Abstract

We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction of special notations and a so-called normal form of directed intervals. As an application, it has been shown that both interval systems can be used for the computation of tight inner and outer inclusions of ranges of functions and consequently for the development of software for automatic computation of ranges of functions.

Citations

42	Interval Analysis in the Extended Interval Space - Kaucher - 1980
26	Theory of interval algebra and its application to numerical analysis - Sunaga - 1958
13	Uber Eigenschaften und Anwendungsmoglichkeiten der erweiterten Intervall-rechnung und des hyperbolische Fastkorpers uber R - Kaucher - 1977
11	Calculus for interval functions of a real variable, Computing 22 - Markov - 1979
11	Validated solution of initial value problems for ODE - Stetter - 1990
8	On the presentation of ranges of monotone functions using interval arithmetic", Interval Computations - Markov - 1992
8	M.: \Some Interpolation Problems Involving Interval Data - Markov - 1993
6	Some Applications of the Extended Interval Arithmetic to Interval Iterations"; Computing Suppl - Markov - 1980
4	E.: \Interval Analysis"; Prentice-Hall, Englewood Cli s - Moore - 1966
3	Using Markov's Interval Arithmetic to Evaluate Bessel-Ricatti Functions"; NOC - Bartholomew-Biggs, Zakovic - 1994
3	Uber die Distributivgesetze der erweiterten Intervallarithmetik"; Computing 24 - Dimitrova - 1980
3	Uber einige intervallarithmetische Grundbegri e - Ratschek - 1969
3	Solving the tolerance problem for interval linear equations - Shary - 1994
2	On Validated Newton Type Method for Nonlinear Equations - Dimitrova, Markov - 1994
2	Trepat, A.: "Fundamentals of SIGLA, an Interval Computing System over the Completed Set of Intervals - Gardenes - 1980
2	Extended Interval Arithmetic Involving - Markov - 1992
2	V."How to Use Monotonicity Type Information to Get Better Estimates of The Range of Real-Valued Functions - Nesterov - 1993
2	Teilbarkeitskriterien der Intervallarithmetik - Ratschek - 1972
2	Computer Methods for the Range of Functions", Ellis Horwood Limited - Ratschek, Rokne - 1984
2	P.: On controlled solution set of interval algebraic systems - Shary - 1992
1	L.: "Computing the Range of Derivatives"; In: Computer arithmetic, scienti c computation and mathematical modelling - Corliss, Rall - 1991
1	E.: "Extended Interval Arithmetics: New Results and Applications"; In: Computer arithmetic and enclosure methods (Eds - Dimitrova, Markov, et al. - 1992
1	Extended Interval Arithmetic"; Compt. rend - Markov - 1977
1	Bemerkung zur Di erentiation von intervallwertigen Funktionen"; Computing 26 - Schroder - 1981