# NAG CL Interfaces14aec (psi_​deriv_​real)

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## 1Purpose

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
 double s14aec (double x, Integer k, NagError *fail)
The function may be called by the names: s14aec, nag_specfun_psi_deriv_real or nag_real_polygamma.

## 3Description

s14aec evaluates an approximation to the $k$th derivative of the psi function $\psi \left(x\right)$ given by
 $ψ (k) (x)=dkdxk ψ(x)=dkdxk (ddx loge⁡Γ(x)) ,$
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 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).
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: $\mathbf{x}$double 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: $\mathbf{fail}$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_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 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
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 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

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)