A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, mi, II Ac- b ll2, is proposed. The family is based on incomplete Gram-Schmidt (IGS) factorizations of A. Particular attention has been given to the following members of the family: Incomplete Classical Gram-Schmidt (ICGS), Incomplete Modified Gram-Schmidt (IMGS) and Compressed Incomplete Modified Gram-Schmidt (CIMGS) factorizations. The numerical properties of each of these methods have been considered as well as the relationships between the methods concerning the preservation of sparsity, computational efficiency and the quality of the preconditioner. The imple-mentation of these methods has been investigated and all of the important family members have been coded. One of the important topics in this portion of the dissertation is the careful symbolic analysis of the production of the preconditioner and its use during the incomplete factorization phase to avoid excessive unnecessary work. We have also analyzed some of the implications of viewing the incomplete Cholesky (IC) factorization as a limit case of the IGS family. When certain conditions are imposed on the sparsity pattern of the preconditioning matrix, IC is identical to IMGS. Therefore, a sufficient condition on the sparsity pattern for a stable IC factorization can be derived. Based on this condition, two algorithms for modifying
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