The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,L\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,L\}$. 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 bidiagonalizations. 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 $(A^T,L^T)^T$ 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)过程迭代地将矩阵对$\{A,L\}$同时化为两个双对角形式，可用于计算$\{A,L\}$的偏广义奇异值分解(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的不精确计算内迭代：首先探究内迭代计算误差对外迭代的影响，进而提出重正交化JBD(rJBD)过程以保持部分Lanczos向量的正交性。对rJBD进行误差分析以建立其与Lanczos双对角化的联系，并利用这些结果研究基于rJBD的GSVD计算的收敛性与精度。研究表明：计算得到的GSVD分量的精度取决于内迭代的计算精度以及$(A^T,L^T)^T$的条件数，而收敛速度受影响较小。对于实际基于JBD的GSVD计算，本文结果可为选择合适内迭代计算精度以获取所需精度的近似GSVD分量提供指导。数值实验验证了理论结果。