Ullrich Computer Arithmetic and Self-Validating Numerical Methods

1. Auflage 2014
ISBN: 978-1-4832-6781-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 316 Seiten, Web PDF

Notes and Reports in Mathematics in Science and Engineering, Volume VII: Computer Arithmetic and Self-Validating Numerical Methods compiles papers presented at the first international conference on 'Computer Arithmetic and Self-Validating Numerical Methods, held in Basel from October 2 to 6, 1989. This book begins by providing a tutorial introduction to computer arithmetic with operations of maximum accuracy, differentiation arithmetic and enclosure methods, and programming languages for self-validating numerical methods. The rest of the chapters discuss the determination of guaranteed bounds for eigenvalues by variational methods and guaranteed inclusion of solutions of differential equations. An appendix covering the IMACS-GAMM resolution on computer arithmetic is provided at the end of this publication. This volume is recommended for researchers and professionals working on computer arithmetic and self-validating numerical methods.

Autoren/Hrsg.

Ullrich, Christian

Inhaltsverzeichnis

1;Front Cover;1
2;Computer Arithmetic and Self-Validating Numerical Methods;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;8
6;Preface;10
7;Acknowledgments;12
8;Chapter 1. What Do We Need Beyond IEEE Arithmetic ?;14
8.1;1 Introduction;14
8.2;2 Scalar products and IEEE arithmetic;15
8.3;3 Algorithms for the scalar product;17
8.4;4 Problems and suggestions;21
8.5;5 Designs and implementations;28
8.6;6 Conclusions;39
8.7;References;40
9;Chapter 2. Chips for High Precision Arithmetic;46
9.1;1. Introduction;46
9.2;2. Exploration of the Design Space;47
9.3;3. Architecture of the ARITHMOS Processor;50
9.4;4. Architecture Evaluation;61
9.5;5. Conclusions;65
9.6;Acknowledgement;66
9.7;References;66
10;Chapter 3. Enclosure Methods;68
10.1;1. Introduction;68
10.2;2. Notation;69
10.3;3. Interval arithmetic evaluation;69
10.4;4. Outlook;84
10.5;References;84
11;Chapter 4. Differentiation Arithmetics;86
11.1;1. Evaluation Arithmetics;86
11.2;2. Code List Representation of Functions;87
11.3;3. Formal Power Series Arithmetic;88
11.4;4. Automatic Differentiation;90
11.5;5. Taylor Arithmetics;91
11.6;6. Rounded Taylor Arithmetic;92
11.7;7. Partial Derivatives;95
11.8;8. Gradient and Hessian Arithmetic;96
11.9;9. Serial Computation of Gradients and Hessians;97
11.10;10. Parallel Implementation of Differentiation Arithmetics;100
11.11;References;102
12;Chapter 5. Industrial Applications of Interval Techniques;104
12.1;1. Introduction;104
12.2;2. High Accuracy;105
12.3;3. When should interval techniques be considered?;113
12.4;4. An example - Least squares;114
12.5;5. An example - Nonlinear systems;116
12.6;6. What are some limitations of interval techniques?;118
12.7;7. What should you DO?;123
12.8;Acknowledgments;124
12.9;References;125
13;Chapter 6. Programming Languages for Enclosure Methods;128
13.1;1 Introduction;128
13.2;2 The Role of Arithmetic;130
13.3;3 New Developments;135
13.4;4 New Datatype Dotprecision;137
13.5;5 Scalar Product Expressions;139
13.6;6 Program Parts with Highly Accurate Evaluation of Expressions;142
13.7;7 Final Remarks;146
13.8;References;146
14;Chapter 7. The Determination of Guaranteed Bounds to Eigenvalues with the Use of Variational Methods
I;150
14.1;1. Introduction;150
14.2;2. Eigenvalue problems with bilinear forms;151
14.3;3. Determination of guaranteed bounds to eigenvalues by means of matrix eigenvalue problems;152
14.4;4. Inclusion theorems and variational methods;154
14.5;6. Further numerical tests;164
15;Chapter 8. The Determination of Guaranteed Bounds to Eigenvalues with the Use of Variational Methods
II;168
15.1;1 Introduction;168
15.2;2 Matrix eigenvalue problems;169
15.3;3 An eigenvalue problem with an ordinary differential equation;177
15.4;References;182
16;Chapter 9. Validated Solution of Initial Value Problems for
ODE;184
16.1;Introduction;184
16.2;1 The Method, Areas for Improvement;185
16.3;2 Accuracy Control;187
16.4;3 Minimizing the Effort;190
16.5;4 A priori Inclusion;193
16.6;5 Representation of Inclusion Sets;195
16.7;6 Stiff Systems;196
16.8;References;199
17;Chapter 10. Guaranteed Inclusions of Solutions of some Types of Boundary Value Problems;202
17.1;1. Introduction and operators of monotonie type;202
17.2;2. Choice of a suitable class of approximating functions;203
17.3;3. The algorithm;204
17.4;4. Interval-Analysis;204
17.5;5. Some remarks for the numerical computation;205
17.6;6. Some classes of operators of montonic type;205
17.7;7. Mixed boundary value problems (Kreiss-Lorenz [89]);206
17.8;8. Discontinuous boundary values, Cavity flow;207
17.9;9. A nonlinear delay equation;209
17.10;10. Generalizations and Outlook;209
17.11;References;210
18;Chapter 11. Periodic Solutions: Enclosure, Verification, and Applications;212
18.1;1. Introduction;212
18.2;2. Periodic Solutions via Discretizations of ODEs and Discretization Errors;215
18.3;3. The Logistic Equation;219
18.4;4. The Lotka-Volterra Problem;220
18.5;5. The Lorenz Problem;221
18.6;6. The Restricted Three Body Problem of Celestial Mechanics;234
18.7;7. Periodic Solutions of Mathematical Models for Gear Drive Vibrations;244
18.8;9. Conclusions and Final Remarks;248
18.9;List of References;253
19;Chapter 12. Numerical Algorithms for Existence Proofs and Error Estimates for Two-Point Boundary Value Problems;260
19.1;1. Introduction;260
19.2;2. The construction of T;262
19.3;3. The choice of D;263
19.4;4. Constructing
. by linearization;264
19.5;5. On the procedures A and
E;265
19.6;6. Some results on differential inequalities;267
19.7;7. Veryfying (L) in some special cases;268
19.8;8. Transformation, general remarks;273
19.9;9. Transformation into an integral equation;275
19.10;10. Transformation by using breakpoints;278
19.11;References;280
20;Chapter 13. Aspects of Self-Validating Numerics in Banach Spaces;282
20.1;1. Introduction;283
20.2;2. One Autonomous Nonlinear PDE;286
20.3;3. Systems of First Order Autonomous Nonlinear PDEs;294
20.4;4. Generalized Hyperbolic Nonlinear Systems;300
20.5;5. Remarks on Boundary Conditions and Parallel Computations;302
20.6;6. A First Approach to the Problem;305
20.7;7. Some Algorithmic Aspects;307
20.8;8. Conclusions;310
21;References;311
22;Appendix: IMACS-GAMM Resolution on Computer Arithmetic;314
23;NOTES AND REPORTS IN MATHEMATIC SINCIENCE AND ENGINEERING;316