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An overview of the most successful algorithms and techniques for solving large, sparse systems of equations and some algorithms and strategies for solving optimization problems. The most important topics dealt with concern iterative methods, especially Krylov methods, ordering techniques, and some iterative optimization tools.
The book is a compendium of theoretical and numerical methods for solving large algebraic systems, special emphasis being placed on convergence and numerical behaviour as affected by rounding errors, accuracy in computing solutions for ill-conditioned matrices, preconditioning effectiveness, ordering procedures, stability factors, hybrid procedures and stopping criteria. Recent advances in numerical matrix calculations are presented, especially methods to accelerate the solution of symmetric and unsymmetric linear systems. Convergence analysis of the multi-grid method using a posteriori error estimation in second order elliptic equations are presented. Some inverse problems are also included. Evolution based software is described, such as genetic algorithms and evolution strategies, relations and class hierarchising to improve the exploration of large search spaces and finding near-global optima. Recent developments in messy genetic algorithms are also described.
The tutorial nature of the book makes it suitable for mathematicians, computer scientists, engineers and postgraduates.
Preface. Computational Complexity of Solving Large Sparse and Large Special Linear Systems of Equations; V.Y. Pan. Block Iterative Methods for Reduced Systems of Linear Equations; D.J. Evans. Parallel Implicit Schemes for the Solution of Linear Systems; D.J. Evans. Adaptive Multigrid Methods for Hybrid Finite Elements; L. Ferragut. On Finding and Analyzing the Structure of The Cholesky Factor; A. George. The Go-Away Algorithm for Block Factorization of a Sparse Matrix; P.R. Almeida, J.R. Franco. Renumbering Sparse Matrices by Simulated Annealing; G. Winter, et al. Preconditioned Krylov Subspace Methods; Y. Saad. Preconditioning Krylov Methods; A. Suárez, et al. Convergence and Numerical Behaviour of the Krylov Space Methods; Z. Strakos. Look-ahead Block-CG Algorithms; C.G. Broyden. Iterative Bi-CG Type Methods and Implementation Aspects; H. Van der Vorst, G.L.G. Sleijpen. Problems of Breakdown and Near-Breakdown in Lanczos-Based Algorithms; C. Brezinski, et al. Hybrid Methods for Solving Systems of Equations; C. Brezinski. ABS Algorithms for Linear Equations and Applications to Optimization; E. Spedicato, et al. Solving Inverse Thermal Problems Using Krylov Methods; G. Montero. An Introduction on Global Optimization by Genetic Algorithms; G. Winter, et al. Blackbox and Non-blackbox Optimization: A Common Perspective; H. Kargupta. Messy Genetic Algorithms: Recent Developments; H. Kargupta. List of Contributors. Index.