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Iterative Refinement for Diagonalizable Non-Hermitian Eigendecompositions

Takeshi Terao

Abstract

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.

Iterative Refinement for Diagonalizable Non-Hermitian Eigendecompositions

Abstract

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 -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.

Paper Structure

This paper contains 25 sections, 5 theorems, 75 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting and notation
  3. Updates for simple eigenvalues
  4. Right-only case: first-order derivation
  5. Exact residual identity in the right-only case
  6. Exact perturbation formula for the driving matrix
  7. Exact update for the biorthogonality error
  8. Local estimate for the $W$-method
  9. Stabilization for clustered eigenvalues
  10. Computational realization
  11. Use cases
  12. Biorthogonalization preprocessing
  13. Finite-precision realizations in the right-only regime
  14. Numerical experiments
  15. Metrics
  16. ...and 10 more sections

Key Result

Proposition 1

Setup. Let $A\in\mathbb{C}^{n\times n}$, let $\widehat{V}\in\mathbb{C}^{n\times n}$ be invertible, and let $\widehat{D}=\mathrm{diag}(\widehat{d}_1,\dots,\widehat{d}_n)$ be diagonal. Assume that Algorithm alg:right is executed in exact arithmetic and that the simple-eigenvalue branch is taken. Defin Write $\widetilde{D}=\mathrm{diag}(\widetilde{d}_1,\dots,\widetilde{d}_n)$, and let $\widetilde{E}=

Figures (8)

  • Figure 1: Residual histories for the left-right $W$-method and for two right-only realizations of Algorithm \ref{['alg:right']}.
  • Figure 2: Refinement history for Algorithm \ref{['alg:both']} on a diagonalizable matrix with complex eigenvalues.
  • Figure 3: Effect of the biorthogonalization preprocessing on Algorithm \ref{['alg:both']}.
  • Figure 4: Comparison of two left-right refinements on the clustered test: the naive componentwise update and the cluster-aware update. Both use the same exact biorthogonalization after each iteration.
  • Figure 5: Sensitivity of the naive and cluster-aware updates to the conditioning of the cluster basis.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 1: Exact residual identity and quadratic estimate in the right-only case
  • proof
  • Proposition 2: Exact perturbation representation of $Y_W$
  • proof
  • Proposition 3: Exact update formula for the biorthogonality error
  • proof
  • Theorem 4: Local quadratic-type estimate for the $W$-method in the simple-eigenvalue case
  • proof
  • Remark 5
  • Proposition 6: Clusterwise re-diagonalization preserves the approximate invariant subspace
  • ...and 2 more