Cholesky-Like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

It is well-known that orthogonalization of column vectors in a rectangular matrix B with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix A can be seen as its factorization B = QR that is equivalent to the Cholesky-like factorization in the form BTAB = RTΩR, where Ω is some signature matrix. Under the assumption of nonzero principal minors of the matrix M = BTAB we give bounds for the conditioning of the triangular factor R in terms of extremal singular values ofM and of only those principal submatrices ofM, where there is a change of sign inΩ. Using these results we study the numerical behavior of two types of orthogonalization schemes and we give the worst-case bounds for quantities computed in finite precision arithmetic. In particular, we analyze the implementation based on the Cholesky-like factorization of M and the Gram-Schmidt process with respect to the bilinear form induced by the matrix A. To improve the accuracy of computed results we consider also the Gram-Schmidt process with reorthogonalization and show that its behavior is similar to the scheme based on the Cholesky-like factorization with one step of iterative refinement.