A generalized skew-symmetric Lanczos bidiagonalization method for computing several extreme eigenpairs of a large skew-symmetric/symmetric positive definite matrix pair

TL;DR

This paper introduces a generalized Lanczos bidiagonalization method tailored for efficiently computing multiple extreme eigenpairs of large skew-symmetric and symmetric positive definite matrix pairs, with rigorous convergence analysis and practical algorithm enhancements.

Contribution

It proposes a novel GSSLBD method with convergence analysis, accuracy estimates, and an efficient IRGSSLBD algorithm for large matrix pairs.

Findings

The method accurately computes eigenvalues in finite precision.

Semi-$B$-orthogonality suffices for accurate eigenpair computation.

Numerical experiments demonstrate robustness and efficiency.

Abstract

A generalized skew-symmetric Lanczos bidiagonalization (GSSLBD) method is proposed to compute several extreme eigenpairs of a large matrix pair $(A,B)$, where $A$ is skew-symmetric and $B$ is symmetric positive definite. The underlying GSSLBD process produces two sets of $B$-orthonormal generalized Lanczos basis vectors that are also $B$-biorthogonal and a series of bidiagonal matrices whose singular values are taken as the approximations to the imaginary parts of the eigenvalues of $(A,B)$ and the corresponding left and right singular vectors premultiplied with the left and right generalized Lanczos basis matrices form the real and imaginary parts of the associated approximate eigenvectors. A rigorous convergence analysis is made on the desired eigenspaces approaching the Krylov subspaces generated by the GSSLBD process and accuracy estimates are made for the approximate eigenpairs. In finite precision arithmetic, it is shown that the semi-$B$-orthogonality and semi-$B$-biorthogonality of the computed left and right generalized Lanczos vectors suffice to compute the eigenvalues accurately. An efficient partial reorthogonalization strategy is adapted to GSSLBD in order to maintain the desired semi-$B$-orthogonality and semi-$B$-biorthogonality. To be practical, an implicitly restarted GSSLBD algorithm, abbreviated as IRGSSLBD, is developed with partial $B$-reorthogonalizations. Numerical experiments illustrate the robustness and overall efficiency of the IRGSSLBD algorithm.