An algorithm for the matrix Lambert W function

An algorithm is proposed for computing primary matrix Lambert $W$ functions of a square matrix $A$, which are solutions of the matrix equation $We^W = A$. The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton's method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton's method for the Lambert $W$ function is shown to be numerically unstable. By reorganizing the iteration a new Newton variant is constructed that is proved to be numerically stable. Numerical experiments demonstrate that the algorithm is able to compute the branches of the matrix Lambert $W$ function in a numerically reliable way.