Accélération de la convergence en analyse numérique / C. Brezinski

Auteur: Brezinski, Claude (1941-) - AuteurType de document: MonographieCollection: Lecture notes in mathematics ; 584Langue: françaisPays: AllemagneÉditeur: Berlin : Springer-Verlag, 1977Description: 1 vol. (313 p.) ; 25 cm ISBN: 3540082417 ; br. Résumé: This book is an introduction to methods of acceleration of convergence in numerical analysis. Acceleration of convergence is an important part of numerical analysis which, as the author states, has been little investigated. A large number of methods used in numerical analysis and in applied mathematics are iterative methods, but they converge too slowly, so that one must use methods of acceleration. This volume is designed for mathematicians who wish to study this field, as well as for all those who wish to use these methods of acceleration. There are algorithms of acceleration that open new methods in numerical analysis, with connections to the initial subject, namely, solution of systems of linear and nonlinear equations, calculation of the eigenvalues of a matrix, numerical quadratures, etc. This book is the result of a course that the author gave for a number of years at the University of Lille. He states that this volume is mainly theoretical, but that he is preparing a work containing numerous applications and FORTRAN programs of the algorithms. The topics discussed are as follows: (1) comparison of convergent sequences; order of a sequence; comparison of two sequences; comparison theorems; index of comparison; asymptotic development of a series; (2) the processes of summation: general formulation of the problem; study of several processes; the process of extrapolation of Richardson; interpretation of certain processes of summation; (3) the epsilon-algorithm: the Δ2 process of Aitken; the transformation of Shanks and of the epsilon-algorithm; properties of the epsilon-algorithm; interpretation of the epsilon-algorithm and the epsilon-algorithm and the Padé table; theorems of convergence; application to numerical quadrature; (4) study of different algorithms for the acceleration of convergence: the process of Overhalt; various other algorithms; generalizations of the epsilon-algorithm; the problem of acceleration of convergence; formalization of the processes of acceleration of convergence; use of the algorithms of acceleration of convergence; (5) transformation of more general sequences: epsilon-algorithm applied to sequences of square matrices; transformation of sequences in Banach space; transformation of a sequence of vectors by the epsilon-algorithm; solution of systems of nonlinear equations by the vector epsilon-algorithm; calculation of proper values of a matrix by the vector epsilon-algorithm; the topological epsilon-algorithm; (6) estimation algorithms: the first confluent form of the epsilon-algorithm; study of the convergence; the problem of the acceleration of convergence; confluent form of the topological epsilon-algorithm; development in Taylor series; confluent form of the Overholt process; rational transformation of a function; applications of the confluent epsilon-algorithm and of the ρ-algorithm; (7) continued fractions: definitions and properties; transformation of a series into a continued fraction; contraction; associated and corresponding continued fractions; the q−d algorithm; convergence; interpolation formulas; interpolation of Hermite by rational fractions; (8) conclusions; (9) references. (MathSciNet).Bibliographie: Bibliogr. p. [298]-311. Index. Sujets MSC: 65Bxx Numerical analysis -- Acceleration of convergence
65-02 Numerical analysis -- Research exposition (monographs, survey articles)
40Axx Sequences, series, summability -- Convergence and divergence of infinite limiting processes
En-ligne: Springerlink | MathSciNet
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Bibliogr. p. [298]-311. Index

This book is an introduction to methods of acceleration of convergence in numerical analysis. Acceleration of convergence is an important part of numerical analysis which, as the author states, has been little investigated. A large number of methods used in numerical analysis and in applied mathematics are iterative methods, but they converge too slowly, so that one must use methods of acceleration. This volume is designed for mathematicians who wish to study this field, as well as for all those who wish to use these methods of acceleration. There are algorithms of acceleration that open new methods in numerical analysis, with connections to the initial subject, namely, solution of systems of linear and nonlinear equations, calculation of the eigenvalues of a matrix, numerical quadratures, etc. This book is the result of a course that the author gave for a number of years at the University of Lille. He states that this volume is mainly theoretical, but that he is preparing a work containing numerous applications and FORTRAN programs of the algorithms.
The topics discussed are as follows: (1) comparison of convergent sequences; order of a sequence; comparison of two sequences; comparison theorems; index of comparison; asymptotic development of a series; (2) the processes of summation: general formulation of the problem; study of several processes; the process of extrapolation of Richardson; interpretation of certain processes of summation; (3) the epsilon-algorithm: the Δ2 process of Aitken; the transformation of Shanks and of the epsilon-algorithm; properties of the epsilon-algorithm; interpretation of the epsilon-algorithm and the epsilon-algorithm and the Padé table; theorems of convergence; application to numerical quadrature; (4) study of different algorithms for the acceleration of convergence: the process of Overhalt; various other algorithms; generalizations of the epsilon-algorithm; the problem of acceleration of convergence; formalization of the processes of acceleration of convergence; use of the algorithms of acceleration of convergence; (5) transformation of more general sequences: epsilon-algorithm applied to sequences of square matrices; transformation of sequences in Banach space; transformation of a sequence of vectors by the epsilon-algorithm; solution of systems of nonlinear equations by the vector epsilon-algorithm; calculation of proper values of a matrix by the vector epsilon-algorithm; the topological epsilon-algorithm; (6) estimation algorithms: the first confluent form of the epsilon-algorithm; study of the convergence; the problem of the acceleration of convergence; confluent form of the topological epsilon-algorithm; development in Taylor series; confluent form of the Overholt process; rational transformation of a function; applications of the confluent epsilon-algorithm and of the ρ-algorithm; (7) continued fractions: definitions and properties; transformation of a series into a continued fraction; contraction; associated and corresponding continued fractions; the q−d algorithm; convergence; interpolation formulas; interpolation of Hermite by rational fractions; (8) conclusions; (9) references. (MathSciNet)

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