## 2014.58: An Algorithm for the Matrix Lambert W Function

2014.58:
Massimiliano Fasi, Nicholas J. Higham and Bruno Iannazzo
(2015)
*An Algorithm for the Matrix Lambert W Function.*
SIAM Journal on Matrix Analysis and Applications, 36 (2).
pp. 669-685.
ISSN 1095-7162

*This is the latest version of this eprint.*

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DOI: 10.1137/140997610

## 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: | Article |
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Uncontrolled Keywords: | Lambert $W$ function, primary matrix function, Newton method, matrix iteration, numerical stability, Schur--Parlett method |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2014.58 |

Deposited By: | Nick Higham |

Deposited On: | 12 July 2015 |

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- An Algorithm for the Matrix Lambert W Function (deposited 12 July 2015)
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