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For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or$Ax=b$ with $b$ contaminated by white noise, the Golub-Kahan bidiagonalizationbased LSQR method and its mathematically equivalent CGLS, the ConjugateGradient (CG) method applied to $A^TAx=A^Tb$, are most commonly used. They haveintrinsic regularizing effects, where the iteration number $k$ plays the roleof regularization parameter. The long-standing fundamental question is: {\emCan LSQR and CGLS find 2-norm filtering best possible regularized solutions}?The author has given definitive answers to this question for severely andmoderately ill-posed problems when the singular values of $A$ are simple. Thispaper extends the results to the multiple singular value case, and studies theapproximation accuracy of Krylov subspaces, the quality of low rankapproximations generated by Golub-Kahan bidiagonalization and the convergenceproperties of Ritz values. For the two kinds of problems, we prove that LSQRfinds 2-norm filtering best possible regularized solutions at semi-convergence.Particularly, we consider some important and untouched issues on best, nearbest and general rank $k$ approximations to $A$ for the ill-posed problems withthe singular values $\sigma_k=\mathcal{O}(k^{-\alpha})$ with $\alpha>0$, andthe relationships between them and their nonzero singular values. Numericalexperiments confirm our theory. The results on general rank $k$ approximationsand the properties of their nonzero singular values apply to several Krylovsolvers, including LSQR, CGME, MINRES, MR-II, GMRES and RRGMRES.

The Krylov Subspaces, Low Rank Approximations and Ritz Values of LSQR for Linear Discrete Ill-Posed Problems: the Multiple Singular Value Case