A new scaling for Newton’s iteration for the polar decomposition and its backward stability. (English) Zbl 1170.65019

The authors propose a scaling scheme for Newton’s iterations for calculating the polar decomposition. It is pointed out that the scheme is costless and simplifies iterations. For matrices, with proper condition number bounds, no more than nine iterations are needed for convergence to the unitary polar factor with a convergence tolerance roughly equal to the machine epsilon. It is shown that by employing the bi-diagonal factorization for matrix inversion with Hingam’s scaling [cf. N. J. Higham, Linear Algebra Appl. 103, 103–118 (1988; Zbl 0649.65026)] or with a Frobenius norm scaling, leads to backward stability, if the iterations are not too large.


65F05 Direct numerical methods for linear systems and matrix inversion


Zbl 0649.65026
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