#19405: Add lyapunov_rank() method for polyhedral cones
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Reporter: mjo | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.10
Component: geometry | Keywords:
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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The Lyapunov rank measures the length of a `lyapunov_like_basis()` and
quantifies how many equations you can get out of the complementarity
condition `<x,s> = 0`. Counting a basis is easy but unnecessarily slow
when, for example, you have a single ray in a 100-dimensional space.
The new `lyapunov_rank()` method computes the rank using a shortcut based
on considering that single ray as living in its own span. Then the
remaining 99-dimensions worth of zeros are easily dealt with.
The operation of "considering the ray as living in its own span" is a bit
tricky, and can only be done up to linear isomorphism. A private helper
method `_restrict_to_space()` handles that.
This is new stuff, but I've presented it to the department
(http://michael.orlitzky.com/presentations
/the_lyapunov_rank_of_an_improper_cone_-_part_i_-_algorithms.pdf) and a
few people have even read the preprint. Perhaps more importantly, there's
a doctest that checks the answer against the naive algorithm:
`len(K.lyapunov_like_basis())`. So I'm Pretty Sure it works.
--
Ticket URL: <http://trac.sagemath.org/ticket/19405>
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