Before revision: The joint bidiagonalization process of a large matrix pair A,L ^2+\lambda^2\|Lx\|_{2}^2\right\} or the essentially equivalent one \min\|Lx\|_{2} \ s.t. \ x\inS = \{x|\ \|Ax-b\|_{2}\leq \eta\|e\|_{2}\}, where e is a Gaussian white noise, L is a regularization matrix and \eta>1 slightly.
After revision: The joint bidiagonalization process of a large 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} \ s.t. \ x\inS = \{x|\ \|Ax-b\|_{2}\leq \eta\|e\|_{2}\}, where e is a Gaussian white noise, L is a regularization matrix and \eta>1 slightly.