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For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or$Ax=b$ with $b$ contaminated by a white noise, Lanczos bidiagonalization basedLSQR and its mathematically equivalent CGLS are most commonly used. They haveintrinsic regularizing effects, where the number $k$ of iterations plays therole of regularization parameter. However, hitherto there has been no answer tothe long-standing fundamental concern of Bj\"{o}rck and Eld\'{e}n in 1979: {\emfor which kinds of problems LSQR and CGLS can find best possible regularizedsolutions}? Here a best possible regularized solution means that it is at leastas accurate as the best regularized solution obtained by the truncated singularvalue decomposition (TSVD) method or by standard-form Tikhonov regularizationand cannot be improved under certain conditions. In this paper we make adetailed analysis on the regularization of LSQR for severely, moderately andmildly ill-posed problems. For the first two kinds of problems, we prove thatLSQR finds best possible solutions at semi-convergence and the followingresults hold until semi-convergence: (i) the $k$-step Lanczos bidiagonalizationalways generates a near best rank $k$ approximation to $A$; (ii) the $k$ Ritzvalues always approximate the first $k$ large singular values of $A$ in naturalorder; (iii) the $k$-step LSQR always captures the $k$ dominant SVD componentsof $A$; (iv) the diagonals and subdiagonals of the bidiagonal matricesgenerated by Lanczos bidiagonalization decay as fast as the singular values of$A$. However, for the third kind of problem, the above results do not holdgenerally. We also analyze the regularization of the other two Krylov solversLSMR and CGME, proving that LSMR has similar regularizing effects to LSQR foreach kind of problem and both are superior to CGME. Numerical experimentsconfirm our theory on LSQR.

The Regularization Theory of the Krylov Iterative Solvers LSQR, CGLS, LSMR and CGME For Linear Discrete Ill-Posed Problems