new recurrence relations for rational approximations based on inverse…

[MikaelSlevinsky]

… factorial series

• add a warning to all rational algorithms if kmax is attained

• change tolerance to 8eps because 8 is a power of two.

A simple benchmark: the ₃F₂ on the readme used to take 18 microseconds in double precision and 14 milliseconds in BigFloat:

julia> using HypergeometricFunctions, BenchmarkTools
[ Info: Precompiling HypergeometricFunctions [34004b35-14d8-5ef3-9330-4cdb6864b03a]

julia> α = (1, 2+im, 2.5)
(1, 2 + 1im, 2.5)

julia> β = (3.5, 4)
(3.5, 4)

julia> z = exp(im*π/3)
0.5000000000000001 + 0.8660254037844386im

julia> @btime pFq($α, $β, $z)
  17.868 μs (14 allocations: 2.23 KiB)
0.8434434031615785 + 0.34175507615462697im

julia> z = exp(im*big(π)/3)
0.5000000000000000000000000000000000000000000000000000000000000000000000000000086 + 0.8660254037844386467637231707529361834714026269051903140279034897259665084543988im

julia> @btime pFq($α, $β, $z)
  14.245 ms (583770 allocations: 31.17 MiB)
0.8434434031615690763389963048175253868863156451003855955719081209861492349279575 + 0.3417550761546319732614495656712509723030350666571102474299311122586948108418733im

After merging this pull request, it will take only 4.2 microseconds in double precision and 5.2 milliseconds in BigFloat and is a couple digits more accurate:

julia> using HypergeometricFunctions, BenchmarkTools
[ Info: Precompiling HypergeometricFunctions [34004b35-14d8-5ef3-9330-4cdb6864b03a]

julia> α = (1, 2+im, 2.5)
(1, 2 + 1im, 2.5)

julia> β = (3.5, 4)
(3.5, 4)

julia> z = exp(im*π/3)
0.5000000000000001 + 0.8660254037844386im

julia> @btime pFq($α, $β, $z)
  4.266 μs (5 allocations: 832 bytes)
0.843443403161569 + 0.3417550761546328im

julia> z = exp(im*big(π)/3)
0.5000000000000000000000000000000000000000000000000000000000000000000000000000086 + 0.8660254037844386467637231707529361834714026269051903140279034897259665084543988im

julia> @btime pFq($α, $β, $z)
  5.231 ms (184913 allocations: 9.87 MiB)
0.8434434031615690763389963048175253868863156451003855955719081209861492349265671 + 0.3417550761546319732614495656712509723030350666571102474299311122586948108410529im

[codecov]

Codecov Report

Patch coverage: 69.06% and project coverage change: -2.25 ⚠️

Comparison is base (7e82207) 73.35% compared to head (c5c7ef2) 71.10%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #64      +/-   ##
==========================================
- Coverage   73.35%   71.10%   -2.25%     
==========================================
  Files           7        7              
  Lines        1246     1194      -52     
==========================================
- Hits          914      849      -65     
- Misses        332      345      +13     

Impacted Files	Coverage Δ
src/drummond.jl	72.10% <36.84%> (+0.37%)	⬆️
src/specialfunctions.jl	54.72% <38.70%> (-1.53%)	⬇️
src/weniger.jl	96.94% <100.00%> (-0.67%)	⬇️

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