1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.

Constraint: $x$ must not be ‘too close’ (see Section 6) to a non-positive integer.
2: $k$ – Integer Input: On entry: the function $ψ^{(k)} (x)$ to be evaluated.

Constraint: $0 \leq k \leq 6$ .
3: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $k = 〈value〉$ .
Constraint: $k \leq 6$ .

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

On entry, x is ‘too close’ to a non-positive integer: $x = 〈value〉$ and $nint (x) = 〈value〉$ .

$ifail = 2$: Evaluation abandoned due to likelihood of underflow.

$ifail = 3$: Evaluation abandoned due to likelihood of overflow.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

All constants in s14adf 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

s14aef is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates

ψ^{(2)} (x)

x = 2.5

, and prints the results.

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s14ae: FL CL AD

NAG FL Interfaces14aef (psi_​deriv_​real)

▸▿ 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

NAG FL Interface
s14aef (psi_deriv_real)