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

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: $branch$ – 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 $Δ z$ 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: $ifail$ – 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 $ifail \neq 0$ 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

- 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 $- 1$ : $Re (z) = ⟨ value ⟩$ 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 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

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.

c05bb: FL CL CPP AD

NAG FL Interfacec05bbf (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

NAG FL Interface
c05bbf (lambertw_complex)