# NAG C Library Function Document

## 1Purpose

nag_polygamma_deriv (s14adc) returns a sequence of values of scaled derivatives of the psi function $\psi \left(x\right)$ (also known as the digamma function).

## 2Specification

 #include #include
 void nag_polygamma_deriv (double x, Integer n, Integer m, double ans[], NagError *fail)

## 3Description

nag_polygamma_deriv (s14adc) 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 function is derived from the function 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:    $\mathbf{x}$doubleInput
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 $\mathit{k}=n,\dots ,n+m-1$.
Constraint: ${\mathbf{m}}\ge 1$.
4:    $\mathbf{ans}\left[{\mathbf{m}}\right]$doubleOutput
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:    $\mathbf{fail}$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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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_INTERNAL_WORKSPACE
There is not enough internal workspace to continue computation. m is probably too large.
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
Computation abandoned due to the likelihood of overflow.
NE_REAL
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.
NE_UNDERFLOW_LIKELY
Computation abandoned due to the likelihood of underflow.

## 7Accuracy

All constants in nag_polygamma_deriv (s14adc) 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 nag_polygamma_deriv (s14adc), 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 nag_polygamma_deriv (s14adc) is approximately proportional to $m$, plus a constant. In general, it is much cheaper to call nag_polygamma_deriv (s14adc) 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 nag_polygamma_deriv (s14adc).

## 10Example

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