NAG FL Interface
c05bbf (lambertw_​complex)

1 Purpose

c05bbf computes the values of Lambert's W function Wz.

2 Specification

Fortran Interface
Subroutine c05bbf ( branch, offset, z, w, resid, ifail)
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 <nag.h>
void  c05bbf_ (const Integer *branch, const logical *offset, const Complex *z, Complex *w, double *resid, Integer *ifail)
The routine may be called by the names c05bbf or nagf_roots_lambertw_complex.

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
fw = wew   for   wC .  
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.
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=.TRUE. on entry you inform c05bbf 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 Integer Input
On entry: the branch required.
2: offset Logical Input
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 Δz from -exp-1 of the intended argument to W; that is, Wβ is computed, where β=-exp-1+Δz.
If offset=.FALSE., z is the argument z of the function; that is, Wβ is computed, where β=z.
4: w Complex (Kind=nag_wp) Output
On exit: the value Wβ: see also the description of z.
5: resid Real (Kind=nag_wp) Output
On exit: the residual WβexpWβ-β: see also the description of z.
6: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value -1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases c05bbf may return useful information.
ifail=1
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.
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 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.
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Figure 2: realW0z
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 gnuplot_plot_1