NAG Library Function Document

nag_lambertW_complex (c05bbc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_lambertW_complex (c05bbc) computes the values of Lambert's W function Wz.

2
Specification

#include <nag.h>
#include <nagc05.h>
void  nag_lambertW_complex (Integer branch, Nag_Boolean offset, Complex z, Complex *w, double *resid, NagError *fail)

3
Description

nag_lambertW_complex (c05bbc) calculates an approximate value for Lambert's W function (sometimes known as the ‘product log’ or ‘Omega’ function), which is the inverse function of
fw = wew   for   wC .  
The function f is many-to-one, and so, except at 0, W is multivalued. nag_lambertW_complex (c05bbc) 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.
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −3p −2p -p 0 π 2p 3p −10 −5 0 5 10 Branch −3 Branch −2 Branch −1 Principal Branch / Branch 0 Branch 1 Branch 2 Branch 3 gnuplot_plot_1 gnuplot_plot_2 gnuplot_plot_3 gnuplot_plot_4 gnuplot_plot_5 gnuplot_plot_6 gnuplot_plot_7 gnuplot_plot_8 gnuplot_plot_9 gnuplot_plot_10 gnuplot_plot_11 gnuplot_plot_12 gnuplot_plot_13 gnuplot_plot_14
Figure 1: Ranges of the branches of Wz
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+Δz rather than Wz: by setting offset=Nag_TRUE on entry you inform nag_lambertW_complex (c05bbc) that you are providing Δz, 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 Δz from -exp-1 of the intended argument to W; that is, Wβ is computed, where β=-exp-1+Δz.
If offset=Nag_FALSE, z is the argument z of the function; that is, Wβ is computed, where β=z.
4:     w Complex *Output
On exit: the value Wβ: see also the description of z.
5:     resid double *Output
On exit: the residual WβexpWβ-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: Rez=value and Imz=value.
z is close to -exp-1. Enter z as an offset to -exp-1 for greater accuracy: Rez=value and Imz=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.
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 gnuplot_plot_1 gnuplot_plot_2 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 −1 −0.5 0 0.5 1 1.5 2
Figure 2: realW0z
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 gnuplot_plot_1