NAG Library Routine Document

Integer, Intent (In)	::	branch
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Out)	::	resid
Complex (Kind=nag_wp), Intent (In)	::	z
Complex (Kind=nag_wp), Intent (Out)	::	w
Logical, Intent (In)	::	offset

C Header Interface

#include <nagmk26.h>

void	c05bbf_ (const Integer branch, const logical offset, const Complex z, Complex w, double resid, Integer ifail)

3

Description

c05bbf calculates an approximate value for Lambert's

W

function (sometimes known as the ‘product log’ or ‘Omega’ function), which is the inverse function of

f (w) = w e^{w} for w \in C .

The function

f

is many-to-one, and so, except at

0

W

is multivalued. c05bbf allows you to specify the branch of

W

on which you would like the results to lie by using the argument 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

Arguments

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 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, because for this routine the values of the output arguments 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:

$ifail = 1$: For the given offset $z$ , $W$ is negligibly different from $- 1$ : $Re (z) = 〈value〉$ and $Im (z) = 〈value〉$ .

$z$ is close to $- \exp (- 1)$ . Enter $z$ as an offset to $- \exp (- 1)$ for greater accuracy: $Re (z) = 〈value〉$ and $Im (z) = 〈value〉$ .

$ifail = 2$: The iterative procedure used internally did not converge in $〈value〉$ iterations. Check the value of resid for the accuracy of w.

$ifail = - 99$: 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

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

Parallelism and Performance

c05bbf is not threaded in any implementation.

9

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

10

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 Document

c05bbf (lambertw_complex)

▸▿ 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