  The joint bidiagonalization process of a matrix pair $\{A,L\}$ can be used to
develop iterative regularization algorithms for large scale ill-posed problems
in general-form Tikhonov regularization
$\min_x\left\{\|Ax-b\|_{2}^2+\lambda^2\|Lx\|_{2}^2\right\}$ or the essentially
equivalent one $\min\|Lx\|_{2} \ \mbox{{\rm s.t.}} \ x\in\mathcal{S} = \{x|\
\|Ax-b\|_{2}\leq \eta\|e\|_{2}\}$, where $e$ is a Gaussian white noise, $L$ is
a regularization matrix and $\eta>1$ slightly. A bottleneck of the algorithms
is that a large scale inner least squares problem with $(A^{T}, L^{T})^{T}$ as
the coefficient matrix must be solved at each outer iteration, which may be
costly, especially when the solution accuracy of these problems is high. In
this paper, we give a detailed investigation on the solution accuracy
requirement on the inner least squares problems and propose a practical
stopping criterion for inner iterations. The results show that for ill-posed
problems with not too small noise levels, the solution accuracy of the inner
least squares problems can be relaxed considerably while it will not reduce the
accuracy of regularized solutions, thus the overall efficiency of the
algorithms can be improved substantially. Numerical experiments are made to
illustrate our results and show some numerical performance of the algorithms.
