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Background

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.





David P. Koester
Sun Oct 22 16:27:33 EDT 1995