LDU decomposition - sparse matrix

In summary, to find the number of negative eigenvalues in a large sparse symmetric matrix, one must perform an LDLT decomposition and count the number of negative diagonal entries in the D matrix. However, if only the D matrix is needed and the lower matrix L cannot be stored, efficient algorithms such as Cholesky decomposition may be helpful.
  • #1
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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  • #2
Maybe algorithms for the similar Cholesky decomposition can help. They are at least stable.
 

FAQ: LDU decomposition - sparse matrix

What is LDU decomposition for a sparse matrix?

LDU decomposition is a method used to break down a sparse matrix into three components: L (lower triangular matrix), D (diagonal matrix), and U (upper triangular matrix). It is a variant of LU decomposition specifically designed for sparse matrices.

How is LDU decomposition different from LU decomposition?

While both LDU and LU decompositions aim to factorize a matrix into lower and upper triangular matrices, LDU decomposition also includes a diagonal matrix. This allows for more efficient computation and storage of sparse matrices.

What are the advantages of using LDU decomposition for sparse matrices?

LDU decomposition is advantageous for sparse matrices because it reduces the amount of computation required to solve a linear system. It also allows for better memory management by storing only the non-zero elements of the matrix, resulting in a more compact representation.

Can LDU decomposition be applied to non-square matrices?

Yes, LDU decomposition can be applied to non-square matrices as long as the matrix is invertible. However, the resulting L, D, and U matrices may not be square.

Are there any limitations to using LDU decomposition for sparse matrices?

One limitation of LDU decomposition is that it only works for matrices with non-zero diagonal elements. Additionally, if the matrix is not strictly diagonally dominant, the decomposition may not produce a unique solution.

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