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Eigenvalues of Matrices Françoise Chatelin, Laboratoire Central de Recherches, Thomson-CSF, Orsay, France With exercises by Françoise Chatelin and Mario Ahués, Universite; de Saint-Etienne, France Translated with additional material by Walter Ledermann, University of Sussex The calculation of eigenvalues of matrices is a problem of great practical and theoretical importance with many different types of application. This book provides a modern and complete guide to this subject, at an elementary level, by presenting in matrix notation the fundamental aspects of the theory of linear operators in finite dimensions. This volume is a combination of two books; translations of Professor Chatelin's original and the corresponding book of exercises by Professor Ahués. The exercises are an indispensable complement to the main text. Solutions are furnished for some of the exercises. The book will be of particular value to undergraduate students following courses on numerical analysis and for researchers and practitioners with an interest in this area.
Algebra --- Eigenwaarden --- Eigenvalues --- Matrices --- 512.64 --- 512.64 Linear and multilinear algebra. Matrix theory --- Linear and multilinear algebra. Matrix theory --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Eigenvalues. --- Matrices. --- Valeurs propres --- Itération (mathématiques) --- Analyse numérique. --- Numerical analysis --- Iterative methods (Mathematics) --- Algèbre linéaire. --- Algebras, Linear --- Itération (mathématiques) --- Analyse numérique. --- Algèbre linéaire.
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Assume that after preconditioning we are given a fixed point problem x = Lx + f (*) where L is a bounded linear operator which is not assumed to be symmetric and f is a given vector. The book discusses the convergence of Krylov subspace methods for solving fixed point problems (*), and focuses on the dynamical aspects of the iteration processes. For example, there are many similarities between the evolution of a Krylov subspace process and that of linear operator semigroups, in particular in the beginning of the iteration. A lifespan of an iteration might typically start with a fast but slowing phase. Such a behavior is sublinear in nature, and is essentially independent of whether the problem is singular or not. Then, for nonsingular problems, the iteration might run with a linear speed before a possible superlinear phase. All these phases are based on different mathematical mechanisms which the book outlines. The goal is to know how to precondition effectively, both in the case of "numerical linear algebra" (where one usually thinks of first fixing a finite dimensional problem to be solved) and in function spaces where the "preconditioning" corresponds to software which approximately solves the original problem.
Convergence --- Equations --- Iterative methods (Mathematics) --- Convergence (Mathématiques) --- Itération (Mathématiques) --- Numerical solutions --- Solutions numériques --- 517.98 --- 519.6 --- 681.3*G13 --- Functional analysis and operator theory --- Computational mathematics. Numerical analysis. Computer programming --- Numerical linear algebra: conditioning; determinants; Eigenvalues; error analysis; linear systems; matrix inversion; pseudoinverses; sparse and very largesystems --- 681.3*G13 Numerical linear algebra: conditioning; determinants; Eigenvalues; error analysis; linear systems; matrix inversion; pseudoinverses; sparse and very largesystems --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 517.98 Functional analysis and operator theory --- Convergence (Mathématiques) --- Itération (Mathématiques) --- Solutions numériques --- Convergence. --- Numerical solutions.
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