# NAG Library Function Document

## 1Purpose

nag_lambertW (c05bac) returns the real values of Lambert's $W$ function $W\left(x\right)$.

## 2Specification

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

## 3Description

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
 $fw = wew for w∈C .$
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 $x\ge -\mathrm{exp}\left(-1\right)$ and which is double valued on the interval $\left(-\mathrm{exp}\left(-1\right),0\right)$. This double-valued function is split into two real-valued branches according to the sign of $W\left(x\right)+1$. We denote by ${W}_{0}$ the branch satisfying ${W}_{0}\left(x\right)\ge -1$ for all real $x$, and by ${W}_{-1}$ the branch satisfying ${W}_{-1}\left(x\right)\le -1$ 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 $-\mathrm{exp}\left(-1\right)$ greater accuracy comes from evaluating $W\left(-\mathrm{exp}\left(-1\right)+\Delta x\right)$ rather than $W\left(x\right)$: by setting ${\mathbf{offset}}=\mathrm{Nag_TRUE}$ on entry you inform nag_lambertW (c05bac) that you are providing $\Delta x$, not $x$, in x.

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: if ${\mathbf{offset}}=\mathrm{Nag_TRUE}$, x is the offset $\Delta x$ from $-\mathrm{exp}\left(-1\right)$ of the intended argument to $W$; that is, $W\left(\beta \right)$ is computed, where $\beta =-\mathrm{exp}\left(-1\right)+\Delta x$.
If ${\mathbf{offset}}=\mathrm{Nag_FALSE}$, x is the argument $x$ of the function; that is, $W\left(\beta \right)$ is computed, where $\beta =x$.
Constraints:
• if ${\mathbf{branch}}=0$, $-\mathrm{exp}\left(-1\right)\le \beta$;
• if ${\mathbf{branch}}=-1$, $-\mathrm{exp}\left(-1\right)\le \beta <0.0$.
2:    $\mathbf{branch}$IntegerInput
On entry: the real branch required.
${\mathbf{branch}}=0$
The branch ${W}_{0}$ is selected.
${\mathbf{branch}}=-1$
The branch ${W}_{-1}$ is selected.
Constraint: ${\mathbf{branch}}=0$ or $-1$.
3:    $\mathbf{offset}$Nag_BooleanInput
On entry: controls whether or not x is being specified as an offset from $-\mathrm{exp}\left(-1\right)$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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, ${\mathbf{branch}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{branch}}=0$ 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.
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, ${\mathbf{branch}}=-1$, ${\mathbf{offset}}=\mathrm{Nag_FALSE}$ and ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{branch}}=-1$ and ${\mathbf{offset}}=\mathrm{Nag_FALSE}$ then ${\mathbf{x}}<0.0$.
On entry, ${\mathbf{branch}}=-1$, ${\mathbf{offset}}=\mathrm{Nag_TRUE}$ and ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{branch}}=-1$ and ${\mathbf{offset}}=\mathrm{Nag_TRUE}$ then ${\mathbf{x}}<\mathrm{exp}\left(-1.0\right)$.
On entry, ${\mathbf{offset}}=\mathrm{Nag_TRUE}$ and ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{offset}}=\mathrm{Nag_TRUE}$ then ${\mathbf{x}}\ge 0.0$.
On entry, ${\mathbf{offset}}=\mathrm{Nag_FALSE}$ and ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{offset}}=\mathrm{Nag_FALSE}$ then ${\mathbf{x}}\ge -\mathrm{exp}\left(-1.0\right)$.
NW_REAL
For the given offset ${\mathbf{x}}$, $W$ is negligibly different from $-1$: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
${\mathbf{x}}$ is close to $-\mathrm{exp}\left(-1\right)$. Enter ${\mathbf{x}}$ as an offset to $-\mathrm{exp}\left(-1\right)$ for greater accuracy: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

For a high percentage of legal ${\mathbf{x}}$ 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 ${\mathbf{x}}$. This depends on the accuracy of the base-$10$ logarithm on your system.

## 8Parallelism and Performance

nag_lambertW (c05bac) is not threaded in any implementation.

None.

## 10Example

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 $-\mathrm{exp}\left(-1\right)$, and the arguments ${\mathbf{x}}$ themselves. It then evaluates the function for these sets of input data ${\mathbf{x}}$ and prints the results.

### 10.1Program Text

Program Text (c05bace.c)

### 10.2Program Data

Program Data (c05bace.d)

### 10.3Program Results

Program Results (c05bace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017