  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.
