On Two-Stage Householder Orthogonalization
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908906455826432 |
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| author | He, Zhuang-Ao Shao, Meiyue |
| author_facet | He, Zhuang-Ao Shao, Meiyue |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14449 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Two-Stage Householder Orthogonalization He, Zhuang-Ao Shao, Meiyue Numerical Analysis 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. |
| title | On Two-Stage Householder Orthogonalization |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2602.14449 |