NDLI: A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Byers, Ralph Xu, Hongguo
Copyright Year	2008
Abstract	We propose a scaling scheme for Newton's iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can, for example, be the Frobenius norms of the matrix and its inverse. In exact arithmetic, for matrices with condition number no greater than $10^{16}$, with this scaling scheme no more than 9 iterations are needed for convergence to the unitary polar factor with a convergence tolerance roughly equal to $10^{-16}$. It is proved that if matrix inverses computed in finite precision arithmetic satisfy a backward-forward error model, then the numerical method is backward stable. It is also proved that Newton's method with Higham's scaling or with Frobenius norm scaling is backward stable.
Starting Page	822
Ending Page	843
Page Count	22
File Format	PDF
ISSN	08954798
DOI	10.1137/070699895
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	2
Volume Number	30
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-09-04
Access Restriction	Subscribed
Subject Keyword	scaling Direct methods for linear systems and matrix inversion Newton's method matrix sign function polar decomposition numerical stability singular value decomposition (SVD)
Content Type	Text
Resource Type	Article
Subject	Analysis

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