NAG Library Function Document

nag_lambertW (c05bac)

+− 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

nag_lambertW (c05bac) returns the real values of Lambert's

W

function

W (x)

2 Specification

#include <nag.h>

#include <nagc05.h>

double

nag_lambertW (double x, Integer branch, Nag_Boolean offset, NagError *fail)

3 Description

nag_lambertW (c05bac) calculates an approximate value for the real branches of 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. nag_lambertW (c05bac) restricts

W

and its argument

x

to be real, resulting in a function defined for

x \geq - \exp (- 1)

and which is double valued on the interval

(- \exp (- 1), 0)

. This double-valued function is split into two real-valued branches according to the sign of

W (x) + 1

. We denote by

W_{0}

the branch satisfying

W_{0} (x) \geq - 1

for all real

x

, and by

W_{- 1}

the branch satisfying

W_{- 1} (x) \leq - 1

for all real

x

. You may select your branch of interest using the argument branch.

The precise method used to approximate

W

is described fully in Barry et al. (1995). For

x

close to

- \exp (- 1)

greater accuracy comes from evaluating

W (- \exp (- 1) + Δ x)

rather than

W (x)

: by setting

offset = Nag_TRUE

on entry you inform nag_lambertW (c05bac) that you are providing

Δ x

, not

x

, in x.

4 References

Barry D J, Culligan–Hensley P J, and Barry S J (1995) Real values of the

W

-function ACM Trans. Math. Software 21(2) 161–171

5 Arguments

1: x – doubleInput

On entry: if

offset = Nag_TRUE

, x is the offset

Δ x

from

- \exp (- 1)

of the intended argument to

W

; that is,

W (β)

is computed, where

β = - \exp (- 1) + Δ x

offset = Nag_FALSE

, x is the argument

x

of the function; that is,

W (β)

is computed, where

β = x

Constraints:

if $branch = 0$ , $- \exp (- 1) \leq β$ ;
if $branch = - 1$ , $- \exp (- 1) \leq β < 0.0$ .

2: branch – IntegerInput

On entry: the real branch required.

$branch = 0$: The branch $W_{0}$ is selected.
$branch = - 1$: The branch $W_{- 1}$ is selected.

Constraint:

branch = 0

- 1

3: offset – Nag_BooleanInput

On entry: controls whether or not x is being specified as an offset from

- \exp (- 1)

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT: On entry, $branch = ⟨value⟩$ .
Constraint: $branch = 0$ or $- 1$ .
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.
NE_REAL: On entry, $branch = - 1$ , $offset = Nag_FALSE$ and $x = ⟨value⟩$ .
Constraint: if $branch = - 1$ and $offset = Nag_FALSE$ then $x < 0.0$ .
On entry, $branch = - 1$ , $offset = Nag_TRUE$ and $x = ⟨value⟩$ .
Constraint: if $branch = - 1$ and $offset = Nag_TRUE$ then $x < \exp (- 1.0)$ .
On entry, $offset = Nag_TRUE$ and $x = ⟨value⟩$ .
Constraint: if $offset = Nag_TRUE$ then $x \geq 0.0$ .
On entry, $offset = Nag_FALSE$ and $x = ⟨value⟩$ .
Constraint: if $offset = Nag_FALSE$ then $x \geq - \exp (- 1.0)$ .
NW_REAL: For the given offset $x$ , $W$ is negligibly different from $- 1$ : $x = ⟨value⟩$ .
$x$ is close to $- \exp (- 1)$ . Enter $x$ as an offset to $- \exp (- 1)$ for greater accuracy: $x = ⟨value⟩$ .

7 Accuracy

For a high percentage of legal

x

on input, nag_lambertW (c05bac) 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 implementations and for a small proportion of such

x

. This depends on the accuracy of the base-

10

logarithm on your system.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads from a file the values of the required branch, whether or not the arguments to

W

are to be considered as offsets to

- \exp (- 1)

, and the arguments

x

themselves. It then evaluates the function for these sets of input data

x

and prints the results.

NAG Library Function Documentnag_lambertW (c05bac)