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Preface | ||
1 | Informal Introduction: Data Processing, Interval Computations, and Computational Complexity | 1 |
2 | The Notions of Feasibility and NP-Hardness: Brief Introduction | 23 |
3 | In The General Case, The Basic Problem of Interval Computations is Intractable | 41 |
4 | Basic Problem of Interval Computations for Polynomials of a Fixed Number of Variables | 53 |
5 | Basic Problem of Interval Computations for Polynomials of Fixed Order | 71 |
6 | Basic Problem of Interval Computations for Polynomials with Bounded Coefficients | 79 |
7 | Fixed Data Processing Algorithms, Varying Data: Still NP-Hard | 83 |
8 | Fixed Data, Varying Data Processing Algorithms: Still Intractable | 85 |
9 | What If We Only Allow Some Arithmetic Operations in Data Processing? | 87 |
10 | For Fractionally-Linear Functions, A Feasible Algorithm Solves the Basic Problem of Interval Computations | 91 |
11 | Solving Interval Linear Systems is NP-Hard | 99 |
12 | Interval Linear Systems: Search for Feasible Classes | 111 |
13 | Physical Corollary: Prediction is Not Always Possible, Even for Linear Systems with Known Dynamics | 143 |
14 | Engineering Corollary: Signal Processing is NP-Hard | 153 |
15 | Bright Sides of NP-Hardness of Interval Computations I: NP-Hard Means That Good Interval Heuristics Can Solve Other Hard Problems | 159 |
16 | If Input Intervals are Narrow Enough, Then Interval Computations are Almost Always Easy | 161 |
17 | Optimization - A First Example of a Numerical Problem in Which Interval Methods are Used: Computational Complexity and Feasibility | 173 |
18 | Solving Systems of Equations | 197 |
19 | Approximation of Interval Functions | 207 |
20 | Solving Differential Equations | 219 |
21 | Properties of Interval Matrices I: Main Results | 225 |
22 | Properties of Interval Matrices II: Proofs and Auxiliary Results | 257 |
23 | Non-Interval Uncertainty I: Ellipsoid Uncertainty and Its Generalizations | 289 |
24 | Non-Interval Uncertainty II: Multi-Intervals and Their Generalizations | 309 |
25 | What If Quantities are Discrete? | 325 |
26 | Error Estimation for Indirect Measurements: Interval Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem | 331 |
A | In Case of Interval (or More General) Uncertainty, No Algorithm Can Choose the Simplest Representative | 347 |
B | Error Estimation for Indirect Measurements: Case of Approximately Known Functions | 365 |
C | From Interval Computations to Modal Mathematics | 381 |
D | Beyond NP: Two Roots Good, One Root Better | 395 |
E | Does "NP-Hard" Really Mean "Intractable"? | 401 |
F | Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will? | 405 |
G | The Worse, The Better: Paradoxical Computational Complexity of Interval Computations and Data Processing | 409 |
References | 413 | |
Index | 451 |
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