The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,B\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,B\}$. The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiabonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of $(A^T,B^T)^T$ while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.

翻译：联合双对角化(JBD)过程通过迭代方式将矩阵对$\{A,B\}$同步缩减为两个双对角形式，可用于计算$\{A,B\}$的部分广义奇异值分解(GSVD)。该过程具有嵌套的内-外迭代结构，其中内迭代通常无法精确计算。本文研究JBD中不精确计算的内迭代问题：首先分析内迭代计算误差对外迭代的影响，继而提出一种重正交化JBD(rJBD)过程以保持部分Lanczos向量的正交性。通过对rJBD进行误差分析，建立其与Lanczos双对角化的联系。进一步利用这些结果研究基于rJBD的GSVD计算的收敛性与精度。结果表明，所计算GSVD分量的精度取决于内迭代的计算精度及$(A^T,B^T)^T$的条件数，而收敛速率受影响较小。对于实际基于JBD的GSVD计算，本文结果可为选择合适的内迭代计算精度以获取期望精度的近似GSVD分量提供指导。数值实验验证了理论结果。