A quasi-orthogonal iterative method for eigenvalue problems

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.