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A new scaling for Newton's iteration for the polar decomposition and its backward stability
Byers, Ralph Xu, Hongguo
Byers, Ralph
Xu, Hongguo
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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.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/070699895.
Date
2008-04-27
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Society for Industrial and Applied Mathematics
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Keywords
matrix sign function, polar decomposition, singular value decomposition (SVD), Newton's method, numerical stability, scaling
Citation
Byers, Ralph & Xu, Hongguo. "A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability." SIAM. J. Matrix Anal. & Appl., 30(2), 822–843. (22 pages) 2008. http://dx.doi.org/10.1137/070699895.