Stable iterations for the matrix square root

Higham, Nicholas J. (1997) Stable iterations for the matrix square root. Numerical Algorithms, 15 (2). pp. 227-242. ISSN 1572-9265

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Abstract

Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.

Item Type: Article
Uncontrolled Keywords: matrix square root - matrix logarithm - matrix sign function - M-matrix - symmetric positive definite matrix - Padé approximation - numerical stability - Newton''s method - Schulz method - 65F30
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: Ms Lucy van Russelt
Date Deposited: 06 Jul 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/367

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