nag_lambertW_complex (c05bbc) calculates an approximate value for Lambert's function (sometimes known as the ‘product log’ or ‘Omega’ function), which is the inverse function of
The function is many-to-one, and so, except at , is multivalued. nag_lambertW_complex (c05bbc) allows you to specify the branch of 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 are summarised in Figure 1.
Figure 1: Ranges of the branches of
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 .
The precise method used to approximate is as described in Corless et al. (1996). For close to greater accuracy comes from evaluating rather than : by setting on entry you inform nag_lambertW_complex (c05bbc) that you are providing , not , in z.
Corless R M, Gonnet G H, Hare D E G, Jeffrey D J and Knuth D
E (1996) On the Lambert function Advances in Comp. Math.3 329–359
On entry: the branch required.
On entry: controls whether or not z is being specified as an offset from .
On entry: if , z is the offset from of the intended argument to ; that is, is computed, where .
If , z is the argument of the function; that is, is computed, where .
– Complex *Output
On exit: the value : see also the description of z.
– double *Output
On exit: the residual : see also the description of z.
– NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 18.104.22.168 in the Essential Introduction for further information.
On entry, argument had an illegal value.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
For the given offset , is negligibly different from : and .
is close to . Enter as an offset to for greater accuracy: and .
The iterative procedure used internally did not converge in iterations. Check the value of resid for the accuracy of w.
For a high percentage of , 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 . This depends on the accuracy of the base- logarithm on your system.
8 Parallelism and Performance
9 Further Comments
The following figures show the principal branch of .