Projection Methods for Linear Systems

The aim of this paper is to give a uniied framework for deriving projection methods for solving systems of linear equations. We shall show that all these methods follow from a unique minimization problem. The particular cases of the methods of steepest descent, Richardson and conjugate gradients will be treated in details. Projection acceleration procedures for accelerating the convergence of an arbitrary iterative method will also be proposed and discussed. The aim of this paper is to give a uniied framework for deriving projection methods for solving systems of linear equations. Usually, these methods are obtained either by projection on various planes or by minimizing the next residual or, when the matrix of the system is symmetric and positive deenite, by minimizing a quadratic functional. We shall show that all these approaches follow, in fact, from a unique minimization problem based on a variational formulation and that all the projection methods can be derived from it. The particular cases of the methods of steepest descent, Richardson and conjugate gradients will be treated in details. Based on the previous ideas, projection acceleration procedures for accelerating the convergence of an arbitrary iterative method will also be proposed and discussed. Extensions of the minimal residual smoothing algorithm 43, 42] and of the hybrid procedure 8] will also be obtained. More details can be found in 7]. 1 Variational formulation Let X and Y be subspaces of an inner product space on IR and M : X ?! Y a linear symmetric positive deenite operator. It is well know (see, for example 11, 37]) that the solution x of the equation Mx = c, where c 2 Y , is also the unique minimum of the strictly convex quadratic functional