Solving Normal Equations in Linear Least Squares Problems

Laboratory for High Performance Scientific Computing and Computer Simulation

Department of Computer Science

University of Kentucky

Lexington, KY 40506-0046, USA

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

**Mathematics Subject Classification**: 65F10, 65F20, 65F25, 65F50.

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This research work supported in part by the U.S. National Science Foundation under the grant CCR-9902022, CCR-9988165, CCR-0092532.