# NAG FL Interfaces14aef (psi_​deriv_​real)

## 1Purpose

s14aef returns the value of the $k$th derivative of the psi function $\psi \left(x\right)$ for real $x$ and $k=0,1,\dots ,6$, via the function name.

## 2Specification

Fortran Interface
 Function s14aef ( x, k,
 Real (Kind=nag_wp) :: s14aef Integer, Intent (In) :: k Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s14aef_ (const double *x, const Integer *k, Integer *ifail)
The routine may be called by the names s14aef or nagf_specfun_psi_deriv_real.

## 3Description

s14aef evaluates an approximation to the $k$th derivative of the psi function $\psi \left(x\right)$ given by
 $ψ k x=dkdxk ψx=dkdxk ddx loge⁡Γx ,$
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
 $ψ k x+1=ψ k x+dkdxk 1x$
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).

## 4References

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

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}$ must not be ‘too close’ (see Section 6) to a non-positive integer.
2: $\mathbf{k}$Integer Input
On entry: the function ${\psi }^{\left(k\right)}\left(x\right)$ to be evaluated.
Constraint: $0\le {\mathbf{k}}\le 6$.
3: $\mathbf{ifail}$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, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\le 6$.
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, x is ‘too close’ to a non-positive integer: ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and $\mathrm{nint}\left({\mathbf{x}}\right)=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=2$
Evaluation abandoned due to likelihood of underflow.
${\mathbf{ifail}}=3$
Evaluation abandoned due to likelihood of overflow.
${\mathbf{ifail}}=-99$
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. 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.

## 8Parallelism and Performance

s14aef is not threaded in any implementation.

None.

## 10Example

This example evaluates ${\psi }^{\left(2\right)}\left(x\right)$ at $x=2.5$, and prints the results.

### 10.1Program Text

Program Text (s14aefe.f90)

### 10.2Program Data

Program Data (s14aefe.d)

### 10.3Program Results

Program Results (s14aefe.r)