Some properties of LSQR for large sparse linear least squares problems

It is well-known that many Krylov solvers for linear systems, eigenvalue problems, and singular value decomposition problems have very simple and elegant formulas for residual norms. These formulas not only allow us to further understand the methods theoretically but also can be used as cheap stopping criteria without forming approximate solutions and residuals at each step before convergence takes place. LSQR for large sparse linear least squares problems is based on the Lanczos bidiagonalization process and is a Krylov solver. However, there has not yet been an analogously elegant formula for residual norms. This paper derives such kind of formula. In addition, the author gets some other properties of LSQR and its mathematically equivalent CGLS.