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 = Nag_TRUE

on entry you inform nag_lambertW_complex (c05bbc) 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: $offset$ – Nag_BooleanInput: On entry: controls whether or not z is being specified as an offset from $- \exp (- 1)$ .
3: $z$ – ComplexInput: On entry: if $offset = Nag_TRUE$ , 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 $offset = Nag_FALSE$ , z is the argument $z$ of the function; that is, $W (β)$ is computed, where $β = z$ .
4: $w$ – Complex *Output: On exit: the value $W (β)$ : see also the description of z.
5: $resid$ – double *Output: On exit: the residual $|W (β) \exp (W (β)) - β|$ : see also the description of z.
6: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_REAL: 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 $Im (z) = 〈value〉$ .
NW_TOO_MANY_ITER: The iterative procedure used internally did not converge in $〈value〉$ iterations. Check the value of resid for the accuracy of w.

7

Accuracy

For a high percentage of

z

, nag_lambertW_complex (c05bbc) is accurate to the number of decimal digits of precision on the host machine (see nag_decimal_digits (X02BEC)). 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

nag_lambertW_complex (c05bbc) 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 Function Document

nag_lambertW_complex (c05bbc)

▸▿ 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