A quasi-orthogonal iterative method for eigenvalue problems

TL;DR

This paper introduces a quasi-orthogonal iterative method for large-scale eigenvalue problems that avoids explicit orthogonalization, improving computational efficiency, robustness, and scalability while maintaining high-precision orthogonality.

Contribution

The paper presents a novel quasi-orthogonal iterative method that inherently preserves quasi-orthogonality without explicit orthogonalization, offering a more efficient and stable approach for large-scale eigenvalue problems.

Findings

Method inherently preserves quasi-orthogonality.

Enhanced robustness against numerical perturbations.

Validated efficiency and stability through numerical experiments.

Abstract

For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit orthogonalization. To address these challenges, we propose a quasi-orthogonal iterative method that dispenses with explicit orthogonalization and orthogonal initial data. It inherently preserves quasi-orthogonality (the iterates asymptotically tend to be orthogonal) and enhances robustness against numerical perturbations. Rigorous analysis confirms its energy-decay property and convergence of energy, gradient, and iterate. Numerical experiments validate the theoretical results, demonstrate key advantages of strong robustness and high-precision numerical orthogonality preservation, and thereby position our iterative method as an efficient, stable alternative for large-scale eigenvalue computations.