| Тойм: | 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 and 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 factorization 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.
|
|---|