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