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 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 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 noninteger 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 nag_polygamma_deriv (s14adc), which is based on the function 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 $ψ^{(k)} (x)$ 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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \leq 6$ .

On entry, $k = 〈value〉$ .
Constraint: $k \geq 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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: $x = 〈value〉$ and $nint (x) = 〈value〉$ .
NE_UNDERFLOW_LIKELY: Evaluation abandoned due to likelihood of underflow.

7 Accuracy

All constants in nag_polygamma_deriv (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 = \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 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example evaluates

ψ^{(2)} (x)

x = 2.5

, and prints the results.

NAG Library Function Documentnag_real_polygamma (s14aec)