public static double s14ae(
	double x,
	int k,
	out int ifail
)

Public Shared Function s14ae ( _
	x As Double, _
	k As Integer, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s14ae(
	double x, 
	int k, 
	[OutAttribute] int% ifail
)

static member s14ae : 
        x : float * 
        k : int * 
        ifail : int byref -> float 

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

s14ae evaluates an approximation to the $k$th derivative of the psi function $\psi \left(x\right)$ given by where $x$ is real with $x\ne 0,-1,-2,\dots$ and $k=0,1,\dots ,6$. For negative noninteger values of $x$, the recurrence relationship 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).

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

Errors or warnings detected by the method:

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 $p=\min \left(t,18\right)$. 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 $x=1.46\dots$, where only absolute accuracy can be obtained.

None.

None.

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