

julia> pFq([1/2], [1., 3/2], x)
1.022415813179649929832430194462446585213636487335708107714185911e+83

julia> pFq([1/2], [1., 3/2], -x)
2618540298.085577180549025083135444297331203397443340991447512761

julia> x = ArbReal(10000, digits=256)
10000.0

julia> Float64(pFq([1/2], [1., 3/2], -x))
0.004728870002692929

I imagine that Mathematica is adaptively increasing the working precision.

Versions:

  Version 1.5.3-pre.0 (2020-09-24)
  [7e558dbc] ArbNumerics v1.2.1
  [34004b35] HypergeometricFunctions v0.3.3

[Mikael Slevinsky]

The referenced commit fixes this by switching algorithms for 1F2 outside the disk of centre 0 and radius 0.72.