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.

翻译：本文提出了一种广义斜对称Lanczos双对角化（GSSLBD）方法，用于计算大规模矩阵对$(A,B)$的若干极端特征对，其中$A$为斜对称矩阵，$B$为对称正定矩阵。该GSSLBD过程生成两组$B$-标准正交的广义Lanczos基向量（同时满足$B$-双正交性）以及一系列双对角矩阵，这些双对角矩阵的奇异值被用作$(A,B)$特征值虚部的近似值，而对应的左右奇异向量分别左乘左右广义Lanczos基矩阵后，构成相应近似特征向量的实部与虚部。本文对GSSLBD过程生成的Krylov子空间逼近目标特征空间的过程进行了严格的收敛性分析，并对近似特征对的精度给出了估计。在有限精度运算中，研究表明计算得到的左右广义Lanczos向量所具备的半$B$-正交性与半$B$-双正交性足以保证特征值的计算精度。为维持所需的半$B$-正交性与半$B$-双正交性，一种高效的部分重正交化策略被适配至GSSLBD方法中。为提升实用性，本文进一步开发了结合部分$B$-重正交化的隐式重启GSSLBD算法（简称IRGSSLBD）。数值实验验证了IRGSSLBD算法的鲁棒性与整体计算效率。