1Purpose

s14adf returns a sequence of values of scaled derivatives of the psi function $\psi \left(x\right)$ (also known as the digamma function).

2Specification

Fortran Interface
 Subroutine s14adf ( x, n, m, ans,
 Integer, Intent (In) :: n, m Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x Real (Kind=nag_wp), Intent (Out) :: ans(m)
#include <nag.h>
 void s14adf_ (const double *x, const Integer *n, const Integer *m, double ans[], Integer *ifail)
The routine may be called by the names s14adf or nagf_specfun_polygamma_deriv.

3Description

s14adf computes $m$ values of the function
 $wk,x=-1k+1ψ k x k! ,$
for $x>0$, $k=n$, $n+1,\dots ,n+m-1$, where $\psi$ is the psi function
 $ψx=ddx ln⁡Γx=Γ′x Γx ,$
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
 $wk,x∼εk,x+12xk+1 +1xk∑j=1∞B2j2j+k-1! 2j!k!x2j$
for large $x$ greater than a machine-dependent value ${x}_{\mathrm{min}}$, followed by backward recurrence using
 $wk,x=wk,x+1+x-k-1$
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
 $wk,x=∑j=1∞1x+jk+1,$
which converges rapidly for large $k$, is used instead.

4References

NIST Digital Library of Mathematical Functions
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502

5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{n}$Integer Input
On entry: the index of the first member $n$ of the sequence of functions.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{m}$Integer Input
On entry: the number of members $m$ required in the sequence $w\left(\mathit{k},x\right)$, for $\mathit{k}=n,\dots ,n+m-1$.
Constraint: ${\mathbf{m}}\ge 1$.
4: $\mathbf{ans}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: the first $m$ elements of ans contain the required values $w\left(\mathit{k},x\right)$, for $\mathit{k}=n,\dots ,n+m-1$.
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value 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 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).

6Error 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, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=4$
Computation abandoned due to the likelihood of underflow.
${\mathbf{ifail}}=5$
Computation abandoned due to the likelihood of overflow.
${\mathbf{ifail}}=6$
There is not enough internal workspace to continue computation. m is probably too large.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7Accuracy

All constants in s14adf are given to approximately $18$ digits of precision. Calling the number of digits of precision in the floating-point 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, 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.

8Parallelism and Performance

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 $\mathit{k}=n,\dots ,n+m-1$, rather than to make $m$ separate calls of s14adf.

10Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.