Primary Audience for the Book Specialists in numerical computations who are interested in algorithms with automatic result verification. Engineers, scientists, and practitioners who desire results with automatic verification and who would therefore benefit from the experience of suc cessful applications. Students in applied mathematics and computer science who want to learn these methods. Goal Of the Book This book contains surveys of applications of interval computations, i. e. , appli cations of numerical methods with automatic result verification, that were pre sented at an international workshop on the subject in EI Paso, Texas, February 23-25, 1995. The purpose of this book is to disseminate detailed and surveyed information about existing and potential applications of this new growing field. Brief Description of the Papers At the most fundamental level, interval arithmetic operations work with sets: The result of a single arithmetic operation is the set of all possible results as the operands range over the domain. For example, [0. 9,1. 1] + [2. 9,3. 1] = [3. 8,4. 2], where [3. 8,4. 2] = {x + ylx E [0. 9,1. 1] and y E [3. 8,4. 2]}. The power of interval arithmetic comes from the fact that (i) the elementary operations and standard functions can be computed for intervals with formulas and subroutines; and (ii) directed roundings can be used, so that the images of these operations (e. g.
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ISBN-13
978-1-4613-3440-8 (9781461334408)
DOI
10.1007/978-1-4613-3440-8
Schweitzer Klassifikation
1 Applications of Interval Computations: An Introduction.- 1 What are Interval Computations?.- 2 International Workshop on Applications of Interval Computations: How This Book Originated.- 3 General Optimization Problems.- 4 General Systems of Equations and Inequalities.- 5 Linear Interval Problems.- 6 Interval Computations Can Also Handle Possible Additional Information About the Input Data.- 7 Software and Hardware Support for Interval Computations.- References.- 2 A Review of Techniques in the Verified Solution of Constrained Global Optimization Problems.- 1 Introduction, Basic Ideas and Literature.- 2 On Constrained Optimization Problems.- 3 On Use of Interval Newton Methods.- 4 Applications.- 5 Summary and Present Work.- References.- 3 The Shape of the Symmetric Solution Set.- 1 Introduction.- 2 Notation.- 3 Results.- 4 Examples.- References.- 4 Linear Interval Equations: Computing Enclosures with Bounded Relative Overestimation is NP-Hard.- 1 Introduction.- 2 The Result.- 3 The Symmetric Case.- 4 Concluding Remark.- References.- 5 Quality Improvement via Optimization of Tolerance Intervals During the Design Stage.- 1 Introduction.- 2 Some Basic Models, and their Origins.- 3 Model of Performance Characteristic is Known Beforehand.- 4 Model Parameters Estimated in Controlled Conditions.- 5 Controlled Conditions are Unavailable.- 6 Temperature Controller.- 7 Conclusions.- References.- 6 Applications of Interval Computations to Regional Economic Input-Output Models.- 1 Economic Input-Output Models.- 2 Technical Coefficients are Only Known with Uncertainty.- 3 Statistical Methods are Not Directly Applicable, Hence, Interval Computations May Be Useful.- 4 Computational Algorithms.- 5 An Example.- References.- 7 Interval Arithmetic in Quantum Mechanics.- 1 Quantum Mechanics.- 2 Computer-Assisted Set-up.- 3 The Thomas-Fermi Equation.- 4 The Aperiodicity Inequality.- References.- 8 Interval Computations on the Spreadsheet.- 1 Limitations of Spreadsheet Computing.- 2 Extended IA on a Spreadsheet.- 3 Global IA on a Spreadsheet.- 4 Interval Constraint Spreadsheets.- 5 Discussion.- References.- 9 Solving Optimization Problems with Help of the UniCalc Solver.- 1 Introduction.- 2 The UniCalc Solver.- 3 The Algorithm of Sub definite Calculations.- 4 Solving Integer Programming Problems.- 5 Real-Valued Optimization.- 6 Future Developments.- References.- 10 Automatically Verified Arithmetic on Probability Distributions and Intervals.- 1 Introduction.- 2 Correctly Representing PDFs and Intervals with Histograms.- 3 Arithmetic Operations.- References.- 11 Nested Intervals and Sets: Concepts, Relations to Fuzzy Sets, and Applications.- 1 Introduction.- 2 Nested Intervals and Nested Sets.- 3 Other Problems Where Nested Sets and Nested Intervals Can Be Used: Identification, Optimization, Control, and Decision Making.- 4 Applications of Nested Sets and Nested Intervals.- Appendix A Proofs.- References.- 12 Fuzzy Interval Inference Utilizing the Checklist Paradigm and BK-Relational Products.- 1 Introduction.- 2 Many-Valued Logics for Interval Fuzzy Inference Based on the Checklist Paradigm.- 3 Groups of Logic Transformations of Interval Connectives.- 4 Special Types of Compositions of Relations.- 5 An Application: The Basic Knowledge Handling Mechanisms of CLIN AID by Means of Relational Inference.- 6 Toward Successful Utilization of Interval Methods in Soft Computing.- References.- 13 Computing Uncertainty in Interval Based Sets.- 1 Introduction.- 2 Evidence Sets as a Description of Uncertainty.- 3 Different Measures of Uncertainty, and How to Describethem Numerically.- 4 L-Fuzzy Sets, Interval Based L-Fuzzy Sets, and L-Evidence Sets.- 5 3-D Uncertainty Unit Cube.- References.- 14 Software and Hardware Techniques for Accurate, Self-Validating Arithmetic.- 1 Introduction.- 2 Software Tools.- 3 Hardware Designs.- 4 A Variable-Precision, Interval Arithmetic Coprocessor.- 5 Conclusions and Areas for Future Research.- References.- 15 Stimulating Hardware and Software Support for Interval Arithmetic.- 1 Introduction.- 2 The Participants.- 3 Stable Equilibrium.- 4 The Interval Paradigm Shift.- 5 System Supplier Demand.- 6 End-user Demand.- 7 Action Plan.- References.
