Numerical Methods for the Matrix Lambert W Function

The ``Lambert W'' is a not-so-widely-known special function implicitly defined by the equation z = W(z) exp (W(z)). Although its simple definition, which recalls the logarithm, its computation is made non-trivial by the lack of regularity and logarithm-like properties. Thus, iterative methods are a natural choice, which works very well in practice and can be easily extended to the matrix case. Nevertheless, turning to matrices poses some new challenges which require a non-trivial extension of the original definition and the development of a quite involved algorithm.