Stability analysis of block LDLT factorization for symmetric indefinite matrices

摘要

We consider block LDL~T factorization for symmetric indefinite matrices in the form LDL~T, where L is unit lower triangular and D is block diagonal with each diagonal block having dimension 1 or 2. The stability of this factorization and its application to solving symmetric indefinite linear systems has been well studied. On the other hand, while all rounding error analysis of block LDL~T factorization in the literature relies on the outer product form, this paper gives a novel componentwise backward error analysis based on the inner product form. The new results include a condition under which block LDL~T factorization in inexact arithmetic is guaranteed to preserve the inertia and a reliability analysis of rank estimation and inertia estimation of symmetric indefinite matrices by block LDL ~T factorization.