Iterative Refinement for Diagonalizable Non-Hermitian Eigendecompositions
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918426379812864 |
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| author | Terao, Takeshi |
| author_facet | Terao, Takeshi |
| contents | This paper develops matrix-multiplication-based iterative refinement for diagonalizable non-Hermitian eigendecompositions. The main theory concerns simple eigenvalues and distinguishes two input regimes. In the right-only regime, where only approximate right eigenvectors and eigenvalues are available, a first-order derivation selects the update and the resulting post-update residual identity is exact, yielding a quadratic residual bound. In the left-right regime, where approximate left and right eigenvectors are both available, the computable driving matrix is an exact perturbation of the inverse-based one and the biorthogonality correction satisfies an exact Newton--Schulz-type error identity. Under a small biorthogonality error, these relations yield a local second-order estimate for the resulting $W$-method. Clustered eigenvalues are handled separately by a stabilization extension based on clusterwise re-diagonalization and suppression of intracluster corrections, whose effect is verified on controlled matrices with ill-conditioned cluster bases. The method is intended as post-processing for an already accurate eigendecomposition. The attraction region is not analyzed, and no complete theory is given for the clustered case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02840 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Iterative Refinement for Diagonalizable Non-Hermitian Eigendecompositions Terao, Takeshi Numerical Analysis This paper develops matrix-multiplication-based iterative refinement for diagonalizable non-Hermitian eigendecompositions. The main theory concerns simple eigenvalues and distinguishes two input regimes. In the right-only regime, where only approximate right eigenvectors and eigenvalues are available, a first-order derivation selects the update and the resulting post-update residual identity is exact, yielding a quadratic residual bound. In the left-right regime, where approximate left and right eigenvectors are both available, the computable driving matrix is an exact perturbation of the inverse-based one and the biorthogonality correction satisfies an exact Newton--Schulz-type error identity. Under a small biorthogonality error, these relations yield a local second-order estimate for the resulting $W$-method. Clustered eigenvalues are handled separately by a stabilization extension based on clusterwise re-diagonalization and suppression of intracluster corrections, whose effect is verified on controlled matrices with ill-conditioned cluster bases. The method is intended as post-processing for an already accurate eigendecomposition. The attraction region is not analyzed, and no complete theory is given for the clustered case. |
| title | Iterative Refinement for Diagonalizable Non-Hermitian Eigendecompositions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.02840 |