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Projection Methods for Systems of Equations,

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Projection Methods for Systems of Equations,

C. Brezinski

ISBN 0444827773
Pages 408

The solutions of systems of linear and nonlinear equations occurs in many situations and is therefore a question of major interest. Advances in computer technology has made it now possible to consider systems exceeding several hundred thousands of equations. However, there is a crucial need for more efficient algorithms.

The main focus of this book (except the last chapter, which is devoted to systems of nonlinear equations) is the consideration of solving the problem of the linear equation Ax = b by an iterative method. Iterative methods for the solution of this question are described which are based on projections. Recently, such methods have received much attention from researchers in numerical linear algebra and have been applied to a wide range of problems.

The book is intended for students and researchers in numerical analysis and for practitioners and engineers who require the most recent methods for solving their particular problem.

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. Series: Studies in Computational Mathematics