Reorthogonalized Block Classical Gram--Schmidt

TL;DR

This paper introduces a new reorthogonalized block classical Gram-Schmidt algorithm that improves numerical stability and accuracy in QR factorization of full column rank matrices.

Contribution

It proposes a novel reorthogonalized block classical Gram-Schmidt algorithm with improved error bounds and stability over previous methods.

Findings

Produces orthogonal Q with  I - Q^T Q _2 = O(b1) in floating point arithmetic.

Achieves b1 d O(b1) imes b2 d ext{approximation error} in QR factorization.

Improves previous bounds on the CGS2 algorithm by Giraud et al. (2005).

Abstract

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.