Abstract
This paper is a brief introduction to how the techniques of computational complexity can be applied to real analysis—integration in particular. We investigate how the difficulty of computing a function relates to the difficulty of computing its integral. Our comments are directed to an audience that is more familiar with traditional analysis and numerical methods than it is with complexity theory.
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References
O. Aberth. Computable Analysis. 1980. McGraw-Hill.
E. Bishop. Foundations of Constructive Analysis. 1967. McGraw-Hill.
J. M. Borwein and P. B. Borwein. On the complexity of familiar functions and numbers. Draft. Private Communication, 1986.
H. Friedman. The computational complexity of maximization and integration. Adv. in Math., 53, 1984, 80–98.
M. R. Garey and D. S. Johnson. Computers and Intractability. Freeman, 1979.
J. von zur Gathen. Feasible arithmetic computations: Valiant’s hypothesis. J. Symb. Comp., 4, 1987, 137–172.
J. von zur Gathen and G. Seroussi. Boolean circuits versus arithmetic circuits. Proc. 6th Int. Conf. in Computer Science, Santiago, Chile. 1986, 171–184.
A. Grzegorczyk. On the definition of computable real continuous functions. Fund. Math., 44, 1957, 61–71.
H. J. Hoover. Feasibly constructive analysis. Ph.D. thesis and Technical Report 206/87, Department of Computer Science, University of Toronto, 1987.
H. J. Hoover. Feasible real functions and arithmetic circuits. Technical Report TR 88-16, August 1988, Department of Computing Science, University of Alberta, Edmonton, Canada.
K. Ko and H. Friedman. Computational complexity of real functions. Theoret. Comput. Sci., 20, 1982, 323–352.
A. M. Turing. On computable numbers, with an application to the entscheidungsproblem. Proc. London Math. Soc. (2), 42, 1936/37, 230–265.
A. M. Turing. A correction. Proc. London Math. Soc. (2), 43, 1937, 544–546.
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© 1989 Springer-Verlag New York Inc.
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Hoover, H.J. (1989). Why Integration is Hard. In: Kaltofen, E., Watt, S.M. (eds) Computers and Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9647-5_22
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DOI: https://doi.org/10.1007/978-1-4613-9647-5_22
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97019-6
Online ISBN: 978-1-4613-9647-5
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