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psi
Psi (digamma) function.SYNOPSIS:
double x, y, psi();
y = psi(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 this 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:
Relative error (except absolute when |psi| < 1):
arithmetic domain # trials peak rms
DEC 0,30 2500 1.7e-16 2.0e-17
IEEE 0,30 30000 1.3e-15 1.4e-16
IEEE -30,0 40000 1.5e-15 2.2e-16
ERROR MESSAGES:
message condition value returned
singularity x integer <=0 MAXNUM