Representations of intervals and optimal error bounds

Author:
L. B. Rall

Journal:
Math. Comp. **41** (1983), 219-227

MSC:
Primary 65G10; Secondary 41A50

MathSciNet review:
701636

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Abstract: Classical methods of numerical analysis lead to approximate values and error bounds for computed results, while interval analysis provides lower and upper bounds for the exact values. Equivalence of these approaches is investigated on the basis of a general definition of an error function, and a way is given to find optimal approximation points in an interval *I* and the corresponding minimum error bound . Conversely, it is shown that , can be used as alternative coordinates for the representation of the interval *I*, and the rules for interval arithmetic can be formulated in the resulting system. In particular, the arithmetic, harmonic, and geometric means of the endpoints of an interval turn out to be the optimal approximation points corresponding to absolute error, relative error, and relative precision, respectively. The rules for interval arithmetic are given explicitly in the resulting coordinate systems. It is also shown that calculations performed in interval arithmetic generally lead to smaller intervals than predicted by certain a priori bounds.

**[1]**G. H. Hardy, J. E. Littlewood, and G. Pólya,*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395****[2]**Ramon E. Moore,*Interval analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR**0231516****[3]**Ramon E. Moore,*Methods and applications of interval analysis*, SIAM Studies in Applied Mathematics, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1979. MR**551212****[4]**F. W. J. Olver,*A new approach to error arithmetic*, SIAM J. Numer. Anal.**15**(1978), no. 2, 368–393. MR**0483379****[5]**H. Ratschek,*Representation of interval operations by coordinates*, Computing**24**(1980), no. 2-3, 93–96 (English, with German summary). MR**620078**, 10.1007/BF02281714

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701636-9

Keywords:
Optimal approximations,
optimal error bounds,
interval arithmetic,
absolute error,
relative error,
percentage error,
relative precision,
elementary means,
excess width

Article copyright:
© Copyright 1983
American Mathematical Society