Stable and efficient computation of generalized polar decompositions

Journal article

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.

Authors: Benner, P; Nakatsukasa, Y; Penke, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Matrix Analysis and Applications
• Volume: 43
• Issue: 3
• Pages: 1058-1083
• Publication date: 2022-07-11
• Acceptance date: 2022-03-17
• DOI: 10.1137/21M1411986
• EISSN: 1095-7162
• ISSN: 0895-4798
• Language: English
• Keywords: FFR