Higham, Nicholas J. and Mackey, D. Steven and Mackey, Niloufer and Tisseur, Françoise (2005) Functions Preserving Matrix Groups and Iterations for the Matrix Square Root. SIAM Journal On Matrix Analysis and Applications, 26 (3). pp. 849877. ISSN 10957162
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Abstract
For any matrix automorphism group $\G$ associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of $A\in\G$ also lie in $\G$; moreover, the polar factors of $A$ lie in $\G$ if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and pseudoorthogonal groups. This work is concerned with exploiting the structure of $\G$ when computing the polar and matrix sign decompositions of matrices in $\G$. We give sufficient conditions for a matrix iteration to preserve the group structure and show that a family of globally convergent rational Pad\'ebased iterations of Kenney and Laub satisfy these conditions. The wellknown scaled Newton iteration for computing the unitary polar factor does not preserve group structure, but we show that the approach of the iterates to the group is precisely tethered to the approach to unitarity, and that this forces a different and exploitable structure in the iterates. A similar relation holds for the Newton iteration for the matrix sign function. We also prove that the number of iterations needed for convergence of the structurepreserving methods can be precisely predicted by running an associated scalar iteration. Numerical experiments are given to compare the cubically and quintically converging iterations with Newton's method and to test stopping criteria. The overall conclusion is that the structurepreserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problemdependent.
Item Type:  Article 

Uncontrolled Keywords:  automorphism group, bilinear form, sesquilinear form, scalar product, adjoint, Fr\'echet derivative, stability analysis, perplectic matrix, pseudoorthogonal matrix, generalized polar decomposition, matrix sign function, matrix $p$th root, matrix square root, structure preservation, matrix iteration, Newton iteration 
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:  27 Oct 2006 
Last Modified:  20 Oct 2017 14:12 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/636 
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Functions Preserving Matrix Groups and Iterations for the Matrix Square Root. (deposited 24 Nov 2005)
 Functions Preserving Matrix Groups and Iterations for the Matrix Square Root. (deposited 27 Oct 2006) [Currently Displayed]
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