sparse matrix LU decomposition

Sparse matrix LU decomposition is a technique used to factorize a sparse matrix into two matrices: a lower triangular matrix (L) and an upper triangular matrix (U). This factorization can be used to solve systems of linear equations more efficiently than using the original matrix.

The LU decomposition of a sparse matrix involves finding a permutation matrix (P) such that PA = LU, where A is the original matrix. The permutation matrix is used to re-order the rows of the matrix to reduce fill-in, or the creation of non-zero entries in the factorized matrices.

Once the permutation matrix is found, the LU decomposition can be computed using a variant of Gaussian elimination algorithm. This algorithm operates on the matrix in a row-wise fashion, systematically eliminating entries from the matrix to create the L and U matrices.

Sparse matrix LU decomposition is particularly useful for large sparse matrices, where the computation of the full LU decomposition would be too costly in terms of time and memory.