Complexity Theory of Real Functions

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Springer Science & Business Media, Dec 6, 2012 - Computers - 310 pages
Starting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some times also yield interesting new practical algorithms. A typical exam ple is the application of the ellipsoid algorithm to combinatorial op timization problems (see, for example, Lovasz [1986]). On the other hand, it has a strong influence on many different branches of mathe matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per mutation groups. In the area of numerical computation, there are also two tradi tionally independent approaches: recursive analysis and numerical analysis.
 

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Contents

Introduction
1
Basics in Discrete Complexity Theory
12
Computational Complexity of Real
40
Maximization
71
Roots and Inverse Functions
107
Measure and Integration
159
Differentiation
190
Ordinary Differential Equations
215
Approximation by Polynomials
247
An Optimization Problem in Control
274
Bibliography 291
290
Index
301
Copyright

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