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-/* igam.c
- *
- * Incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * double a, x, y, igam();
- *
- * y = igam( a, x );
- *
- * DESCRIPTION:
- *
- * The function is defined by
- *
- * x
- * -
- * 1 | | -t a-1
- * igam(a,x) = ----- | e t dt.
- * - | |
- * | (a) -
- * 0
- *
- *
- * In this implementation both arguments must be positive.
- * The integral is evaluated by either a power series or
- * continued fraction expansion, depending on the relative
- * values of a and x.
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 200000 3.6e-14 2.9e-15
- * IEEE 0,100 300000 9.9e-14 1.5e-14
- */
- /* igamc()
- *
- * Complemented incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * double a, x, y, igamc();
- *
- * y = igamc( a, x );
- *
- * DESCRIPTION:
- *
- * The function is defined by
- *
- *
- * igamc(a,x) = 1 - igam(a,x)
- *
- * inf.
- * -
- * 1 | | -t a-1
- * = ----- | e t dt.
- * - | |
- * | (a) -
- * x
- *
- *
- * In this implementation both arguments must be positive.
- * The integral is evaluated by either a power series or
- * continued fraction expansion, depending on the relative
- * values of a and x.
- *
- * ACCURACY:
- *
- * Tested at random a, x.
- * a x Relative error:
- * arithmetic domain domain # trials peak rms
- * IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15
- * IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15
- */
-
-/*
-Cephes Math Library Release 2.8: June, 2000
-Copyright 1985, 1987, 2000 by Stephen L. Moshier
-*/
-
-#include <math.h>
-#ifdef ANSIPROT
-extern double lgam ( double );
-extern double exp ( double );
-extern double log ( double );
-extern double fabs ( double );
-extern double igam ( double, double );
-extern double igamc ( double, double );
-#else
-double lgam(), exp(), log(), fabs(), igam(), igamc();
-#endif
-
-extern double MACHEP, MAXLOG;
-static double big = 4.503599627370496e15;
-static double biginv = 2.22044604925031308085e-16;
-
-double igamc( a, x )
-double a, x;
-{
-double ans, ax, c, yc, r, t, y, z;
-double pk, pkm1, pkm2, qk, qkm1, qkm2;
-
-if( (x <= 0) || ( a <= 0) )
- return( 1.0 );
-
-if( (x < 1.0) || (x < a) )
- return( 1.0 - igam(a,x) );
-
-ax = a * log(x) - x - lgam(a);
-if( ax < -MAXLOG )
- {
- mtherr( "igamc", UNDERFLOW );
- return( 0.0 );
- }
-ax = exp(ax);
-
-/* continued fraction */
-y = 1.0 - a;
-z = x + y + 1.0;
-c = 0.0;
-pkm2 = 1.0;
-qkm2 = x;
-pkm1 = x + 1.0;
-qkm1 = z * x;
-ans = pkm1/qkm1;
-
-do
- {
- c += 1.0;
- y += 1.0;
- z += 2.0;
- yc = y * c;
- pk = pkm1 * z - pkm2 * yc;
- qk = qkm1 * z - qkm2 * yc;
- if( qk != 0 )
- {
- r = pk/qk;
- t = fabs( (ans - r)/r );
- ans = r;
- }
- else
- t = 1.0;
- pkm2 = pkm1;
- pkm1 = pk;
- qkm2 = qkm1;
- qkm1 = qk;
- if( fabs(pk) > big )
- {
- pkm2 *= biginv;
- pkm1 *= biginv;
- qkm2 *= biginv;
- qkm1 *= biginv;
- }
- }
-while( t > MACHEP );
-
-return( ans * ax );
-}
-
-
-
-/* left tail of incomplete gamma function:
- *
- * inf. k
- * a -x - x
- * x e > ----------
- * - -
- * k=0 | (a+k+1)
- *
- */
-
-double igam( a, x )
-double a, x;
-{
-double ans, ax, c, r;
-
-if( (x <= 0) || ( a <= 0) )
- return( 0.0 );
-
-if( (x > 1.0) && (x > a ) )
- return( 1.0 - igamc(a,x) );
-
-/* Compute x**a * exp(-x) / gamma(a) */
-ax = a * log(x) - x - lgam(a);
-if( ax < -MAXLOG )
- {
- mtherr( "igam", UNDERFLOW );
- return( 0.0 );
- }
-ax = exp(ax);
-
-/* power series */
-r = a;
-c = 1.0;
-ans = 1.0;
-
-do
- {
- r += 1.0;
- c *= x/r;
- ans += c;
- }
-while( c/ans > MACHEP );
-
-return( ans * ax/a );
-}