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I've recently posted this question. To summarise, I'm dealing with supposedly Symmetric Positive Definite(SPD) matrices, but due to machine-precision they end up not being SPD. In a comment, a user suggested to use the lower triangular portion of the matrix.

  1. I would like to be sure in what way we can use that? What algorithm is he possibly alluding to? I thought of the Cholesky Decomposition...
  2. With that method, or another which uses the lower triangular part of the matrix, will I be able to take any 'almost' SPD matrix, and return an SPD matrix?
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    $\begingroup$ Is the matrix still symmetric or does it lose its symmetry due to machine-precision? Also, what computations do you do to update the symmetric matrix? If the updates can be represented as a sequence of rank one updates, then you could do the updates directly to the Cholesky factor for your matrix and that might help ensure your matrix remains PD. $\endgroup$
    – spektr
    May 19, 2021 at 20:12
  • $\begingroup$ What the answerer suggests is to define a macro/function which can be roughly expressed as "func Afromlower(i,j) : if i>j then return A(j,i), otherwise return A(i,j)". Then you can just use that function handle in place of the matrix and the regular algorithms can be applied directly (out-of-place of course, with this approach you would have to redesign the algorithms to do in-place computations). $\endgroup$ May 19, 2021 at 23:54
  • $\begingroup$ W.r.t. your original question, you only need to add $c$ to the nonpositive eigenvalues. $\endgroup$ May 20, 2021 at 17:59

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