Consider the direct solution of the linear system
where A is an sparse matrix. For this research, it has been
assumed that this matrix is neither symmetric nor position symmetric,
however, the algorithms can be extended to Choleski factorization of
symmetric positive definite matrices with minimal modifications to the
mathematics or the software implementation described in
section 5. Additional discussions on the state of the
literature for Choleski factorization are presented below. The sparse
matrix A can be numerically factored into two separate triangular
matrices, one sparse matrix being lower triangular, L, and the other
sparse matrix being upper triangular, U:
A lower triangular matrix, L, has all zeros above the diagonal and an upper triangular matrix, U, has all zeros below the diagonal. Triangular linear systems can be readily solved numerically by solving for the first value in the triangular linear system and substituting that value into subsequent equations. This procedure is repeated for all equations in the linear system.