Efficient Computation of Dominant Eigenvalues Using Adaptive Block Lanczos with Chebyshev Filtering

TL;DR

This paper introduces an efficient adaptive block Lanczos method combined with Chebyshev filtering for computing dominant eigenvalues of large matrices, improving stability and spectral separation.

Contribution

The paper proposes a novel adaptive block Lanczos algorithm with Chebyshev filtering that enhances numerical stability and spectral separation for eigenvalue computations.

Findings

Effective on dense and sparse matrices

Improves numerical stability through SVD-based biorthogonality

Enhances spectral separation with Chebyshev filtering

Abstract

We present an efficient method for computing dominant eigenvalues of large, nonsymmetric, diagonalizable matrices based on an adaptive block Lanczos algorithm combined with Chebyshev polynomial filtering. The proposed approach improves numerical stability through two key components: (i) the Adaptive Block Lanczos (ABLE) method, which maintains biorthogonality using SVD based stabilization, and (ii) Chebyshev filtering, which enhances spectral separation via iterative polynomial filtering. Numerical experiments on dense and sparse test problems confirm the effectiveness of the ABLE Chebyshev algorithm.