An alternative full-pivoting algorithm for the factorization of indefinite symmetric matrices

This paper presents an algorithm for the factorization of indefinite symmetric matrices that factors any symmetric matrix A into the form LDLT, with D diagonal and L triangular, with its subdiagonal filled with zeros. The algorithm is based on Jacobi rotations, as opposed to the widely used permutation methods (Aasen, Bunch-Parlett, and Bunch-Kaufman). The method introduces little increase in computational cost and provides a bound on the elements of the reduced matrices of order 2nf(n), which is smaller than that of the Bunch-Parlett method (≃3nf(n)), and similar to that of Gaussian elimination with full pivoting (nf(n)). Furthermore, the factorization method is not blocked. Although the method presented is formulated in a full-pivoting scheme, it can easily be adapted to a scheme similar to that of the Bunch-Kaufman approach. A backward error analysis is also presented, showing that the elements of the error matrix can be bounded in terms of the elements of the reduced matrices.