NAG Library Routine Document
s14adf (polygamma_deriv)
1
Purpose
s14adf returns a sequence of values of scaled derivatives of the psi function $\psi \left(x\right)$ (also known as the digamma function).
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n, m  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  x  Real (Kind=nag_wp), Intent (Out)  ::  ans(m) 

C Header Interface
#include nagmk26.h
void 
s14adf_ (const double *x, const Integer *n, const Integer *m, double ans[], Integer *ifail) 

3
Description
s14adf computes
$m$ values of the function
for
$x>0$,
$k=n$,
$n+1,\dots ,n+m1$, where
$\psi $ is the psi function
and
${\psi}^{\left(k\right)}$ denotes the
$k$th derivative of
$\psi $.
The routine is derived from the routine PSIFN in
Amos (1983). The basic method of evaluation of
$w\left(k,x\right)$ is the asymptotic series
for large
$x$ greater than a machinedependent value
${x}_{\mathrm{min}}$, followed by backward recurrence using
for smaller values of
$x$, where
$\epsilon \left(k,x\right)=\mathrm{ln}x$ when
$k=0$,
$\epsilon \left(k,x\right)=\frac{1}{k{x}^{k}}$ when
$k>0$, and
${B}_{2j}$,
$j=1,2,\dots $, are the Bernoulli numbers.
When
$k$ is large, the above procedure may be inefficient, and the expansion
which converges rapidly for large
$k$, is used instead.
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502
5
Arguments
 1: $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the argument $x$ of the function.
Constraint:
${\mathbf{x}}>0.0$.
 2: $\mathbf{n}$ – IntegerInput

On entry: the index of the first member $n$ of the sequence of functions.
Constraint:
${\mathbf{n}}\ge 0$.
 3: $\mathbf{m}$ – IntegerInput

On entry: the number of members $m$ required in the sequence
$w\left(\mathit{k},x\right)$, for $\mathit{k}=n,\dots ,n+m1$.
Constraint:
${\mathbf{m}}\ge 1$.
 4: $\mathbf{ans}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the first
$m$ elements of
ans contain the required values
$w\left(\mathit{k},x\right)$, for
$\mathit{k}=n,\dots ,n+m1$.
 5: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{x}}\le 0.0$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{n}}<0$. 
 ${\mathbf{ifail}}=3$

On entry,  ${\mathbf{m}}<1$. 
 ${\mathbf{ifail}}=4$

No results are returned because underflow is likely. Either
x or
${\mathbf{n}}+{\mathbf{m}}1$ is too large. If possible, reduce the value of
m and call
s14adf again.
 ${\mathbf{ifail}}=5$

No results are returned because overflow is likely. Either
x is too small, or
${\mathbf{n}}+{\mathbf{m}}1$ is too large. If possible, reduce the value of
m and call
s14adf again.
 ${\mathbf{ifail}}=6$

No results are returned because there is not enough internal workspace to continue computation.
${\mathbf{n}}+{\mathbf{m}}1$ may be too large. If possible, reduce the value of
m and call
s14adf again.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
All constants in s14adf are given to approximately $18$ digits of precision. Calling the number of digits of precision in the floatingpoint arithmetic being used $t$, then clearly the maximum number of correct digits in the results obtained is limited by $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$. Empirical tests of s14adf, taking values of $x$ in the range $0.0<x<50.0$, and $n$ in the range $1\le n\le 50$, have shown that the maximum relative error is a loss of approximately two decimal places of precision. Tests with $n=0$, i.e., testing the function $\psi \left(x\right)$, have shown somewhat better accuracy, except at points close to the zero of $\psi \left(x\right)$, $x\simeq 1.461632$, where only absolute accuracy can be obtained.
8
Parallelism and Performance
s14adf is not threaded in any implementation.
The time taken for a call of s14adf is approximately proportional to $m$, plus a constant. In general, it is much cheaper to call s14adf with $m$ greater than $1$ to evaluate the function $w\left(\mathit{k},x\right)$, for $\mathit{k}=n,\dots ,n+m1$, rather than to make $m$ separate calls of s14adf.
10
Example
This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
10.1
Program Text
Program Text (s14adfe.f90)
10.2
Program Data
Program Data (s14adfe.d)
10.3
Program Results
Program Results (s14adfe.r)