LDU decomposition - sparse matrix

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SUMMARY

The discussion centers on performing an LDLT decomposition of a large sparse symmetric matrix to determine the number of its negative eigenvalues. The user seeks an efficient method to obtain the diagonal matrix D without storing the lower triangular matrix L, as the matrix's size and sparsity make it impractical to retain L. The conversation suggests exploring algorithms similar to those used in Cholesky decomposition, which are known for their stability.

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  • Sparse matrix representation and storage techniques
  • LDLT decomposition methodology
  • Understanding of eigenvalues and eigenvectors
  • Cholesky decomposition principles
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  • Research efficient algorithms for LDLT decomposition in sparse matrices
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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.
 

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