Kaucher / Miranker / Rheinboldt Self-Validating Numerics for Function Space Problems

Computation with Guarantees for Differential and Integral Equations
1. Auflage 2014
ISBN: 978-1-4832-7377-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 256 Seiten, Web PDF

Self-Validating Numerics for Function Space Problems describes the development of computational methods for solving function space problems, including differential, integral, and function equations. This seven-chapter text highlights three approaches, namely, the E-methods, ultra-arithmetic, and computer arithmetic. After a brief overview of the different self-validating approaches, this book goes on introducing the mathematical preliminaries consisting principally of fixed-point theorems and the computational context for the development of validating methods in function spaces. The subsequent chapters deals with the development and application of point of view of ultra-arithmetic and the constructs of function-space arithmetic spaces, such as spaces, bases, rounding, and approximate operations. These topics are followed by discussion of the iterative residual correction methods for function problems and the requirements of a programming language needed to make the tools and constructs of the methodology available in actual practice on a computer. The last chapter describes the techniques for adapting the methodologies to a computer, including the self-validating results for specific problems. This book will prove useful to mathematicians and advance mathematics students.

Autoren/Hrsg.

Kaucher, Edgar W.

Miranker, Willard L.

Rheinboldt, Werner

Inhaltsverzeichnis

1;Front Cover;1
2;Self-Validating Numerics for Function Space Problems;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Acknowledgments;14
8;Chapter
1. Introduction;16
8.1;E-Methods;18
8.2;Ultra-arithmetic;19
8.3;Computer Arithmetic;20
8.4;Suggestions to the Reader;23
9;Chapter

2. Mathematical Preliminaries;24
9.1;2.1 Basic Formulation of Self-Validating Methods in M;25
9.2;2.2 A Broader Setting for Self-Validating Methods;27
10;Chapter

3. Ultra-arithmetic and Roundings;43
10.1;3.1 Spaces, Bases, Roundings, and Approximate Operations;44
10.2;3.2 Spaces, Bases, and Roundings for Validation;80
11;Chapter

4. Methods for Functional Equations;99
11.1;4.1 Methods for Linear Equations;104
11.2;4.2 Methods for Nonlinear Function Equations;143
12;Chapter

5. Iterative Residual Correction;157
12.1;5.1 Arithmetic Implications of IRC;159
12.2;5.2 IRC for Initial-Value Problems and Volterra Integral Equations;167
12.3;5.3 Iterative Residual Correction with Carry;174
12.4;5.4 A Formalism for IRC in Function Space;186
13;Chapter

6. Comments on Programming Language;206
14;Chapter

7. Application and Illustrative Computation;211
14.1;7.1 Review of the Computational Process;212
14.2;7.2 Illustrative Computation;233
15;Glossaries;262
16;References;269