The Stability of Block Variants of Classical Gram--Schmidt
Erin Carson, Kathryn A. Lund, Miroslav Rozložńık · SIAM Journal on Matrix Analysis and Applications · 2021
The block version of the classical Gram--Schmidt (\tt BCGS) method is often employed to efficiently compute orthogonal bases for Krylov subspace methods and eigenvalue solvers, but a rigorous proof of its stability behavior has not yet been established. It is shown that the usual implementation of \tt BCGS can lose orthogonality at a rate worse than $O(\varepsilon) \kappa^{2}({$\mathcalX$})$, where $\mathcal{X}$ is the input matrix and $\varepsilon$ is the unit roundoff. A useful intermediate quantity denoted as the Cholesky residual is given special attention and, along with a block generalization of the Pythagorean theorem, this quantity is used to develop more stable variants of \tt BCGS. These variants are proven to have $O(\varepsilon) \kappa^2({$\mathcalX$})$ loss of orthogonality with relatively relaxed conditions on the intrablock orthogonalization routine satisfied by the most commonly used algorithms. A variety of numerical examples illustrate the theoretical bounds.