Optimizing Halley's Iteration for Computing the Matrix Polar Decomposition
Yuji Nakatsukasa, Zhaojun Bai, François Gygi · SIAM Journal on Matrix Analysis and Applications · 2010
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 $10^{16}$, 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.