Computer Arithmetic and Self-Validating Numerical Methods

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 Methods

1. 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 Accuracy

3. 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 Methods

1. 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 Methods

168

1 Introduction

2 Matrix eigenvalue problems

3 An eigenvalue problem with an ordinary differential equation

References

Chapter 9. Validated Solution of Initial Value Problems for

184

Introduction

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

7. Mixed boundary value problems (Kreiss-Lorenz [89])

8. Discontinuous boundary values, Cavity flow

9. A nonlinear delay equation

10. Generalizations and Outlook

References

Chapter 11. Periodic Solutions: Enclosure, Verification, and Applications

1. Introduction

2. Periodic Solutions via Discretizations of ODEs and Discretization Errors

3. The Logistic Equation

4. The Lotka-Volterra Problem

5. The Lorenz Problem

6. The Restricted Three Body Problem of Celestial Mechanics

7. Periodic Solutions of Mathematical Models for Gear Drive Vibrations

9. Conclusions and Final Remarks

List of References

Chapter 12. Numerical Algorithms for Existence Proofs and Error Estimates for Two-Point Boundary Value Problems

1. Introduction

2. The construction of T

3. The choice of D

4. Constructing

264

5. On the procedures A and

6. Some results on differential inequalities

7. Veryfying (L) in some special cases

8. Transformation, general remarks

9. Transformation into an integral equation

10. Transformation by using breakpoints

References

Chapter 13. Aspects of Self-Validating Numerics in Banach Spaces

1. Introduction

2. One Autonomous Nonlinear PDE

3. Systems of First Order Autonomous Nonlinear PDEs

4. Generalized Hyperbolic Nonlinear Systems

5. Remarks on Boundary Conditions and Parallel Computations

6. A First Approach to the Problem

7. Some Algorithmic Aspects

8. Conclusions

References

Appendix: IMACS-GAMM Resolution on Computer Arithmetic

NOTES AND REPORTS IN MATHEMATIC SINCIENCE AND ENGINEERING