beta.lisp

;;;; Copyright (c) 2015 Russell Andrew Edson
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;;;; Code for approximations of the beta function.
;;;; Date: 22/09/2015

(in-package :cl-mathspecialfunctions)

;;; The beta function is related closely to the gamma function,
;;; with the identity B(x,y) = G(x)G(y)/G(x+y).
(defun beta-gamma-identity (x y)
  "Returns the beta function approximation using the Gamma function."
  (/ (* (gamma x) (gamma y))
     (gamma (+ x y))))

;;; The beta function can be approximated using an asymptotic formula
;;; by James Stirling, particularly when x and y are large in magnitude.
(defun stirling-approximation (x y)
  "Approximates the beta function using Stirling's approximation."
  (* (sqrt (* 2 pi))
     (expt (/ x (+ x y)) x)
     (expt (/ y (+ x y)) y)
     (expt (/ (* x y) (+ x y)) -1/2)))

;;; The beta function. Here we use the gamma function identity for
;;; 'small' values of x and y, and use the Stirling asymptotic
;;; formula for large x, y.
;;;
;;; Code Usage Examples:
;;;
;;;   Computing beta(1,1/2) = 2
;;;   > (beta 1 1/2)
;;;   1.9888264803704336d0
;;;
;;;   Computing beta(80,65) = 2.0452135320392d-44
;;;   > (beta 80 65)
;;;   2.041639548717661d-44
;;;
;;;   Computing beta(200,201) = 4.856608623805591d-122
;;;   > (beta 200 201)
;;;   4.853581858408593d-122
(defun beta (x y)
  "Returns an approximation to beta(x,y), with the gamma/Stirling formulae."
  (let ((bignum 15))
    (if (and (< x bignum) (< y bignum))
        (beta-gamma-identity x y)
        (stirling-approximation x y))))