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We introduce a dynamically weighted Halley (DWH) iteration for computing the polar decomposition of a matrix, and we prove that the new method is globally and asymptotically cubically convergent. For matrices with condition number no greater than 10¹⁶, the DWH method needs at most six iterations for convergence with the tolerance 10⁻¹⁶. The Halley iteration can be implemented via QR decompositions without explicit matrix inversions. Therefore, it is an inverse free communication friendly algorithm for the emerging multicore and hybrid high performance computing systems.
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Nakatsukasa et al. (2010) studied this question.
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