<?
# Visit http://www.johndcook.com/stand_alone_code.html for the source of this code and more like it.
# Note that the functions Gamma and LogGamma are mutually dependent.
function gamma($x) {
if ($x <= 0.0)
{
die("Invalid input argument $x. Argument must be positive");
}
# Split the function domain into three intervals:
# (0, 0.001), [0.001, 12), and (12, infinity)
###########################################################################
# First interval: (0, 0.001)
#
# For small x, 1/Gamma(x) has power series x + gamma x^2 - ...
# So in this range, 1/Gamma(x) = x + gamma x^2 with error on the order of x^3.
# The relative error over this interval is less than 6e-7.
$gamma = 0.577215664901532860606512090; # Euler's gamma constant
if ($x < 0.001) {
return 1.0/($x*(1.0 + $gamma*$x));
}
###########################################################################
# Second interval: [0.001, 12)
if ($x < 12.0)
{
# The algorithm directly approximates gamma over (1,2) and uses
# reduction identities to reduce other arguments to this interval.
$y = $x;
$n = 0;
$arg_was_less_than_one = ($y < 1.0);
# Add or subtract integers as necessary to bring y into (1,2)
# Will correct for this below
if ($arg_was_less_than_one)
{
$y += 1.0;
}
else
{
$n = floor($y) - 1; # will use n later
$y -= $n;
}
# numerator coefficients for approximation over the interval (1,2)
$p =
array(
-1.71618513886549492533811E+0,
2.47656508055759199108314E+1,
-3.79804256470945635097577E+2,
6.29331155312818442661052E+2,
8.66966202790413211295064E+2,
-3.14512729688483675254357E+4,
-3.61444134186911729807069E+4,
6.64561438202405440627855E+4
);
# denominator coefficients for approximation over the interval (1,2)
$q =
array(
-3.08402300119738975254353E+1,
3.15350626979604161529144E+2,
-1.01515636749021914166146E+3,
-3.10777167157231109440444E+3,
2.25381184209801510330112E+4,
4.75584627752788110767815E+3,
-1.34659959864969306392456E+5,
-1.15132259675553483497211E+5
);
$num = 0.0;
$den = 1.0;
$z = $y - 1;
for ($i = 0; $i < 8; $i++)
{
$num = ($num + $p[$i])*$z;
$den = $den*$z + $q[$i];
}
$result = $num/$den + 1.0;
# Apply correction if argument was not initially in (1,2)
if ($arg_was_less_than_one)
{
# Use identity gamma(z) = gamma(z+1)/z
# The variable "result" now holds gamma of the original y + 1
# Thus we use y-1 to get back the orginal y.
$result /= ($y-1.0);
}
else
{
# Use the identity gamma(z+n) = z*(z+1)* ... *(z+n-1)*gamma(z)
for ($i = 0; $i < $n; $i++) {
$result *= $y++;
}
}
return $result;
}
###########################################################################
# Third interval: [12, infinity)
if ($x > 171.624)
{
# Correct answer too large to display.
return Double.POSITIVE_INFINITY;
}
return exp(log_gamma($x));
}
function log_gamma($x) {
if ($x <= 0.0)
{
die("Invalid input argument $x. Argument must be positive");
}
if ($x < 12.0)
{
return log(abs(gamma($x)));
}
# Abramowitz and Stegun 6.1.41
# Asymptotic series should be good to at least 11 or 12 figures
# For error analysis, see Whittiker and Watson
# A Course in Modern Analysis (1927), page 252
$c =
array(
1.0/12.0,
-1.0/360.0,
1.0/1260.0,
-1.0/1680.0,
1.0/1188.0,
-691.0/360360.0,
1.0/156.0,
-3617.0/122400.0
);
$z = 1.0/($x*$x);
$sum = $c[7];
for ($i=6; $i >= 0; $i--)
{
$sum *= $z;
$sum += $c[$i];
}
$series = $sum/$x;
$halfLogTwoPi = 0.91893853320467274178032973640562;
$logGamma = ($x - 0.5)*log($x) - $x + $halfLogTwoPi + $series;
return $logGamma;
}
?>