#12966: Indefinite factorization for exact matrices
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Reporter: rbeezer | Owner: jason, was
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-5.1
Component: linear algebra | Resolution:
Keywords: sd40.5 | Work issues:
Report Upstream: N/A | Reviewers: Andrey Novoseltsev
Authors: Rob Beezer | Merged in:
Dependencies: | Stopgaps:
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Old description:
> Almost any square symmetric (or Hermitian) matrix A over a field can be
> decomposed into a lower triangular matrix L and a diagonal matrix D such
> that A = L*D*L-transpose, suitably adjusted in the Hermitian case.
>
> 1) This is of interest for its own sake (eg for solving systems).
>
> 2) If the field has square roots and the diagonal matrix has positive
> entries, then the Cholesky decomposition is easy. This would fix #11274.
>
> 3) This will give a good way to tell if a matrix is positive definite.
>
> '''Apply''':
> 1. [attachment:trac_12966-indefinite-factorization-v1.patch]
New description:
Almost any square symmetric (or Hermitian) matrix A over a field can be
decomposed into a lower triangular matrix L and a diagonal matrix D such
that A = L*D*L-transpose, suitably adjusted in the Hermitian case.
1) This is of interest for its own sake (eg for solving systems).
2) If the field has square roots and the diagonal matrix has positive
entries, then the Cholesky decomposition is easy. This would fix #11274.
3) This will give a good way to tell if a matrix is positive definite.
'''Apply''':
1. [attachment:trac_12966-indefinite-factorization-v2.patch]
--
Comment (by rbeezer):
Reworked with a cleaner separation. Error-checking and tests are in the
underscore method, making the regular call much shorter an I think the two
following routines will work better also.
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12966#comment:8>
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