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A skew-symmetric Lanczos bidiagonalization method for computing several largest eigenpairs of a large skew-symmetric matrix

Jinzhi Huang, Zhongxiao Jia

TL;DR

This work develops a skew-symmetric Lanczos bidiagonalization (SSLBD) framework to compute several largest conjugate eigenpairs of a large real skew-symmetric matrix $A$ by converting the problem to a structured SVD and performing operations in real arithmetic. It establishes rigorous convergence and accuracy results, showing that semi-orthogonality and semi-biorthogonality suffice to accurately compute singular values, and it introduces an efficient implicitly restarted SSLBD algorithm with partial reorthogonalization to control ghost artifacts. The method leverages a half-sized bidiagonal decomposition and Rayleigh-Ritz projection to obtain Ritz triplets $(\theta,\tilde u,\tilde v)$ that recover eigenpairs $(\pm i\theta, (\tilde u \pm i\tilde v)/\sqrt{2})$, with cheap residual checks guiding termination. Numerical experiments demonstrate strong performance and reliability, often outperforming standard eigensolvers like ${\sf eigs}$ and ${\sf svds}$ on large skew-symmetric matrices while maintaining real arithmetic throughout.

Abstract

The spectral decomposition of a real skew-symmetric matrix $A$ can be mathematically transformed into a specific structured singular value decomposition (SVD) of $A$. Based on such equivalence, a skew-symmetric Lanczos bidiagonalization (SSLBD) method is proposed for the specific SVD problem that computes extreme singular values and the corresponding singular vectors of $A$, from which the eigenpairs of $A$ corresponding to the extreme conjugate eigenvalues in magnitude are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of basis vectors and the semi-biorthogonality of two sets of basis vectors suffice to compute the singular values accurately. A commonly used efficient partial reorthogonalization strategy is adapted to maintaining the needed semi-orthogonality and semi-biorthogonality. For a practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm.

A skew-symmetric Lanczos bidiagonalization method for computing several largest eigenpairs of a large skew-symmetric matrix

TL;DR

This work develops a skew-symmetric Lanczos bidiagonalization (SSLBD) framework to compute several largest conjugate eigenpairs of a large real skew-symmetric matrix by converting the problem to a structured SVD and performing operations in real arithmetic. It establishes rigorous convergence and accuracy results, showing that semi-orthogonality and semi-biorthogonality suffice to accurately compute singular values, and it introduces an efficient implicitly restarted SSLBD algorithm with partial reorthogonalization to control ghost artifacts. The method leverages a half-sized bidiagonal decomposition and Rayleigh-Ritz projection to obtain Ritz triplets that recover eigenpairs , with cheap residual checks guiding termination. Numerical experiments demonstrate strong performance and reliability, often outperforming standard eigensolvers like and on large skew-symmetric matrices while maintaining real arithmetic throughout.

Abstract

The spectral decomposition of a real skew-symmetric matrix can be mathematically transformed into a specific structured singular value decomposition (SVD) of . Based on such equivalence, a skew-symmetric Lanczos bidiagonalization (SSLBD) method is proposed for the specific SVD problem that computes extreme singular values and the corresponding singular vectors of , from which the eigenpairs of corresponding to the extreme conjugate eigenvalues in magnitude are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of basis vectors and the semi-biorthogonality of two sets of basis vectors suffice to compute the singular values accurately. A commonly used efficient partial reorthogonalization strategy is adapted to maintaining the needed semi-orthogonality and semi-biorthogonality. For a practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm.
Paper Structure (11 sections, 9 theorems, 95 equations, 4 tables, 3 algorithms)

This paper contains 11 sections, 9 theorems, 95 equations, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The SSLBD method
  4. The $m$-step SSLBD process
  5. The SSLBD method
  6. A convergence analysis
  7. An implicitly restarted SSLBD algorithm with partial reorthogonalization
  8. Partial reorthogonalization
  9. An implicitly restarted algorithm
  10. Numerical examples
  11. Conclusions

Key Result

Theorem 2.1

\newlabelthm1 The spectral decomposition of the nonsingular skew-symmetric $A\in\mathbb{R}^{n\times n}$ with $n=2\ell$ is with where $U\in\mathbb{R}^{n\times \ell}$ and $V\in\mathbb{R}^{n\times \ell}$ are orthonormal and biorthogonal: and $\Sigma=\mathop{\operator@font diag}\nolimits\{\sigma_1,\dots,\sigma_{\ell}\} \in\mathbb{R}^{\ell\times \ell}$ with $\sigma_1,\dots,\sigma_\ell>0$. The SVD of

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Remark 1
  • Theorem 2.2
  • proof
  • Remark 2
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 11 more