Unpivoted dense LDL¶
Though Cholesky factorizations are ideal for Hermitian Positive-Definite matrices, unpivoted \(LDL^T\) and \(LDL^H\) factorizations are frequently used for significantly larger classes of matrices, with the most notable example being Symmetric Quasidefinite matrices, which were introduced by [Vanderbei1995] and subsequently analyzed as a simple rescaling of a nonsymmetric positive-definite matrix in [GSS1996]. While the stability properties of unpivoted \(LDL^H\) factorizations of dense quasidefinite matrices are of intrinsic interest, there are significantly more practical applications for sparse matrices, where dynamic pivoting incurs typically substantially increases both the memory requirements and operation count of a factorization.
References¶
- Vanderbei1995
Robert J. Vanderbei, Symmetric quasi-definite matrices, SIAM Journal on Optimization, Vol. 5, No. 1, pp. 100–113, 1995. DOI: http://dx.doi.org/10.1137/0805005
- GSS1996
Philip E. Gill, Michael A. Saunders, and Joseph R. Shinnerl, On the stability of Cholesky factorization for symmetric quasidefinite systems, SIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 1, pp. 35–46, 1996. DOI: http://dx.doi.org/10.1137/S0895479893252623