The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair $\{A,L\}$, where we propose three approaches to compute approximate generalized singular values and vectors. We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of $\{A,L\}$. The results show that, under some mild conditions, the semiorthogonality of Lanczos type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We also establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.
翻译:联合多角化(JBD)方法被用来计算一个大型常规矩阵配方的某种极端通用单值和矢量,即$A,L ⁇ $,我们在此建议三种方法来计算大致通用单值和矢量。我们对基JBD进程进行数字分析,并确立其与两个数学等效的有限精度Lanczos标数化战略之间的关系。根据数字分析的结果,我们调查了美元A,L ⁇ $的大致通用单值和矢量的趋同性。结果显示,在某些温和条件下,兰卡佐斯类型矢量的半正数性足以提供与完全或正数性相同的近似通用单数值,这意味着只需要为JBD进程寻求有效的半数化战略即可。我们还为近似通用奇数单值和相应近似正值普通矢量的残余规范设定了一条锐线,从而可以可靠地估计剩余规范,而无需在趋同之前明确计算近称近似通用的近似普通矢量矢量值。