New IMGS-based Preconditioners for Least Squares Problems

Convergence acceleration by preconditioning is usually essential when solving the standard least squares problems by an iterative method. IMGS, is an incomplete modiied version of Gram-Schmidt orthogonalization to obtain an incomplete orthogonal factorization pre-conditioner M = R, where A = Q R + E is an approximation of a QR factorization, Q is an orthogonal matrix and R is upper triangular matrix respectively. Based on the IMGS orthogonalization, a relaxed Incomplete Modiied Gram-Schmidt preconditioning and a new recursive selecting strategy of incomplete orthogonal preconditioning which updates the drop tolerance step by step and decides the corresponding value according to the recursive relation are proposed. The numerical experiments show clearly the robustness of this recursive selecting strategy. For the relaxed IMGS preconditioning approach, a suitable relaxation parameter innuences the performance and quality of the preconditioner.