C05BAF 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
The function
f is many-to-one, and so, except at
0,
W is multivalued. C05BAF restricts
W and its argument
x to be real, resulting in a function defined for
x≥-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
Wx+1. We denote by
W0 the branch satisfying
W0x≥-1 for all real
x, and by
W-1 the branch satisfying
W-1x≤-1 for all real
x. You may select your branch of interest using the parameter
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+Δx rather than
Wx: by setting
OFFSET=.TRUE. on entry you inform C05BAF that you are providing
Δx, not
x, in
X.
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
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
For a high percentage of legal
X on input, C05BAF is accurate to the number of decimal digits of precision on the host machine (see
X02BEF). 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.
None.
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.