  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.
