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

TL;DR

This paper introduces a skew-symmetric Lanczos bidiagonalization method for efficiently computing the largest eigenpairs of large skew-symmetric matrices, with proven convergence and practical reorthogonalization strategies.

Contribution

It proposes a novel SSLBD method tailored for skew-symmetric matrices, including convergence analysis, accuracy estimates, and an implicitly restarted algorithm with partial reorthogonalization.

Findings

The method accurately computes extreme eigenpairs in real arithmetic.

Theoretical convergence and accuracy guarantees are established.

Numerical experiments demonstrate the method's effectiveness and efficiency.

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.