A Skew-Symmetric Lanczos Bidiagonalization Method for Computing Several Extremal Eigenpairs of a Large Skew-Symmetric Matrix

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1114-1147, June 2024.
Abstract. The spectral decomposition of a real skew-symmetric matrix is shown to be equivalent to a specific structured singular value decomposition (SVD) of the matrix. Based on such equivalence, we propose a skew-symmetric Lanczos bidiagonalization (SSLBD) method to compute extremal singular values and the corresponding singular vectors of the matrix, from which its extremal conjugate eigenpairs 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 the computed left and right Lanczos basis vectors and the semi-biorthogonality of two sets of basis vectors are needed to compute the singular values accurately and to make the method work as if it does in exact arithmetic. A commonly used efficient partial reorthogonalization strategy is adapted to maintain the desired semi-orthogonality and semi-biorthogonality. For practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm.