# Numerical inaccuracies when constructing correlation matrices

**URL:** <https://discourse.julialang.org/t/numerical-inaccuracies-when-constructing-correlation-matrices/85177>\
**Category:** General Usage\
**Tags:** linearalgebra\
**Created:** [August 2, 2022, 4:18pm UTC](https://discourse.julialang.org/t/numerical-inaccuracies-when-constructing-correlation-matrices/85177 "2022-08-02T16:18:16Z")\
**Posts on this page:** 1\
**Showing post:** 6

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [August 2, 2022, 7:27pm UTC](https://discourse.julialang.org/t/numerical-inaccuracies-when-constructing-correlation-matrices/85177/6 "2022-08-02T19:27:43Z")

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> [@daniel](#):
>
> Is there a method to heal an _almost_ positive semidefinite matrix like this (or tweaks to make the algorithm more robust)?

The basic issue is that trying to make a matrix _exactly_ semidefinite is inherently nonrobust to roundoff errors. Different ways of computing the eigenvalues will give slightly different values due to differences in floating-point effects, so even if you make the eigenvalues \ge 0 for one computation you might get slightly negative eigenvalues from another computation/algorithm with the same matrix.

The question is, what are you trying to _do_ with the matrix `S2` that requires it to be strictly semidefinite? It is generally better to try to fix your _subsequent_ computation to be tolerant of tiny negative eigenvalues in such cases. For example, `sqrt(Symmetric(A))` includes a backwards-stable correction to return a real matrix square root for nearly semidefinite matrices (see [julia#35057](https://github.com/JuliaLang/julia/pull/35057)).

Alternatively, if you don’t care too much about accuracy, you can just do something like `S2 + 1e-8*norm(S2)*I`, of course (sacrificing half of the significant digits).

You should also consider the possibility that you have a bug in your code, or are mis-using the algorithm, or that the source you are using is incorrect. Your linked article claims that the algorithm generates “full-rank” correlation matrices, but your `S2` matrix is extremely rank deficient: it is a 1000x1000 matrix with numerical rank (number of non-negligible eigenvalues) of only **64**. Is this really what you want?

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