An Algorithm for the Matrix Lambert W Function

Fasi, Massimiliano and Higham, Nicholas J. and Iannazzo, Bruno (2014) An Algorithm for the Matrix Lambert W Function. [MIMS Preprint]

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Abstract

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.

Item Type: MIMS Preprint
Uncontrolled Keywords: Lambert $W$ function, primary matrix function, Newton method, matrix iteration, numerical stability, Schur--Parlett method
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 26 Nov 2014
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2196

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