Collatz Functional Analysis and Numerical Mathematics

1. Auflage 2014
ISBN: 978-1-4832-6400-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 494 Seiten, Web PDF

Functional Analysis and Numerical Mathematics focuses on the structural changes which numerical analysis has undergone, including iterative methods, vectors, integral equations, matrices, and boundary value problems. The publication first examines the foundations of functional analysis and applications, including various types of spaces, convergence and completeness, operators in Hilbert spaces, vector and matrix norms, eigenvalue problems, and operators in pseudometric and other special spaces. The text then elaborates on iterative methods. Topics include the fixed-point theorem for a general iterative method in pseudometric spaces; special cases of the fixed-point theorem and change of operator; iterative methods for differential and integral equations; and systems of equations and difference methods. The manuscript takes a look at monotonicity, inequalities, and other topics, including monotone operators, applications of Schauder's theorem, matrices and boundary value problems of monotone kind, discrete Chebyshev approximation and exchange methods, and approximation of functions. The publication is a valuable source of data for mathematicians and researchers interested in functional analysis and numerical mathematics.

Autoren/Hrsg.

Collatz, Lothar

Inhaltsverzeichnis

1;Front Cover;1
2;Functional Analysis and Numerical Mathematics;4
3;Copyright Page;5
4;Table of Contents;12
5;Translator's Note;6
6;Preface to the German Edition;8
7;Notation;20
8;CHAPTER I. Foundations of Functional Analysis and Applications;22
8.1;1. TYPICAL PROBLEMS IN NUMERICAL MATHEMATICS;22
8.2;2. VARIOUS TYPES OF SPACES;36
8.3;3. ORDERINGS;66
8.4;4. CONVERGENCE AND COMPLETENESS;73
8.5;5. COMPACTNESS;97
8.6;6. OPERATORS IN PSEUDOMETRIC AND OTHER SPECIAL SPACES;107
8.7;7. OPERATORS IN HILBERT SPACES;130
8.8;8. EIGENVALUE PROBLEMS;158
8.9;9. VECTOR AND MATRIX NORMS;185
8.10;10. FURTHER THEOREMS ON VECTOR AND MATRIX NORMS;200
9;CHAPTER II. Iterative Methods;221
9.1;11. THE FIXED-POINT THEOREM FOR A GENERAL ITERATIVE METHOD IN PSEUDOMETRIC SPACES;221
9.2;12. SPECIAL CASES OF THE FIXED-POINT THEOREM AND CHANGE OF OPERATOR;233
9.3;13. ITERATIVE METHODS FOR SYSTEMS OF EQUATIONS;244
9.4;14. SYSTEMS OF EQUATIONS AND DIFFERENCE METHODS;254
9.5;15. ITERATIVE METHODS FOR DIFFERENTIAL AND INTEGRAL EQUATIONS;272
9.6;16. DERIVATIVE OF OPERATORS IN SUPERMETRIC SPACES;288
9.7;17. SOME SPECIAL ITERATIVE METHODS;303
9.8;18. THE METHOD OF FALSE POSITION (Regula Falsi);321
9.9;19. NEWTON'S METHOD WITH IMPROVEMENTS;336
9.10;20. MONOTONICITY AND EXTREMUM PRINCIPLES FOR NEWTON'S METHOD;355
10;CHAPTER III. Monotonicity, Inequalities, and Other Topics;371
10.1;21. MONOTONE OPERATORS;371
10.2;22. FURTHER APPLICATIONS OF SCHAUDER'S THEOREM;384
10.3;23. MATRICES AND BOUNDARY VALUE PROBLEMS OF MONOTONE KIND;397
10.4;24. INITIAL VALUE PROBLEMS AND ADDITIONAL THEOREMS ON MONOTONICITY;414
10.5;25. APPROXIMATION OF FUNCTIONS;431
10.6;26. DISCRETE CHEBYSHEV APPROXIMATION AND EXCHANGE METHODS;454
11;APPENDIX;471
12;Remarks on Schauder's Fixed-Point Theorem;471
13;REFERENCES;477
14;Author Index;486
15;Subject Index;489