# NAG CL Interfaces14acc (polygamma)

## 1Purpose

s14acc returns a value of the function $\psi \left(x\right)-\mathrm{ln}x$, where $\psi$ is the psi function $\psi \left(x\right)=\frac{d}{dx}\mathrm{ln}\Gamma \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$.

## 2Specification

 #include
 double s14acc (double x, NagError *fail)
The function may be called by the names: s14acc, nag_specfun_polygamma or nag_polygamma_fun.

## 3Description

s14acc returns a value of the function $\psi \left(x\right)-\mathrm{ln}x$. The psi function is computed without the logarithmic term so that when $x$ is large, sums or differences of psi functions may be computed without unnecessary loss of precision, by analytically combining the logarithmic terms. For example, the difference $d=\psi \left(x+\frac{1}{2}\right)-\psi \left(x\right)$ has an asymptotic behaviour for large $x$ given by $d\sim \mathrm{ln}\left(x+\frac{1}{2}\right)-\mathrm{ln}x+\mathit{O}\left(\frac{1}{{x}^{2}}\right)\sim \mathrm{ln}\left(1+\frac{1}{2x}\right)\sim \frac{1}{2x}$.
Computing $d$ directly would amount to subtracting two large numbers which are close to $\mathrm{ln}\left(x+\frac{1}{2}\right)$ and $\mathrm{ln}x$ to produce a small number close to $\frac{1}{2x}$, resulting in a loss of significant digits. However, using this function to compute $f\left(x\right)=\psi \left(x\right)-\mathrm{ln}x$, we can compute $d=f\left(x+\frac{1}{2}\right)-f\left(x\right)+\mathrm{ln}\left(1+\frac{1}{2x}\right)$, and the dominant logarithmic term may be computed accurately from its power series when $x$ is large. Thus we avoid the unnecessary loss of precision.
The function is derived from the function PSIFN in Amos (1983).

## 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: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_OVERFLOW_LIKELY
Computation halted due to likelihood of overflow. x may be too small. ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
NE_REAL
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.
NE_UNDERFLOW_LIKELY
Computation halted due to likelihood of underflow. x may be too large. ${\mathbf{x}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

All constants in s14acc 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 $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$.
With the above proviso, results returned by this function should be accurate almost to full precision, except at points close to the zero of $\psi \left(x\right)$, $x\simeq 1.461632$, where only absolute rather than relative accuracy can be obtained.

## 8Parallelism and Performance

s14acc is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s14acce.c)

### 10.2Program Data

Program Data (s14acce.d)

### 10.3Program Results

Program Results (s14acce.r)