TABLE OF CONTENTS In finite-precision arithmetic the Lanczos bidiagonalization algorithm loses orthogonality among the Lanczos vectors, i.e., the columns of the matrices k+1 and 
in (6.15). Similarly, the finite-precision CGLS algorithm loses orthogonality among the vectors A T r ( k). Detailed studies of the CG and Lanczos algorithms in finite precision have recently appeared [68], [69], [70], [158], but these studies do not focus on the regularizing properties of CG.
The consequences of the loss of orthogonality in the Lanczos process are extremely important in connection with procedures for computing eigenvalues and singular values, and the subject has been studied extensively; see, e.g., [71] and the references therein. We emphasize the similarity between the Lanczos bidiagonalization process in the LSQR algorithm and the reorthogonalization free Lanczos procedures for computing the SVD in [71, Chapter 5]. However, as we describe in the following, the loss of orthogonality among the Lanczos vectors in LSQR is not nearly as harmful when solving least squares problems as when computing singular triplets.
Recall that the singular values of the bidiagonal matrix B k in (6.15) are the square roots of the Ritz values ? ( k) j. One consequence of finite-precision arithmetic is that, as the number k of iterations increases, those singular values of B k which have converged to isolated (possibly multiple) singular values of A become multiple, [20] and the multiplicity increases with k
