Computational Complexity and Feasibility of Data Processing and Interval Computations
(Sprache: Englisch)
Targeted audience . Specialists in numerical computations, especially in numerical optimiza tion, who are interested in designing algorithms with automatie result ver ification, and who would therefore be interested in knowing how general their algorithms...
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Klappentext zu „Computational Complexity and Feasibility of Data Processing and Interval Computations “
Targeted audience . Specialists in numerical computations, especially in numerical optimiza tion, who are interested in designing algorithms with automatie result ver ification, and who would therefore be interested in knowing how general their algorithms caIi in principle be. . Mathematicians and computer scientists who are interested in the theory 0/ computing and computational complexity, especially computational com plexity of numerical computations. . Students in applied mathematics and computer science who are interested in computational complexity of different numerical methods and in learning general techniques for estimating this computational complexity. The book is written with all explanations and definitions added, so that it can be used as a graduate level textbook. What this book .is about Data processing. In many real-life situations, we are interested in the value of a physical quantity y that is diflicult (or even impossible) to measure directly. For example, it is impossible to directly measure the amount of oil in an oil field or a distance to a star. Since we cannot measure such quantities directly, we measure them indirectly, by measuring some other quantities Xi and using the known relation between y and Xi'S to reconstruct y. The algorithm that transforms the results Xi of measuring Xi into an estimate fj for y is called data processing.
Inhaltsverzeichnis zu „Computational Complexity and Feasibility of Data Processing and Interval Computations “
Preface1. Informal Introduction: Data Processing, Interval Computations, and Computational Complexity
2. The Notions of Feasibility and NP-Hardness: Brief Introduction
3. In the General Case, The Basic Problem of Interval Computations is Intractable
4. Basic Problem of Interval Computations for Polynomials of a Fixed Number of Variables
5. Basic Problem of Interval Computations for Polynomials of Fixed Order
6. Basic Problem of Interval Computations for Polynomials with Bounded Coefficients
7. Fixed Data Processing Algorithms, Varying Data: Still NP-Hard
8. Fixed Data, Varying Data Processing Algorithms: Still Intractable
9. What if we Only Allow Some Arithmetic Operations in Data Processing?
10. For Fractionally-Linear Functions, A Feasible Algorithm Solves the Basic Problem of Interval Computations
11. Solving Interval Linear Systems is NP-Hard
12. Interval Linear Systems: Search for Feasible Classes
13. Physical Corollary: Prediction is Not Always Possible, Even for Linear Systems with Known Dynamics
14. Engineering Corollary: Signal Processing is NP-Hard
15. Bright Sides of NP-Hardness of Interval Computations I: NP-Hard Means that Good Interval Heuristics Can Solve Other Hard Problems
16. If Input Intervals are Narrow Enough, then Interval Computations are Almost Always Easy
17. Optimization - A First Example of a Numerical Problem in Which Interval Methods are Used: Computational Complexity and Feasibility
18. Solving Systems of Equations
19. Approximation of Interval Functions
20. Solving Differential Equations
21. Properties of Interval Matrices I: Main Results
22. Properties of Interval Matrices II: Proofs and Auxiliary Results
23. Non-Interval Uncertainty I: Ellipsoid Uncertainty And its Generalizations
24. Non-Interval Uncertainty II: Multi-Intervals and Their Generalizations
25. What if Quantities are Discrete?
26. Error Estimation for Indirect Measurements: Interval
... mehr
Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem
- A: In Case of Interval (or More General) Uncertainty, No Algorithm can Choose the Simplest Representative
- B: Error Estimation for Indirect Measurements: Case of Approximately Known Functions
- C: From Interval Computations to Modal Mathematics
- D: Beyond NP: Two Roots Good, One Root Better
- E: Does `NP-Hard' Really Mean `Intractable'?
- F: Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will?
- G: The Worse, the Better: Paradoxical Computational Complexity of Interval Computations and Data Processing
- References
- Indexnty And its Generalizations
24. Non-Interval Uncertainty II: Multi-Intervals and Their Generalizations
25. What if Quantities are Discrete?
26. Error Estimation for Indirect Measurements: Interval Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem
- A: In Case of Interval (or More General) Uncertainty, No Algorithm can Choose the Simplest Representative
- B: Error Estimation for Indirect Measurements: Case of Approximately Known Functions
- C: From Interval Computations to Modal Mathematics
- D: Beyond NP: Two Roots Good, One Root Better
- E: Does `NP-Hard' Really Mean `Intractable'?
- F: Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will?
- G: The Worse, the Better: Paradoxical Computational Complexity of Interval Computations and Data Processing
- References
- A: In Case of Interval (or More General) Uncertainty, No Algorithm can Choose the Simplest Representative
- B: Error Estimation for Indirect Measurements: Case of Approximately Known Functions
- C: From Interval Computations to Modal Mathematics
- D: Beyond NP: Two Roots Good, One Root Better
- E: Does `NP-Hard' Really Mean `Intractable'?
- F: Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will?
- G: The Worse, the Better: Paradoxical Computational Complexity of Interval Computations and Data Processing
- References
- Indexnty And its Generalizations
24. Non-Interval Uncertainty II: Multi-Intervals and Their Generalizations
25. What if Quantities are Discrete?
26. Error Estimation for Indirect Measurements: Interval Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem
- A: In Case of Interval (or More General) Uncertainty, No Algorithm can Choose the Simplest Representative
- B: Error Estimation for Indirect Measurements: Case of Approximately Known Functions
- C: From Interval Computations to Modal Mathematics
- D: Beyond NP: Two Roots Good, One Root Better
- E: Does `NP-Hard' Really Mean `Intractable'?
- F: Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will?
- G: The Worse, the Better: Paradoxical Computational Complexity of Interval Computations and Data Processing
- References
... weniger
Bibliographische Angaben
- Autoren: V. Kreinovich , A. V. Lakeyev , P. T. Kahl , J. Rohn
- 1998, 476 Seiten, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer US
- ISBN-10: 0792348656
- ISBN-13: 9780792348658
- Erscheinungsdatum: 31.12.1997
Sprache:
Englisch
Rezension zu „Computational Complexity and Feasibility of Data Processing and Interval Computations “
`The book is very well organized. The book is readable, very detailed, and intelligible. Moreover, it is exciting and gripping because the reader is often surprised by unexpected results. Everyone who is interested in modern aspects of numerical computation will enjoy reading this interesting and informative work. The book is a must for everyone who is doing research in this field.' Mathematical Reviews, 98m
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