@article {IOPORT.05905073,
    author       = {Fang, Haw-Ren},
    title        = {Stability analysis of block $LDL^{T}$ factorization for symmetric indefinite matrices.},
    year         = {2011},
    journal      = {IMA Journal of Numerical Analysis},
    volume       = {31},
    number       = {2},
    issn         = {0272-4979},
    pages        = {528-555},
    publisher    = {Oxford University Press, Oxford},
    doi          = {10.1093/imanum/drp053},
    abstract     = {The author presents a stability analysis of the block $LDL^{T}$ factorization for symmetric indefinite matrices. A novel componentwise backward error analysis is presented and a condition under which the block $LDL^{T}$ factorization in finite precision arithmetic is guaranteed to preserve inertia is given as well. Rank and inertia estimation of symmetric and indefinite matrices by block $LDL^{T}$ factorization is also discussed and numerical experiments are presented in which the rank and inertia of symmetric and rank-deficient matrices are estimated. The paper starts off by briefly reviewing how to compute the $LDL^{T}$ factorization of a symmetric possibly indefinite matrix. Different types of pivoting strategies are discussed and some historical remarks on the stability analysis of block $LDL^{T}$ factorization are given as well. Some basic results of rounding error analysis are discussed next, followed by a stability analysis that leads to a condition for inertia preservation in finite precision arithmetic. Subsequently, a stability analysis of $LDL^{T}$ factorization to estimate the rank and inertia of a symmetric rank-deficient matrix is presented. From this analysis and a number of numerical experiments it follows that block $LDL^{T}$ factorization (with properly chosen pivoting strategies) can provide a reliable way of estimating the rank and inertia of symmetric, indefinite, and possibly rank-deficient matrices.},
    reviewer     = {Robert F. Remis (Delft)},
    identifier   = {05905073},
}