Front Cover; Computer Arithmetic and Self-Validating Numerical Methods; Copyright Page; Table of Contents; Contributors; Preface; Acknowledgments; Chapter 1. What Do We Need Beyond IEEE Arithmetic ?; 1 Introduction; 2 Scalar products and IEEE arithmetic; 3 Algorithms for the scalar product; 4 Problems and suggestions; 5 Designs and implementations; 6 Conclusions; References; Chapter 2. Chips for High Precision Arithmetic; 1. Introduction; 2. Exploration of the Design Space; 3. Architecture of the ARITHMOS Processor; 4. Architecture Evaluation; 5. Conclusions; Acknowledgement; References

Chapter 3. Enclosure Methods1. Introduction; 2. Notation; 3. Interval arithmetic evaluation; 4. Outlook; References; Chapter 4. Differentiation Arithmetics; 1. Evaluation Arithmetics; 2. Code List Representation of Functions; 3. Formal Power Series Arithmetic; 4. Automatic Differentiation; 5. Taylor Arithmetics; 6. Rounded Taylor Arithmetic; 7. Partial Derivatives; 8. Gradient and Hessian Arithmetic; 9. Serial Computation of Gradients and Hessians; 10. Parallel Implementation of Differentiation Arithmetics; References; Chapter 5. Industrial Applications of Interval Techniques; 1. Introduction

2. High Accuracy3. When should interval techniques be considered?; 4. An example -- Least squares; 5. An example -- Nonlinear systems; 6. What are some limitations of interval techniques?; 7. What should you DO?; Acknowledgments; References; Chapter 6. Programming Languages for Enclosure Methods; 1 Introduction; 2 The Role of Arithmetic; 3 New Developments; 4 New Datatype Dotprecision; 5 Scalar Product Expressions; 6 Program Parts with Highly Accurate Evaluation of Expressions; 7 Final Remarks; References

Chapter 7. The Determination of Guaranteed Bounds to Eigenvalues with the Use of Variational MethodsI1. Introduction; 2. Eigenvalue problems with bilinear forms; 3. Determination of guaranteed bounds to eigenvalues by means of matrix eigenvalue problems; 4. Inclusion theorems and variational methods; 6. Further numerical tests; Chapter 8. The Determination of Guaranteed Bounds to Eigenvalues with the Use of Variational MethodsII; 1 Introduction; 2 Matrix eigenvalue problems; 3 An eigenvalue problem with an ordinary differential equation; References

Chapter 9. Validated Solution of Initial Value Problems forODEIntroduction; 1 The Method, Areas for Improvement; 2 Accuracy Control; 3 Minimizing the Effort; 4 A priori Inclusion; 5 Representation of Inclusion Sets; 6 Stiff Systems; References; Chapter 10. Guaranteed Inclusions of Solutions of some Types of Boundary Value Problems; 1. Introduction and operators of monotonie type; 2. Choice of a suitable class of approximating functions; 3. The algorithm; 4. Interval-Analysis; 5. Some remarks for the numerical computation; 6. Some classes of operators of montonic type