On Monday, 31 January 2022 at 15:19:49 UTC-8 Emmanuel Charpentier wrote:
> As advertised, an atempt at a minimal (non-)working example :
>
> # Reproducible minimal example
> with seed(0): M = matrix(AA, 3, 3, lambda u,v: AA.random_element())
> # Working ring
> WR = M.base_ring().algebraic_closure()
> # A variable to carry the eigenvalues
> l = SR.var("l")
> # Vector of unknowns for the eigenvectors
> V =vector(list(var("v", n=2))+[SR(1)])
> # M.eigenvalues does not return. Get them by hand
>
> Actually, for me on 9.5beta9, `M.eigenvalues()` works just fine. So the
problem is perhaps just platform-dependent, or there is a very recent
change that affected this (my M gets just integer entries from {-2..2})
Looking at the example a bit: you'd be forcing sage to work with a huge
compositum if you're actually getting a 3x3 matrix with non-rational
algebraic entries: even if they are just independent quadratics, you'd end
up in an extension of degree 2^9. This will only work in very limited
cases.
One way to get this kind of thing to work is to work with high-precision
floats, use numerically (fairly) stable methods to compute the desired
answer, and then try to recognize it as algebraic. You probably only care
if it is one of fairly low height. You can then try to turn your
computation into proof, possibly by tracing through height bounds and
showing your precision was sufficient to identify the right solution
uniquely.
You could work on automating this kind of thing, but I doubt you'd ever get
it to work on a reasonable range of examples; just because the height
bounds would be rather ill-behaved.
You can still trace the root cause further on this and perhaps improve
arithmetic in AA a bit, but the general shape of the problem you're trying
to deal with does not look promising for generally performant methods.
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