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

Byers, Ralph; Xu, Hongguo

SIAM J. Matrix Anal. Appl. 30, No. 2, 822-843 (2008).

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.

Reviewer: R. P. Tewarson (Stony Brook)

MSC:

65F05

Direct numerical methods for linear systems and matrix inversion

Keywords:

polar decomposition; singular value decomposition (SVD); scaling

Citations:

Zbl 0649.65026