Before revision:  The results show that  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 . We also discuss some details for implementing the joint bidiagonalization based regularization algorithms efficiently and stably .
After revision:  The results show that 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 .