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/* igaml.c
*
* Incomplete gamma integral
*
*
*
* SYNOPSIS:
*
* long double a, x, y, igaml();
*
* y = igaml( 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
* DEC 0,30 4000 4.4e-15 6.3e-16
* IEEE 0,30 10000 3.6e-14 5.1e-15
*
*/
/* igamcl()
*
* Complemented incomplete gamma integral
*
*
*
* SYNOPSIS:
*
* long double a, x, y, igamcl();
*
* y = igamcl( 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:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 2000 2.7e-15 4.0e-16
* IEEE 0,30 60000 1.4e-12 6.3e-15
*
*/
/*
Cephes Math Library Release 2.3: March, 1995
Copyright 1985, 1995 by Stephen L. Moshier
*/
#include <math.h>
#ifdef ANSIPROT
extern long double lgaml ( long double );
extern long double expl ( long double );
extern long double logl ( long double );
extern long double fabsl ( long double );
extern long double gammal ( long double );
long double igaml ( long double, long double );
long double igamcl ( long double, long double );
#else
long double lgaml(), expl(), logl(), fabsl(), igaml(), gammal();
long double igamcl();
#endif
#define BIG 9.223372036854775808e18L
#define MAXGAML 1755.455L
extern long double MACHEPL, MINLOGL;
long double igamcl( a, x )
long double a, x;
{
long double ans, c, yc, ax, y, z, r, t;
long double pk, pkm1, pkm2, qk, qkm1, qkm2;
if( (x <= 0.0L) || ( a <= 0.0L) )
return( 1.0L );
if( (x < 1.0L) || (x < a) )
return( 1.0L - igaml(a,x) );
ax = a * logl(x) - x - lgaml(a);
if( ax < MINLOGL )
{
mtherr( "igamcl", UNDERFLOW );
return( 0.0L );
}
ax = expl(ax);
/* continued fraction */
y = 1.0L - a;
z = x + y + 1.0L;
c = 0.0L;
pkm2 = 1.0L;
qkm2 = x;
pkm1 = x + 1.0L;
qkm1 = z * x;
ans = pkm1/qkm1;
do
{
c += 1.0L;
y += 1.0L;
z += 2.0L;
yc = y * c;
pk = pkm1 * z - pkm2 * yc;
qk = qkm1 * z - qkm2 * yc;
if( qk != 0.0L )
{
r = pk/qk;
t = fabsl( (ans - r)/r );
ans = r;
}
else
t = 1.0L;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if( fabsl(pk) > BIG )
{
pkm2 /= BIG;
pkm1 /= BIG;
qkm2 /= BIG;
qkm1 /= BIG;
}
}
while( t > MACHEPL );
return( ans * ax );
}
/* left tail of incomplete gamma function:
*
* inf. k
* a -x - x
* x e > ----------
* - -
* k=0 | (a+k+1)
*
*/
long double igaml( a, x )
long double a, x;
{
long double ans, ax, c, r;
if( (x <= 0.0L) || ( a <= 0.0L) )
return( 0.0L );
if( (x > 1.0L) && (x > a ) )
return( 1.0L - igamcl(a,x) );
ax = a * logl(x) - x - lgaml(a);
if( ax < MINLOGL )
{
mtherr( "igaml", UNDERFLOW );
return( 0.0L );
}
ax = expl(ax);
/* power series */
r = a;
c = 1.0L;
ans = 1.0L;
do
{
r += 1.0L;
c *= x/r;
ans += c;
}
while( c/ans > MACHEPL );
return( ans * ax/a );
}
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