NAG Library Function Document

nag_real_polygamma (s14aec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_real_polygamma (s14aec) returns the value of the

k

th derivative of the psi function

ψ (x)

for real

x

and

k = 0, 1, \dots, 6

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_real_polygamma (double x, Integer k, NagError *fail)

3 Description

nag_real_polygamma (s14aec) evaluates an approximation to the

k

th derivative of the psi function

ψ (x)

given by

ψ^{(k)} (x) = \frac{d^{k}}{{d x}^{k}} ψ (x) = \frac{d^{k}}{{d x}^{k}} (\frac{d}{d x} \log_{e} Γ (x)),

where

x

is real with

x \neq 0, - 1, - 2, \dots

and

k = 0, 1, \dots, 6

. For negative non-integer values of

x

, the recurrence relationship

ψ^{(k)} (x + 1) = ψ^{(k)} (x) + \frac{d^{k}}{{d x}^{k}} (\frac{1}{x})

is used. The value of

\frac{{(- 1)}^{k + 1} ψ^{(k)} (x)}{k!}

is obtained by a call to a function based on PSIFN in Amos (1983).

Note that

ψ^{(k)} (x)

is also known as the polygamma function. Specifically,

ψ^{(0)} (x)

is often referred to as the digamma function and

ψ^{(1)} (x)

as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).

4 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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: x must not be ‘too close’ (see Section 6) to a non-positive integer.
2: k – IntegerInput: On entry: the function $ψ^{(k)} (z)$ to be evaluated.
Constraint: $0 \leq k \leq 6$ .
3: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT: On entry, $k = 〈value〉$ .
Constraint: $0 \leq k \leq 6$ .
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.
NE_OVERFLOW_LIKELY: The evaluation has been abandoned due to the likelihood of overflow. The result is returned as zero.
NE_REAL: On entry, $x = 〈value〉$ .
Constraint: x must not be ‘too close’ to a non-positive integer. That is, $|x - nint (x)| \geq$ machine precision $\times$ nint $(x)$ .
NE_UNDERFLOW_LIKELY: The evaluation has been abandoned due to the likelihood of underflow. The result is returned as zero.

7 Accuracy

All constants in the underlying functions 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 (t, 18)

. 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

- ψ^{(0)} (x)

have shown somewhat improved accuracy, except at points near the positive zero of

ψ^{(0)} (x)

x = 1.46 \dots

, where only absolute accuracy can be obtained.

8 Further Comments

None.

9 Example

The example program evaluates

ψ^{(2)} (x)

x = 2.5

, and prints the results.

NAG Library Function Documentnag_real_polygamma (s14aec)