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 

Additional Information:  To appear in SIAM Journal on Matrix Analysis and Applications. 
Uncontrolled Keywords:  Lambert $W$ function, primary matrix function, Newton method, matrix iteration, numerical stability, SchurParlett 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:  07 Apr 2015 
Last Modified:  08 Nov 2017 18:18 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2283 
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An Algorithm for the Matrix Lambert W Function. (deposited 26 Nov 2014)
 An Algorithm for the Matrix Lambert W Function. (deposited 07 Apr 2015) [Currently Displayed]
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