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arXiv · 2602.14449

On Two-Stage Householder Orthogonalization

Zhuang-Ao He·Meiyue Shao
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Abstract

Two-stage orthogonalization is essential in numerical algorithms such as Krylov subspace methods. For this task we need to orthogonalize a matrix $A$ against another matrix $V$ with orthonormal columns. A common approach is to employ the block Gram--Schmidt algorithm. However, its stability largely depends on the condition number of $[V,A]$. While performing a Householder orthogonalization on $[V,A]$ is unconditionally stable, it does not utilize the knowledge that $V$ has orthonormal columns. To address these issues, we propose a two-stage Householder orthogonalization algorithm based on the generalized Householder transformation. Instead of explicitly orthogonalizing the entire $V$, our algorithm only needs to orthogonalizes a square submatrix of $V$. Theoretical analysis and numerical experiments demonstrate that our method is also unconditionally stable.

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Zhuang-Ao He, Meiyue Shao. 2026-02-16. On Two-Stage Householder Orthogonalization. https://arxiv.org/abs/2602.14449

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