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.
翻译:一个真实的Skew对称矩阵的光谱分解法($A美元)可以数学地转换成一个特定结构化单值分解(SVD)美元。根据这种等值,为计算极端单值和相应的单向运量($A美元)的特殊SVD问题,提出了一种SVD问题(SSLBD)的光谱分解法。从SVD问题计算出极端单值和相应的单向运量($A$),从SVD问题中可以计算出与极端同级同级的极等值相对应的美元($A美元),在真实的算术中双向地回收。该方法的一些趋同结果已经确定,并给出了近乎奇特三重的精确估计值。在有限的精确算法中,证明了每组基矢量的半垂直度和两组基本矢量的半二次量值足以准确计算单值。通常使用的高效的重新解算法战略被调整,以维持所需的半垂直和半二次的半二次重新定序算法效率。