A Note on the Ldlt Decomposition of Matrices Fromsaddle - Point

Sparse linear systems Kx = b are considered where K is a specially structured symmetric indeenite matrix. These systems arise frequently, e.g., from mixed nite element discretiza-tions of PDE problems. The LDL T factorization of K with diagonal D and unit lower triangular L is known to exist for natural ordering of K but the resulting triangular factors can be rather dense. On the other hand, for a given permutation matrix P, the LDL T factorization of P T KP may not exist. In this paper a new way to obtain a ll-in minimizing permutation based on initial ll-in minimizing ordering is introduced. For an important subclass of matrices arising from mixed and hybrid nite element discretizations the existence of the LDL T factorization of the permuted matrix is proved. Experimental results on practical problems indicate that the amount of computational savings can be substantial when compared with the approach based on Schur complement.