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/* psif.c
*
* Psi (digamma) function
*
*
* SYNOPSIS:
*
* float x, y, psif();
*
* y = psif( x );
*
*
* DESCRIPTION:
*
* d -
* psi(x) = -- ln | (x)
* dx
*
* is the logarithmic derivative of the gamma function.
* For integer x,
* n-1
* -
* psi(n) = -EUL + > 1/k.
* -
* k=1
*
* This formula is used for 0 < n <= 10. If x is negative, it
* is transformed to a positive argument by the reflection
* formula psi(1-x) = psi(x) + pi cot(pi x).
* For general positive x, the argument is made greater than 10
* using the recurrence psi(x+1) = psi(x) + 1/x.
* Then the following asymptotic expansion is applied:
*
* inf. B
* - 2k
* psi(x) = log(x) - 1/2x - > -------
* - 2k
* k=1 2k x
*
* where the B2k are Bernoulli numbers.
*
* ACCURACY:
* Absolute error, relative when |psi| > 1 :
* arithmetic domain # trials peak rms
* IEEE -33,0 30000 8.2e-7 1.2e-7
* IEEE 0,33 100000 7.3e-7 7.7e-8
*
* ERROR MESSAGES:
* message condition value returned
* psi singularity x integer <=0 MAXNUMF
*/
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1984, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
static float A[] = {
-4.16666666666666666667E-3,
3.96825396825396825397E-3,
-8.33333333333333333333E-3,
8.33333333333333333333E-2
};
#define EUL 0.57721566490153286061
extern float PIF, MAXNUMF;
float floorf(float), logf(float), tanf(float);
float polevlf(float, float *, int);
float psif(float xx)
{
float p, q, nz, x, s, w, y, z;
int i, n, negative;
x = xx;
nz = 0.0;
negative = 0;
if( x <= 0.0 )
{
negative = 1;
q = x;
p = floorf(q);
if( p == q )
{
mtherr( "psif", SING );
return( MAXNUMF );
}
nz = q - p;
if( nz != 0.5 )
{
if( nz > 0.5 )
{
p += 1.0;
nz = q - p;
}
nz = PIF/tanf(PIF*nz);
}
else
{
nz = 0.0;
}
x = 1.0 - x;
}
/* check for positive integer up to 10 */
if( (x <= 10.0) && (x == floorf(x)) )
{
y = 0.0;
n = x;
for( i=1; i<n; i++ )
{
w = i;
y += 1.0/w;
}
y -= EUL;
goto done;
}
s = x;
w = 0.0;
while( s < 10.0 )
{
w += 1.0/s;
s += 1.0;
}
if( s < 1.0e8 )
{
z = 1.0/(s * s);
y = z * polevlf( z, A, 3 );
}
else
y = 0.0;
y = logf(s) - (0.5/s) - y - w;
done:
if( negative )
{
y -= nz;
}
return(y);
}
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