s14aef evaluates an approximation to the
$k$th derivative of the psi function
$\psi \left(x\right)$ given by
where
$x$ is real with
$x\ne 0,-1,-2,\dots \text{}$ and
$k=0,1,\dots ,6$. For negative noninteger values of
$x$, the recurrence relationship
is used. The value of
$\frac{{\left(-1\right)}^{k+1}{\psi}^{\left(k\right)}\left(x\right)}{k!}$ is obtained by a call to
s14adf, which is based on the routine PSIFN in
Amos (1983).
Note that
${\psi}^{\left(k\right)}\left(x\right)$ is also known as the
polygamma function. Specifically,
${\psi}^{\left(0\right)}\left(x\right)$ is often referred to as the
digamma function and
${\psi}^{\left(1\right)}\left(x\right)$ as the
trigamma function in the literature. Further details can be found in
Abramowitz and Stegun (1972).
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502
If on entry
${\mathbf{ifail}}=0$ or
$-1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
All constants in
s14adf are given to approximately
$18$ digits of precision. If
$t$ denotes the number of digits of precision in the floating-point arithmetic being used, then clearly the maximum number in the results obtained is limited by
$p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$. Empirical tests by
Amos (1983) have shown that the maximum relative error is a loss of approximately two decimal places of precision. Further tests with the function
$-{\psi}^{\left(0\right)}\left(x\right)$ have shown somewhat improved accuracy, except at points near the positive zero of
${\psi}^{\left(0\right)}\left(x\right)$ at
$x=1.46\dots \text{}$, where only absolute accuracy can be obtained.
None.