TABLE OF CONTENTS Although the CG process itself has a regularizing effect due to the order in which the Ritz values converge, there is no guarantee that this particular convergence always takes place in particular in finite precision. Hence, it is sometimes convenient to combine the Lanczos bidiagonalization process in LSQR with an "inner regularization" algorithm applied to the bidiagonal ( k + 1) k least squares problem min ? B k ? k ? ? 1 e ( k+1) 1 ? 2 (6.18), and with an "inner regularization parameter" that depends on the iteration number k. In this way, we can effectively filter out small Ritz values that may converge before all the large singular values of A have been captured. This is sometimes called a "hybrid method" [171, 7]. The cost of this feature is that we must save the left Lanczos vectors, i.e., the matrix V k; this is not necessary in the LSQR algorithm.
These ideas were first explored by O'Leary and Simmons in their bidiagonalization-regularization procedure [267], and later described by Bj rck [35]. In these papers, TSVD and Tikhonov regularization are considered as "inner regularization" methods. The main difference between the approaches in the two papers is that [267] uses a general start vector for the Lanczos bidiagonalization process, while [35] uses the right-hand side b as start vector and explicitly uses the LSQR algorithm based on the relation T k+1 b = ?
