Command: LambertWm1

Synopsis

double LambertWm1 -> double

References

[1] Corless R. M. Gonnet G. H. Hare D. E. G. Jeffrey D. J. & Knuth D. E.

(1996). On the lambert w function. Advances in Computational Mathematics 5

329--359.

[2] Galassi M. Davies J. Theiler J. Gough B. Jungman G. Booth M.

& Rossi F. (2006). GNU Scientific Library Reference Manual (2nd Ed.).

Network Theory Limited.

[3] Wikipedia (2009). Lambert W function ---wikipedia the free encyclopedia.

Description

The Lambert-W function [1] is the inverse function of x=W*exp(W). For real values of

x and W the function W(x) is defined on [-1/e \infty). On the interval [-1/e 0)

it is double valued. The two branches coincide at W(-1/e)=-1. The so called

principal branch LambertW0 continuously grows (W>=-1) and crosses the origin (0 0).

The non-principal branch LambertWm1 is defined on [-1/e 0) and declines to -\infty for

growing x.

NEST uses the GSL [2] implementations of LambertW0 and LambertWm1 if available and

falls back to to the iterative scheme LambertW suggested in [1] if not.

The GSL interfaces for LambertW0 and LambertWm1 are in the SpecialFunctionsModule

of SLI.

File

lib/sli/mathematica.sli

Author

Diesmann

The Lambert-W function has applications in many areas as described in [1] and [3].

For example the problem of finding the location of the maximum of the postsynaptic

potential generated by an alpha-function shaped current can be reduced to the

equation

exp(s) = 1 + a*s .

Here s is the location of the maximum in scaled time s=b*t where

b= 1/tau_alpha - 1/tau_m and a is the ratio of the time constants tau_m/tau_alpha.

In terms of the Lambda_W function the solution is

s=1/a * (-aW(-exp(-1/a)/a) -1 ) .

The solution is guaranteed to live on the non-principal branch because the scaled time

needs to be positive. This requires W < -1/a which is trivially fulfilled for the

non-principal branch and there is no solution on the principal branch.

Version: 090818

Examples

The Lambert-W function has applications in many areas as described in [1] and [3].

For example the problem of finding the location of the maximum of the postsynaptic

potential generated by an alpha-function shaped current can be reduced to the

equation

exp(s) = 1 + a*s .

Here s is the location of the maximum in scaled time s=b*t where

b= 1/tau_alpha - 1/tau_m and a is the ratio of the time constants tau_m/tau_alpha.

In terms of the Lambda_W function the solution is

s=1/a * (-aW(-exp(-1/a)/a) -1 ) .

The solution is guaranteed to live on the non-principal branch because the scaled time

needs to be positive. This requires W < -1/a which is trivially fulfilled for the

non-principal branch and there is no solution on the principal branch.

Version: 090818