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 $W$-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.

• The paper introduces a matrix-multiplication-based iterative refinement method that ensures quadratic reduction in residual errors for non-Hermitian eigendecompositions.
• It distinguishes between right-only and left-right regimes by leveraging first-order corrections and Newton–Schulz updates to improve biorthogonality.
• Numerical experiments and stabilization techniques validate mixed-precision improvements, effectively handling challenges such as clustered eigenvalues.

Iterative Refinement Strategies for Diagonalizable Non-Hermitian Eigendecompositions

Problem Context and Motivation

The paper presents a rigorous matrix-multiplication-based iterative refinement procedure tailored for the improvement of approximate eigendecompositions of diagonalizable non-Hermitian matrices. The motivation arises from the observation that standard eigensolvers may return decompositions with amplified finite-precision errors, especially in the presence of nonnormality and lack of orthogonality. Such amplification is well-documented in the literature, and there is demand for efficient post-processing methods that achieve significant accuracy enhancement at a computational cost dominated by matrix products ($\mathcal{O}(n^3)$). The present work builds upon classical approaches (including Newton-type iterations), incorporates mixed-precision computational insights, and distinguishes its theoretical contributions by addressing two refinement regimes based on auxiliary information provided by the eigensolver.

Theoretical Contributions

Refinement Regimes

The paper identifies two principal input regimes:

• Right-only regime: Only approximate right eigenvectors and eigenvalues are available. Here, updates are defined by a first-order formula utilizing $\widehat V^{-1}$ for correction selection, leading to an exact post-update residual identity and a quadratic residual bound.
• Left-right regime: Both approximate left and right eigenvectors are available with approximate biorthogonality. The method uses $\widehat W^{\mathrm H}$ as a surrogate for $\widehat V^{-1}$ in driving the correction. The local theory for simple eigenvalues is developed, where the computable driving matrix is shown to be an exact first-order perturbation, and the biorthogonality correction is captured by a Newton--Schulz-type quadratic error identity.

Both regimes are unified within a matrix-multiplication-dominant computational framework aligned with modern mixed-precision and high-performance numerical workflows.

Strong Theoretical Results

• Exact post-update residual identity (right-only regime): After the first-order update, the right residual measured in the $\widehat V^{-1}$-scaled norm satisfies an identity which yields a quadratic bound (in the first-order driving matrix norm) for the residual.
• Exact perturbation formula (left-right regime): The difference between the implementable driving matrix $Y_W = \widehat W^{\mathrm H}R$ and the ideal $Y_\star = \widehat V^{-1}R$ is explicitly given as first order in the biorthogonality error. This establishes a sharp link between practical computable quantities and the underlying theoretical ideal.
• Exact Newton--Schulz biorthogonality update: The left-right regime update formula for biorthogonality error demonstrates quadratic behavior, contributing to local second-order convergence of the $W$-method when the input biorthogonality error is small.

Clustered Eigenvalues: Stabilization Extension

Simple-eigenvalue theory is complemented with a stabilization procedure for cases where eigenvalues are clustered and naive componentwise corrections are numerically unstable. Clusterwise re-diagonalization is executed via projected matrix diagonalization, suppressing intracluster corrections and therefore preserving the invariant subspace. The instability of the naive update is analytically traced to ill-conditioning of the cluster basis rather than to loss of biorthogonality.

Numerical Experiments and Strong Empirical Claims

Experiments were designed to validate the theory in both synthetic and application-derived dense matrices, with metrics focused on relative residual reduction, biorthogonality error, and eigenvalue consistency.

• Simple eigenvalues: Both right-only and left-right methods drive residual to double-precision limits rapidly (within a few iterations) for diagonalizable test families.
• Clustered eigenvalues: The cluster-aware update stabilizes highly ill-conditioned cases, reaching double-precision residual while naive updates diverge.
• Parameter robustness: Sensitivity analyses reveal strong resilience to nonnormality parameter variation; the refinement method's performance remains robust with increasing $\alpha$ in the synthetic family.
• Application benchmarks: On SuiteSparse dense copies, the right-only refinement converges to double-precision residual in three out of four cases; the left-right method converges in a comparable number of steps. Both fail when the initial decomposition is too inaccurate, highlighting the requirement for sufficient initialization accuracy.
• Mixed-precision timing: The workflow combining single-precision initial eigensolve with double-precision refinement is consistently faster than a full double-precision eigensolve (by 31–35%) and achieves comparable accuracy.

The numerical results are organized to isolate the theoretical mechanisms, and strong claims regarding quadratic residual reduction and stabilization of clustered cases are consistently evidenced.

Practical and Theoretical Implications

The analysis demonstrates that iterative refinement methods, when formulated through matrix-multiplication-based updates, offer significant practical advantages in post-processing non-Hermitian eigendecompositions. These strategies are optimal for current architectures favoring matrix product kernels and mixed-precision arithmetic.

The local theory developed for the $W$-method, with explicit bounds and identities, clarifies convergence properties under precise conditions. Stabilization techniques for clusters extend the practical applicability to cases that would otherwise be numerically prohibitive.

Practically, this enables:

• Enhanced accuracy of approximate decompositions without recomputation, especially relevant in workflows involving large dense matrices or mixed precision.
• Reliable post-processing for eigenproblems arising in physics, engineering, and data science involving non-Hermitian operators.

Theoretically, the local second-order convergence and explicit error identities pave the way for more nuanced convergence analyses. The stabilization for clusters motivates future work in comprehensive convergence theory for clustered eigenvalues and generalization to sparse and structured matrix regimes.

Future Directions

Future work will focus on:

• Extending convergence theory to clustered regimes and analyzing the attraction domains for iterative refinement.
• Integration of the refinement strategies with high-performance sparse eigensolvers and scalable computing environments.
• Broader validation on extensive application-derived benchmarks, both dense and sparse, to empirically characterize robustness and range of applicability.

Conclusion

The paper rigorously develops and analyzes iterative refinement algorithms for diagonalizable non-Hermitian eigendecompositions in both right-only and left-right input regimes, substantiated by exact algebraic identities, local convergence estimates, and systematic numerical evidence. The proposed post-processing methods provide efficient, matrix-multiplication-dominant improvements to computed eigendecompositions, with robust stabilization procedures for clustered eigenvalues and strong empirical convergence claims. The framework instruments practical enhancement within modern mixed-precision workflows, and the theoretical results clarify the limitations and capabilities of local refinement in non-Hermitian settings.