Stable and Efficient Computation of Generalized Polar Decompositions

We present methods for computing the generalized polar decomposition of a matrix based on dynamically weighted Halley iteration. This method is well established standard decomposition. A stable implementation available, where inversion avoided and QR decompositions are used instead. establish natural generalization this approach with respect to signature matrices. Again inverse can be by using called hyperbolic However, does not show same favorable stability properties as its orthogonal counterpart. overcome numerical difficulties generalizing CholeskyQR2 method. computes factorization in way via two successive Cholesky factorizations. An even better achieved employing permuted graph bases, yielding residuals order $10^{-14}$ badly conditioned matrices, other fail.