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.
Inhalt
Preface
Acknowledgments
1. Introduction
E-Methods
Ultra-Arithmetic
Computer Arithmetic
Remark on Enumeration
Suggestions to the Reader
2. Mathematical Preliminaries
2.1 Basic Formulation of Self-Validating Methods in M
Theorem (Schauder-Tychonoff)
Computational Context for Fixed Point Theorems
2.2 A Broader Setting for Self-Validating Methods
2.2.1 Basic Fixed Point Theorems
2.2.2 Modified Krasnoselski-Darbo Fixed Point Theorems
3. Ultra-Arithmetic and Roundings
3.1 Spaces, Bases, Roundings, and Approximate Operations
3.1.1 Examples of Bases
3.1.2 Examples of Roundings
3.2 Spaces, Bases, and Roundings for Validation
3.2.1 Examples of Bases
3.2.2 Examples of Roundings
4 Methods for Functional Equations
4.1 Methods for Linear Equations
4.1.1 The Finite Case
4.1.1 The Infinitesimal Case
4.2 Methods for Nonlinear Function Equations
4.2.1 The Finite Case
4.2.2 The Infinitesimal Case
5 Iterative Residual Correction
5.1 Arithmetic Implications of IRC
5.2 IRC for Initial-Value Problems and Volterra Integral Equations
5.3 Iterative Residual Correction with Carry
5.4 A Formalism for IRC in Function Space
5.4.1 The Block-Relaxation Formalism
5.4.2 Application of the Relaxation Formalism
6. Comments on Programming Language
7. Application and Illustrative Computation
7.1 Review of the Computational Process
7.1.1 Validation in R
7.1.2 Validation in RN
7.1.3 Validation in M
7.1.4 Implementation of the Ultra-Arithmetic
7.2 Illustrative Computation
7.2.1 Linear Differential Equations
7.2.2 Nonlinear Differential Equations
7.2.3 IRC
Glossaries
References