  LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and
its mathematically equivalent CGLS applied to normal equations system, are
commonly used for large-scale discrete ill-posed problems. It is well known
that LSQR and CGLS have regularizing effects, where the number of iterations
plays the role of the regularization parameter. However, it has long been
unknown whether the regularizing effects are good enough to find best possible
regularized solutions. Here a best possible regularized solution means that it
is at least as accurate as the best regularized solution obtained by the
truncated singular value decomposition (TSVD) method. In this paper, we
establish bounds for the distance between the $k$-dimensional Krylov subspace
and the $k$-dimensional dominant right singular space. They show that the
Krylov subspace captures the dominant right singular space better for severely
and moderately ill-posed problems than for mildly ill-posed problems. Our
general conclusions are that LSQR has better regularizing effects for the first
two kinds of problems than for the third kind, and a hybrid LSQR with
additional regularization is generally needed for mildly ill-posed problems.
Exploiting the established bounds, we derive an estimate for the accuracy of
the rank $k$ approximation generated by Lanczos bidiagonalization. Numerical
experiments illustrate that the regularizing effects of LSQR are good enough to
compute best possible regularized solutions for severely and moderately
ill-posed problems, stronger than our theory predicts, but they are not for
mildly ill-posed problems and additional regularization is needed.
