Ebook: Complexity Theory of Real Functions
Author: Ker-I Ko (auth.)
- Tags: Math Applications in Computer Science, Real Functions, Algorithm Analysis and Problem Complexity, Algorithms, Applications of Mathematics, Theory of Computation
- Series: Progress in Theoretical Computer Science
- Year: 1991
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-ix
Introduction....Pages 1-11
Basics in Discrete Complexity Theory....Pages 12-39
Computational Complexity of Real Functions....Pages 40-70
Maximization....Pages 71-106
Roots and Inverse Functions....Pages 107-158
Measure and Integration....Pages 159-189
Differentiation....Pages 190-214
Ordinary Differentiation Equations....Pages 215-246
Approximation by Polynomials....Pages 247-273
An Optimization Problem in Control Theory....Pages 274-289
Back Matter....Pages 291-310
Content:
Front Matter....Pages i-ix
Introduction....Pages 1-11
Basics in Discrete Complexity Theory....Pages 12-39
Computational Complexity of Real Functions....Pages 40-70
Maximization....Pages 71-106
Roots and Inverse Functions....Pages 107-158
Measure and Integration....Pages 159-189
Differentiation....Pages 190-214
Ordinary Differentiation Equations....Pages 215-246
Approximation by Polynomials....Pages 247-273
An Optimization Problem in Control Theory....Pages 274-289
Back Matter....Pages 291-310
....