lambertW | R Documentation |
The Lambert-W function is defined as the multivalued inverse of the function
z = f(W) = We^W
. The function has two branches. The
principal branch is defined on the interval
\left[-\frac{1}{e}, +\infty\right)
and is always greater
than or equal to -1
. The second branch is defined on the interval
\left[-\frac{1}{e}, 0\right)
and is always less than or equal
to -1
. The real-valued function is not defined for values less than
-\frac{1}{e}
.
lambertW0(x)
lambertWm1(x)
x |
numeric vector of values |
The Lambert-W function is defined for all real
x \geq -\frac{1}{e}
. It has two values in the interval
\left(-\frac{1}{e}, 0\right)
. The values strictly greater than
-1
are assigned to the “principal” branch, also referred to as
W_0
, and the values strictly less than -1
are assigned to the
“secondary” branch, referred to as W_{-1}
. For non-negative
x
, only the principal branch exists as the other real-valued branch
approaches negative infinity as x
approaches 0
. The algorithms used
to calculate the values predominantly follow those in the reference with some
simplifications. There are many applications in which the Lambert-W function is
useful, such as combinatorics, physics, and hydrology. The interested reader is
directed to the references for more detail.
Both functions return the appropriate values in the intervals for which they are
defined. Outside of those intervals, they will return NaN
, except that
lambertW0(Inf)
will return its limit Inf
and lambertWm1(0)
will return its limit -Inf
.
Avraham Adler Avraham.Adler@gmail.com
Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., Knuth, D. E. 1996 "On the Lambert W function", Advances in Computational Mathematics, 5, 329–359, Springer <doi:10.1007/BF02124750>
Fritsch, F. N.; Shafer, R. E. & Crowley, W. P. 1973 "Solution of the transcendental equation (we^w = x)", Communications of the ACM, 16, 123–124, Association for Computing Machinery (ACM) <doi:10.1145/361952.361970>
This package provides similar functionality to the
Lambert
functions in the gsl package without
having to obtain or install the entire GSL.
lambertW0(exp(1)) ## Should equal 1, as 1 * exp(1) = e.
lambertW0(0) ## Should equal 0, as 0 * exp(0) = 0.
lambertW0(-exp(-1)) ## Should equal -1.
lambertWm1(-exp(-1)) ## Should also equal -1.
A <- -2 * exp(-2)
lambertWm1(A) ## Should equal -2
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