NAG Library Routine Document

C05BBF

on which you would like the results to lie by using the parameter BRANCH. Our choice of branch cuts is as in Corless et al. (1996), and the ranges of the branches of

W

are summarised in Figure 1.

Figure 1: Ranges of the branches of $W (z)$

For more information about the closure of each branch, which is not displayed in Figure 1, see Corless et al. (1996). The dotted lines in the Figure denote the asymptotic boundaries of the branches, at multiples of

π

The precise method used to approximate

W

is as described in Corless et al. (1996). For

z

close to

- \exp (- 1)

greater accuracy comes from evaluating

W (- \exp (- 1) + Δ z)

rather than

W (z)

: by setting

OFFSET = .TRUE.

on entry you inform C05BBF that you are providing

Δ z

, not

z

, in Z.

4 References

Corless R M, Gonnet G H, Hare D E G, Jeffrey D J and Knuth D E (1996) On the Lambert

W

function Advances in Comp. Math. 3 329–359

5 Parameters

1: BRANCH – INTEGERInput: On entry: the branch required.
2: OFFSET – LOGICALInput: On entry: controls whether or not Z is being specified as an offset from $- \exp (- 1)$ .
3: Z – COMPLEX (KIND=nag_wp)Input: On entry: if $OFFSET = .TRUE.$ , Z is the offset $Δ z$ from $- \exp (- 1)$ of the intended argument to $W$ ; that is, $W (β)$ is computed, where $β = - \exp (- 1) + Δ z$ .
If $OFFSET = .FALSE.$ , Z is the argument $z$ of the function; that is, $W (β)$ is computed, where $β = z$ .
4: W – COMPLEX (KIND=nag_wp)Output: On exit: the value $W (β)$ : see also the description of Z.
5: RESID – REAL (KIND=nag_wp)Output: On exit: the residual $|W (β) \exp (W (β)) - β|$ : see also the description of Z.
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, because for this routine the values of the output parameters may be useful even if $IFAIL \neq 0$ on exit, the recommended value is $- 1$ . 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).

Note: C05BBF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: Warning: the actual argument to $W$ was very close to $- \exp (- 1)$ . If $IFAIL = 0$ or $- 1$ on entry, the output message provides more details of the nature of the warning.

$IFAIL = 2$: Warning: the iterative procedure used internally did not appear to be converging. Check the value of RESID for the accuracy of W.

7 Accuracy

For a high percentage of

Z

, C05BBF is accurate to the number of decimal digits of precision on the host machine (see X02BEF). An extra digit may be lost on some platforms and for a small proportion of

Z

. This depends on the accuracy of the base-

10

logarithm on your system.

8 Further Comments

The following figures show the principal branch of

W

Figure 2: $real (W_{0} (z))$

Figure 3: $Im (W_{0} (z))$

Figure 4: $abs (W_{0} (z))$

9 Example

This example reads from a file the value of the required branch, whether or not the arguments to

W

are to be considered as offsets to

- \exp (- 1)

, and the arguments

Z

themselves. It then evaluates the function for these sets of input data

Z

and prints the results.

NAG Library Routine DocumentC05BBF

+− Contents

1 Purpose

2 Specification

3 Description