is large, sums or differences of psi functions may be computed without unnecessary loss of precision, by analytically combining the logarithmic terms. For example, the difference

d = ψ (x + \frac{1}{2}) - ψ (x)

has an asymptotic behaviour for large

x

given by

d \sim \ln (x + \frac{1}{2}) - \ln x + O (\frac{1}{x^{2}}) \sim \ln (1 + \frac{1}{2 x}) \sim \frac{1}{2 x}

Computing

d

directly would amount to subtracting two large numbers which are close to

\ln (x + \frac{1}{2})

and

\ln x

to produce a small number close to

\frac{1}{2 x}

, resulting in a loss of significant digits. However, using this routine to compute

f (x) = ψ (x) - \ln x

, we can compute

d = f (x + \frac{1}{2}) - f (x) + \ln (1 + \frac{1}{2 x})

, and the dominant logarithmic term may be computed accurately from its power series when

x

is large. Thus we avoid the unnecessary loss of precision.

The routine is derived from the routine PSIFN in Amos (1983).

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

Constraint: $x > 0.0$ .
2: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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,

x \leq 0.0

. s14acf returns the value zero.

$ifail = 2$: No result is computed because underflow is likely. The value of x is too large. s14acf returns the value zero.

$ifail = 3$: No result is computed because overflow is likely. The value of x is too small. s14acf returns the value zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

All constants in s14acf are given to approximately

18

digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used

t

, then clearly the maximum number of correct digits in the results obtained is limited by

p = \min (t, 18)

With the above proviso, results returned by this routine should be accurate almost to full precision, except at points close to the zero of

ψ (x)

x ≃ 1.461632

, where only absolute rather than relative accuracy can be obtained.

8

Parallelism and Performance

s14acf is not threaded in any implementation.

9

Further Comments

None.

10

Example

The example program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine Document

s14acf (polygamma)

▸▿ 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

2

Specification

3

Description