Projection Methods for Systems of Equations

Introduction. 1. Preliminaries. Projections and projection methods. Projections. Projection methods. Extrapolation methods. Best approximation. Algorithms for recursive projection. The general interpolation problem. Recursive projection. Solving linear systems by extrapolation. The topological &egr;-algorithm. The S-&bgr;-algorithm. Vector Padé approximants. Vector &thgr;-type transformations. 2. Biorthogonality. Generalities. Biorthogonal polynomials. Hankel and Toeplitz systems. Hankel matrices. Toeplitz matrices. Biorthogonalization processes. 3. Projection Methods for Linear Systems. Variational formulation. Projection iterative methods. Method of conjugate directions. The conjugate gradients algorithm. Row projection methods. Projection acceleration procedures. Preconditioned steepest descent algorithms. Accelerated descent methods. 4. Lanczos-Type Methods. Extrapolation and projection methods. Vorobyev's method of moments. The method of Lanczos. Generalizations of Lanczos' method. Orthores. Lanczos/Orthores. The method of Arnoldi. 5. Hybrid Procedures. The basic procedure. Recursive use of the procedure. Convergence acceleration. Multiple hybrid procedures. Changing the minimilization criterion. 6. Semi-Iterative Methods. Semi-iterative hybrid procedures. More projection methods. Stationary iterative methods. Nonstationary iterative methods. The minimal residual smoothing method. A hybrid minimal residual smoothing method. 7. Around Richardson's Projection. The basic idea. Choice of the search direction. Choice of the preconditioner. Constant preconditioner. Linear iterative preconditioner. Quadratic iterative preconditioner. Direct preconditioner. A sparse preconditioner. Numerical examples. Another choice for the search direction. Two-step methods. Splitting-up methods. Multiparameter extensions. The symmetric positive definitive case. 8. System of Nonlinear Equations. Introduction. Quasi-Newton methods. A preconditioned Newton's method. Barnes secant method. Broyden methods. The &egr;-algorithms. The method of Henrici. The method of Wolfe. Extension of Newton's method. Nonlinear hybrid procedures. The basic procedure. A vector sequence transformation. Application to fixed point problems. Some fixed point methods. The method of Lemaréchal. The method of Marder-Weitzner. The &Dgr;^k. The more stable MW-type scheme. The methods of Barzilai-Borwein. A choice for the search direction. A multiparameter &Dgr;^k method. Definition of the multiparameter scheme. Minimizing the residual. Appendix. Schur's complement. Sylvester's and Schweins' identities. Bibliography. Index.