How to resolve the algorithm Gamma function step by step in the Prolog programming language

Published on 12 May 2024 09:40 PM
#Prolog

How to resolve the algorithm Gamma function step by step in the Prolog programming language

Table of Contents

Problem Statement

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

Γ

{\displaystyle \Gamma }

through numerical integration.

Better suggested methods:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Gamma function step by step in the Prolog programming language

Source code in the prolog programming language

gamma_coefficients(
     [ 1.00000000000000000000000,  0.57721566490153286060651, -0.65587807152025388107701,
      -0.04200263503409523552900,  0.16653861138229148950170, -0.04219773455554433674820,
      -0.00962197152787697356211,  0.00721894324666309954239, -0.00116516759185906511211,
      -0.00021524167411495097281,  0.00012805028238811618615, -0.00002013485478078823865,
      -0.00000125049348214267065,  0.00000113302723198169588, -0.00000020563384169776071,
       0.00000000611609510448141,  0.00000000500200764446922, -0.00000000118127457048702,
       0.00000000010434267116911,  0.00000000000778226343990, -0.00000000000369680561864,
       0.00000000000051003702874, -0.00000000000002058326053, -0.00000000000000534812253,
       0.00000000000000122677862, -0.00000000000000011812593,  0.00000000000000000118669,
       0.00000000000000000141238, -0.00000000000000000022987,  0.00000000000000000001714
    ]).

tolerance(1e-17).

gamma(X, _) :- X =< 0.0, !, fail.
gamma(X, Y) :-
    X < 1.0, small_gamma(X, Y), !.
gamma(1, 1) :- !.
gamma(1.0, 1) :- !.
gamma(X, Y) :-
    X1 is X - 1,
    gamma(X1, Y1),
    Y is X1 * Y1.
    
small_gamma(X, Y) :-
    gamma_coefficients(Cs),
    recip_gamma(X, 1.0, Cs, 1.0, 0.0, Y0),
    Y is 1 / Y0.

recip_gamma(_, _, [], _, Y, Y) :- !.
recip_gamma(_, _, [], X0, X1, Y) :- tolerance(Tol), abs(X1 - X0) < Tol, Y = X1, !. % early exit
recip_gamma(X, PrevPow, [C|Cs], _, X1, Y) :-
    Power is PrevPow * X,
    X2 is X1 + C*Power,
    recip_gamma(X, Power, Cs, X1, X2, Y).

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