# NAG Library Function Document

## 1Purpose

nag_real_polygamma (s14aec) returns the value of the $k$th derivative of the psi function $\psi \left(x\right)$ for real $x$ and $k=0,1,\dots ,6$.

## 2Specification

 #include #include
 double nag_real_polygamma (double x, Integer k, NagError *fail)

## 3Description

nag_real_polygamma (s14aec) 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 nag_polygamma_deriv (s14adc), which is based on the function 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:    $\mathbf{x}$doubleInput
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:    $\mathbf{k}$IntegerInput
On entry: the function ${\psi }^{\left(k\right)}\left(x\right)$ to be evaluated.
Constraint: $0\le {\mathbf{k}}\le 6$.
3:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\le 6$.
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_OVERFLOW_LIKELY
Evaluation abandoned due to likelihood of overflow.
NE_REAL
On entry, x is ‘too close’ to a non-positive integer: ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and $\mathrm{nint}\left({\mathbf{x}}\right)=〈\mathit{\text{value}}〉$.
NE_UNDERFLOW_LIKELY
Evaluation abandoned due to likelihood of underflow.

## 7Accuracy

All constants in nag_polygamma_deriv (s14adc) 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

nag_real_polygamma (s14aec) 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 (s14aece.c)

### 10.2Program Data

Program Data (s14aece.d)

### 10.3Program Results

Program Results (s14aece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017