March 2014 update

Task/Gamma-function/Perl/gamma-function.pl

@@ -1,44 +1,61 @@
-my @coeff = reverse <DATA>;
+use strict;
+use warnings;
+use constant pi => 4*atan2(1, 1);
+use constant e  => exp(1);
+
 sub Gamma {
-    my $x = shift;
-    my $s;
-    $s = $s*($x-1) + $_ for @coeff;
-    return 1 / $s;
+    my $z = shift;
+    my $method = shift // 'lanczos';
+    if ($method eq 'lanczos') {
+        use constant g => 9;
+        $z <  .5 ?  pi / sin(pi * $z) / Gamma(1 - $z, $method) :
+        sqrt(2* pi) *
+        ($z + g - .5)**($z - .5) *
+        exp(-($z + g - .5)) *
+        do {
+            my @coeff = qw{
+            1.000000000000000174663
+        5716.400188274341379136
+      -14815.30426768413909044
+       14291.49277657478554025
+       -6348.160217641458813289
+        1301.608286058321874105
+        -108.1767053514369634679
+           2.605696505611755827729
+          -0.7423452510201416151527e-2
+           0.5384136432509564062961e-7
+          -0.4023533141268236372067e-8
+            };
+            my ($sum, $i) = (shift(@coeff), 0);
+            $sum += $_ / ($z + $i++) for @coeff;
+            $sum;
+        }
+    } elsif ($method eq 'taylor') {
+        $z <  .5 ? Gamma($z+1, $method)/$z     :
+        $z > 1.5 ? ($z-1)*Gamma($z-1, $method) :
+	do {
+	    my $s = 0; ($s *= $z-1) += $_ for qw{
+	    0.00000000000000000002 -0.00000000000000000023 0.00000000000000000141
+	    0.00000000000000000119 -0.00000000000000011813 0.00000000000000122678
+	    -0.00000000000000534812 -0.00000000000002058326 0.00000000000051003703
+	    -0.00000000000369680562 0.00000000000778226344 0.00000000010434267117
+	    -0.00000000118127457049 0.00000000500200764447 0.00000000611609510448
+	    -0.00000020563384169776 0.00000113302723198170 -0.00000125049348214267
+	    -0.00002013485478078824 0.00012805028238811619 -0.00021524167411495097
+	    -0.00116516759185906511 0.00721894324666309954 -0.00962197152787697356
+	    -0.04219773455554433675 0.16653861138229148950 -0.04200263503409523553
+	    -0.65587807152025388108 0.57721566490153286061 1.00000000000000000000
+	    }; 1/$s;
+	}
+    } elsif ($method eq 'stirling') {
+        no warnings qw(recursion);
+        $z < 100 ? Gamma($z + 1, $method)/$z :
+        sqrt(2*pi*$z)*($z/e + 1/(12*e*$z))**$z / $z;
+    } else { die "unknown method '$method'" }
 }
 
-for (1 .. 10) {
-    $_ /= 3;
-    printf "Gamma(%.5f) ≈ %.14e\n", $_, Gamma($_);
+for my $method (qw(lanczos taylor stirling)) {
+    printf "%10s: ", $method;
+    print join(' ', map { sprintf "%.12f", Gamma($_/3, $method) } 1 .. 10);
+    print "\n";
 }
-
-__DATA__
-1.00000000000000000000
-0.57721566490153286061
--0.65587807152025388108
--0.04200263503409523553
-0.16653861138229148950
--0.04219773455554433675
--0.00962197152787697356
-0.00721894324666309954
--0.00116516759185906511
--0.00021524167411495097
-0.00012805028238811619
--0.00002013485478078824
--0.00000125049348214267
-0.00000113302723198170
--0.00000020563384169776
-0.00000000611609510448
-0.00000000500200764447
--0.00000000118127457049
-0.00000000010434267117
-0.00000000000778226344
--0.00000000000369680562
-0.00000000000051003703
--0.00000000000002058326
--0.00000000000000534812
-0.00000000000000122678
--0.00000000000000011813
-0.00000000000000000119
-0.00000000000000000141
--0.00000000000000000023
-0.00000000000000000002