Stability analysis of block LDLT factorization for symmetric indefinite matrices

We consider block LDLT factorization for symmetric indefinite matrices in the form LDLT, 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 LDLT 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 LDLT 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 LDLT factorization.