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https://issues.apache.org/jira/browse/MATH-1045?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13801910#comment-13801910
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Sean Owen commented on MATH-1045:
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That's a good point. If you make the example matrix non-singular, but then
divide elements by 1e12, it will report it as singular. This seems wrong. On
the other hand it seems a bit undesirable to return an 'inverse' in this case
-- it's dominated by the inverse of that tiny eigenvalue, which is huge, and
the result is pretty unreliable.
I'm a bit out of my depth here but I wonder if it's more reasonable to examine
the eigenvalues in sorted order and examine ratio of one to the next. When that
ratio is below epsilon it makes more sense to declare it "0".
I could also see this being a case of "caller beware". That's the more
conservative thing here.
> EigenDecomposition.Solver should consider tiny values 0 for purposes of
> determining singularity
> -----------------------------------------------------------------------------------------------
>
> Key: MATH-1045
> URL: https://issues.apache.org/jira/browse/MATH-1045
> Project: Commons Math
> Issue Type: Bug
> Affects Versions: 3.2
> Reporter: Sean Owen
> Priority: Minor
> Labels: eigenvalue, singular
> Attachments: MATH-1045.patch
>
>
> EigenDecomposition.Solver tests for singularity by comparing eigenvalues to 0
> for exact equality. Elsewhere in the class and in the code, of course, very
> small values are considered 0. This causes the solver to consider some
> singular matrices as non-singular.
> The patch here includes a test as well showing the behavior -- the matrix is
> clearly singular but isn't considered as such since one eigenvalue are ~1e-14
> rather than exactly 0.
> (What I am not sure of is whether we should really be evaluating the *norm*
> of the imaginary eigenvalues rather than real/imag components separately. But
> the javadoc says the solver only supports real eigenvalues anyhow, so it's
> kind of moot since imag=0 for all eigenvalues.)
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