Every Superinterval of the Function Range Can Be an Interval-Computations Enclosure
How good an estimate can we get by applying "naive" interval computations technique to a given function f(x1,...,xn) over given intervals x1,...,xn? This question was raised in several papers and conference presentations by G. Alefeld, R. Lohner, and others.
Recently, several results have been proven which show, crudely speaking, that whatever reformulation g(x1,...,xn) we choose, we cannot hope to get the exact range for all possible input intervals. However, a question remains: it is possible, for each sequence of input intervals, to find a reformulation which leads to exact range for this particular sequence of input intervals? If not, which enclosures can we thus get? In this paper, we show that if we allow min and max, then, for fixed input intervals, we can get an arbitrary superinterval of the actual range (including the exact range itself).
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- 1.Aberth, O.: Precise Numerical Analysis, Wm. C. Brown Publishers, Dubuque, Iowa, 1988.Google Scholar
- 2.Aberth, O.: Precise Numerical Analysis Using C++, Academic Press, Boston, 1998.Google Scholar
- 3.Alefeld, G.: Enclosure Methods, in: Ullrich, C. (ed.), Computer Arithmetic and Self-Validating Numerical Methods, Academic Press, Boston, 1990, pp. 55-72.Google Scholar
- 4.Beeson, M. J.: Foundations of Constructive Mathematics, Springer-Verlag, N.Y., 1985.Google Scholar
- 5.Bishop, E.: Foundations of Constructive Analysis, McGraw-Hill, 1967.Google Scholar
- 6.Bishop, E. and Bridges, D. S.: Constructive Analysis, Springer, N.Y., 1985.Google Scholar
- 7.Bridges, D. S.: Constructive Functional Analysis, Pitman, London, 1979.Google Scholar
- 8.Hertling, P.: A Lower Bound for Range Enclosure in Interval Arithmetic, in: Chesnaux, J.-M., Jezequel, F., Lamotte, J.-L., and Vignes, J. (eds), Proceedings of the Third Real Numbers and Computers Conference, Paris, April 1998, pp. 103-115.Google Scholar
- 9.Hertling, P.: A Quadratic Lower Bound for Range Computation in Interval Arithmetic, in: Proceedings of the International Conference on Interval Methods and Their Applications on Global Optimization INTERVAL'98, Nanjing, April 1998, pp. 49-51.Google Scholar
- 10.Kreinovich, V., Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer, Dordrecht, 1998.Google Scholar
- 11.Kushner, B. A.: Lectures on Constructive Mathematical Analysis, American Mathematical Society, Providence, RI, 1984.Google Scholar
- 12.Nguyen, H. T., Kreinovich, V., Nesterov, V., and Nakamura, M.: On Hardware Support for Interval Computations and for Soft Computing: A Theorem, IEEE Transactions on Fuzzy Systems 5(1) (1997), pp. 108-127.Google Scholar