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2007.9: Computing the Polar Decomposition---with Applications

2007.9: Nicholas J. Higham (1986) Computing the Polar Decomposition---with Applications. SIAM Journal of Scientific and Statistical Computing, 7 (4). pp. 1160-1174. ISSN 1064-8275

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DOI: 10.1137/0907079

Abstract

A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor H of a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to A and 1/2(A / H) is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation bounds for the polar factors are derived. Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.

Item Type:Article
Uncontrolled Keywords:polar decomposition, singular value decomposition, Newton’s method, matrix square root
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2007.9
Deposited By:Nick Higham
Deposited On:25 January 2007

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