LDU decomposition - sparse matrix

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Hassan2
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I have a large sparse symmetric matrix and I'd like to know the number of its negative eigenvalues. To this end, I should perform an LDLT decomposition of the matrix and count the number of negative diagonal entries of the D matrix. This would be equal to the number of negative eigenvalues.

Since I need the diagonal matrix D only, is there an efficient way to acquire D without saving L ? My matrix is large and only the nonzero entries are stored. I can't store the dense lower matrix L.

Your help is appreciated.
 
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Maybe algorithms for the similar Cholesky decomposition can help. They are at least stable.