Stable and efficient computation of generalized polar decompositions
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...
| Main Authors: | , , |
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| 格式: | Journal article |
| 語言: | English |
| 出版: |
Society for Industrial and Applied Mathematics
2022
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| _version_ | 1826308293346197504 |
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| author | Benner, P Nakatsukasa, Y Penke, C |
| author_facet | Benner, P Nakatsukasa, Y Penke, C |
| author_sort | Benner, P |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T07:15:55Z |
| format | Journal article |
| id | oxford-uuid:d03211c8-4f32-461b-a1b0-6b8d3c9918a3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:15:55Z |
| publishDate | 2022 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:d03211c8-4f32-461b-a1b0-6b8d3c9918a32022-08-09T08:00:18ZStable and efficient computation of generalized polar decompositionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d03211c8-4f32-461b-a1b0-6b8d3c9918a3EnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2022Benner, PNakatsukasa, YPenke, CWe 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. |
| spellingShingle | Benner, P Nakatsukasa, Y Penke, C Stable and efficient computation of generalized polar decompositions |
| title | Stable and efficient computation of generalized polar decompositions |
| title_full | Stable and efficient computation of generalized polar decompositions |
| title_fullStr | Stable and efficient computation of generalized polar decompositions |
| title_full_unstemmed | Stable and efficient computation of generalized polar decompositions |
| title_short | Stable and efficient computation of generalized polar decompositions |
| title_sort | stable and efficient computation of generalized polar decompositions |
| work_keys_str_mv | AT bennerp stableandefficientcomputationofgeneralizedpolardecompositions AT nakatsukasay stableandefficientcomputationofgeneralizedpolardecompositions AT penkec stableandefficientcomputationofgeneralizedpolardecompositions |