Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: He, Zhuang-Ao, Shao, Meiyue
Institution: Fudan University
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7129558880
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