Iterative Refinement for Diagonalizable Non-Hermitian Eigendecompositions

TL;DR

This paper introduces an iterative refinement method for diagonalizable non-Hermitian eigendecompositions, providing theoretical insights and stability extensions for different eigenvector regimes.

Contribution

It develops a matrix-multiplication-based iterative refinement technique with new theoretical bounds for both right-only and left-right eigenvector regimes.

Findings

Quadratic residual bound in the right-only regime.

Exact Newton-Schulz-type error identity in the left-right regime.

Stabilization extension effectively handles clustered eigenvalues.

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.