Comments (1)
thank you for your interest in the project and sorry for the long delay in answering, very busy week :D
I think your idea sounds very interesting and would make sense to include something like that in the package (maybe the names of the API can be negotiated :) ). I'll try here to summarize your proposal to make sure I understand correctly and ask a few clarifying questions.
If I understand correctly you are proposing that given a matrix M
, find a non-interval matrix P
and an interval diagonal matrix D
so that M
is contained in P*D*P^{-1}
. I think this would be interesting to explore and maybe it might have some practical utilities e.g. to bound functions of matrices.
-
Do I understand correctly, that the advantage of your proposed approach is that you are relaxing the problem as you are not requiring
P
to bound the eigenvectors, computing this factorization would be more efficient thanverify_eigen
or the approach proposed in the paper in #116 ? I have a couple of scenarios in mind where this could be useful -
I could not understand from your description whether the matrix
M
would be a matrix of floats (single points) or intervals. Which one did you have in mind? These would have slightly different problem formulations:- find
P, D
so thatM ∈ PDP⁻¹
(starting from floating point) - find
P, D
so thatM ⊆ PDP⁻¹
(starting from interval) - The reason I'm asking is because while the latter is more generals and algorithms working for that work also for the former, if you can assume that the starting point is a matrix of floats (or very small intervals), then you can generally find more performant algorithms, so I think it could be good to separate the two cases and start benchmarking only with the former could be an easier start.
- find
-
In your proposal you mention to compute
P⁻¹
, as you probably know matrix inversion cannot be computed exactly with floating point, if you want to work with intervals and want rigorous computations generally there are two options:- Compute
P⁻¹
rigorously, meaning in the factorizationP
would be a matrix of floats,D
an interval matrix andP⁻¹
an interval matrix guaranteed to contain the true inverse ofP
- Slightly modify the problem: Find
P, D, R
so thatM ∈ PDR
, whereD
is interval andP, R
not. Then probably an approximate inverse ofP
would be good enough.
- Compute
-
Did you have in mind an algorithm on how to determine
P
andD
? Can you link some papers I should link to better understand this issue?
If you have further questions / comments, do not hesitate to ping me.
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Related Issues (20)
- think of interface / dastructures to handle linear-PILS HOT 9
- [feature request]: FEM minimal example problem/test HOT 2
- note about adding new docs pages in cotributing guidelines.
- [enhancement]: spectral decomposition of interval matrices
- [bug] generation of documentation freezes HOT 10
- [enhancement]: A is not a squre matrix HOT 4
- [enhancement]: Ship a correctly rounded threaded OpenBLAS as an Artifact HOT 1
- [enhancement]: inv and det (needed by IntervalRootFinding)
- benchmarks about different matrix multiplication algorithms HOT 2
- [bug]: issues with complex interval matrices multiplication HOT 11
- [enhancement]: format references with APA style HOT 1
- [enhancement]: Add hertz method for eigenvalues of symmetric interval matrices
- write short tutorial about eigenvalues functionalities
- TagBot trigger issue HOT 6
- video not correctly embedded in documentation HOT 3
- [bug]: don't use subset to check if interval vector is in the interior of the other HOT 2
- is the current CI an overkill HOT 1
- [enhancement]: determinant of interval matrices HOT 1
- Taking parametric interval linear system seriuosly HOT 1
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