Task

Implement one algorithm (or more) to compute the Gamma (



Γ


{\displaystyle \Gamma }

) function (in the real field only).

If your language has the function as built-in or you know a library which has it, compare your implementation's results with the results of the built-in/library function.

The Gamma function can be defined as:

  



Γ
(
x
)
=


∫

0


∞



t

x
−
1



e

−
t


d
t



{\displaystyle \Gamma (x)=\displaystyle \int _{0}^{\infty }t^{x-1}e^{-t}dt}


This suggests a straightforward (but inefficient) way of computing the 



Γ


{\displaystyle \Gamma }

 through numerical integration.



Better suggested methods:

 Lanczos approximation
 Stirling's approximation


# Taylor Series
def a
   [ 1.00000_00000_00000_00000,  0.57721_56649_01532_86061, -0.65587_80715_20253_88108,
    -0.04200_26350_34095_23553,  0.16653_86113_82291_48950, -0.04219_77345_55544_33675,
    -0.00962_19715_27876_97356,  0.00721_89432_46663_09954, -0.00116_51675_91859_06511,
    -0.00021_52416_74114_95097,  0.00012_80502_82388_11619, -0.00002_01348_54780_78824,
    -0.00000_12504_93482_14267,  0.00000_11330_27231_98170, -0.00000_02056_33841_69776,
     0.00000_00061_16095_10448,  0.00000_00050_02007_64447, -0.00000_00011_81274_57049,
     0.00000_00001_04342_67117,  0.00000_00000_07782_26344, -0.00000_00000_03696_80562,
     0.00000_00000_00510_03703, -0.00000_00000_00020_58326, -0.00000_00000_00005_34812,
     0.00000_00000_00001_22678, -0.00000_00000_00000_11813,  0.00000_00000_00000_00119,
     0.00000_00000_00000_00141, -0.00000_00000_00000_00023,  0.00000_00000_00000_00002 ]
end
 
def taylor_gamma(x)
  y = x.to_f - 1
  1.0 / a.reverse.reduce(0) { |sum, an| sum * y + an }
end
 
# Lanczos Method
def p
  [ 0.99999_99999_99809_93, 676.52036_81218_851, -1259.13921_67224_028, 
    771.32342_87776_5313, -176.61502_91621_4059,  12.50734_32786_86905, 
    -0.13857_10952_65720_12, 9.98436_95780_19571_6e-6, 1.50563_27351_49311_6e-7 ]
end
 
def lanczos_gamma(z)
  # Reflection formula
  z = z.to_f
  if z < 0.5
    Math::PI / (Math.sin(Math::PI * z) * lanczos_gamma(1 - z))
  else
    z -= 1
    x = p[0]
    (1..p.size - 1).each { |i| x += p[i] / (z + i) }
    t = z + p.size - 1.5
    Math.sqrt(2 * Math::PI) * t**(z + 0.5) * Math.exp(-t) * x
  end
end
 
puts "                Taylor Series         Lanczos Method        Builtin Function"
(1..27).each { |i| n = i/3.0; puts "gamma(%.2f) = %.14e  %.14e  %.14e" % [n, taylor_gamma(n), lanczos_gamma(n), Math.gamma(n)] }
 