GR decompositions and their relations to Cholesky‐like factorizations

Abstract

For a given matrix, we are interested in computing GR decompositions A = GR, where G is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the LDL T and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based approaches.

A Structure-Preserving Divide-and-Conquer Method for Pseudosymmetric Matrices

“…Computing it via a column-elimination approach is notoriously unstable. This is why [14,15] exploit a link to the LDL T factorization and introduces the LDLIQR2 algorithm. Algorithm 8 is the pseudocode of a Zolotarev-based computation of the generalized polar factor.…”

Stable and Efficient Computation of Generalized Polar Decompositions

“…Remark 3.4. If the right side of the equivalence in (7) with nonsingular R is given, H = AR −1 can be recovered from A and R. In the case of signature matrices, the right side can be computed from an LDL T decomposition A * ΣA = LDL * , where L is unit lower triangular, D is real diagonal. Then R := |D| The LDL T factorization with a strictly diagonal D is typically not used in modern algorithms, as it becomes unstable when small diagonal values appear [2].…”

“…In [52], the CholeskyQR2 algorithm is formulated and following these ideas we derive the indefinite variant (see also [7]). We call the algorithm LDLIQR2, standing for LDL T -based computation of the Indefinte QR decomposition, applied twice.…”