s14ae returns the value of the $k$th derivative of the psi function $\psi \left(x\right)$ for real $x$ and $k=0,1,\dots ,6$.

# Syntax

C#
```public static double s14ae(
double x,
int k,
out int ifail
)```
Visual Basic
```Public Shared Function s14ae ( _
x As Double, _
k As Integer, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s14ae(
double x,
int k,
[OutAttribute] int% ifail
)```
F#
```static member s14ae :
x : float *
k : int *
ifail : int byref -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}$ must not be ‘too close’ (see [Error Indicators and Warnings]) to a non-positive integer.
k
Type: System..::..Int32
On entry: the function ${\psi }^{\left(k\right)}\left(x\right)$ to be evaluated.
Constraint: $0\le {\mathbf{k}}\le 6$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s14ae returns the value of the $k$th derivative of the psi function $\psi \left(x\right)$ for real $x$ and $k=0,1,\dots ,6$.

# Description

s14ae evaluates an approximation to the $k$th derivative of the psi function $\psi \left(x\right)$ given by
 $ψkx=dkdxkψx=dkdxkddxloge Γx,$
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
 $ψkx+1=ψkx+dkdxk1x$
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 s14ad, which is based on the method 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).

# References

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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{k}}<0$, or ${\mathbf{k}}>6$, or x is ‘too close’ to a non-positive integer. That is, .
${\mathbf{ifail}}=2$
The evaluation has been abandoned due to the likelihood of underflow. The result is returned as zero.
${\mathbf{ifail}}=3$
The evaluation has been abandoned due to the likelihood of overflow. The result is returned as zero.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

All constants in s14ad 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.

None.

None.

# Example

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

Example program (C#): s14aee.cs

Example program data: s14aee.d

Example program results: s14aee.r