TABLE OF CONTENTS From the above discussion we see that a fundamental task in connection with a theoretical investigation of the regularizing properties of CG iterations is to determine the convergence properties of the Ritz values, given the particular features of discrete ill-posed problems mentioned in 2.1. This is a difficult problem, and it has not been solved satisfactorily yet.
To heuristically explain why the Lanczos bidiagonalization process tends to capture the largest singular values of A first, we switch to the normal equations A TA x = A Tb and to the basis of the right singular vectors v i of A, and again we assume without loss of generality that x (0) = 0. Then a basis for the Krylov subspace 
is given by the columns of the n k matrix DW k with
and the jth column of D W k has elements ? 2 j ?1 i u T ib, i = 1, , n. As long as the discrete Picard condition is satisfied, the elements of each column decay, on the average, with index i. Hence, the first unit vectors e ( n) 1, e ( n) 2, , which are eigenvectors of ? 2, can be approximated well by the columns of D W k.
