Reorthogonalized Block Classical Gram--Schmidt
For numerical linear algebra practitioners, this provides a more stable block Gram-Schmidt method with rigorous error bounds.
This paper proposes a reorthogonalized block classical Gram-Schmidt algorithm that, under floating-point arithmetic, produces Q and R with orthogonality error O(ε) and residual error O(ε||A||), improving previous bounds for the CGS2 algorithm.
A new reorthogonalized block classical Gram--Schmidt algorithm is proposed that factorizes a full column rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular and nonsingular. With appropriate assumptions on the diagonal blocks of $R$, the algorithm, when implemented in floating point arithmetic with machine unit $\macheps$, produces $Q$ and $R$ such that $\| I- Q^{T} Q \|_2 =O(\macheps)$ and $\| A-QR \|_2 =O(\macheps \| A \|_2)$. The resulting bounds also improve a previous bound by Giraud et al. [Num. Math., 101(1):87-100,\ 2005] on the CGS2 algorithm originally developed by Abdelmalek [BIT, 11(4):354--367,\ 1971]. \medskip Keywords: Block matrices, Q--R factorization, Gram-Schmidt process, Condition numbers, Rounding error analysis.