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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 2.7 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_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.

An unexpected error has been triggered by this function. Please contact NAG.
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.
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

nag_lambertW (c05bac) is not threaded in any implementation.

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)