The Lambert W function of matrices

Massimiliano Fasi (The University of Manchester)

ATB Frank Adams 1,

Despite being implicitly defined by the simple equation \(z = W(z) \exp(W(z))\), the so-called "Lambert W" function is far from being straightforward to compute, mainly because of the lack of regularity and logarithm-like properties. The Newton method works sufficiently well for scalars, but a trivial extension to the matrix case suffers from numerical instability, and the choice of the starting value requires additional attention when the matrix iteration is considered. Both issues can be fixed by adopting a numerically stable coupled variant of the Newton iteration and by employing a blocked Schur decomposition.

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