Integration of Interval Functions. II. The Finite Case.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Caprani, Madsen, and Rall have shown previously that the use of interval values leads to a simple theory of integration in which all functions, interval and real, are integrable. Here, a simplified construction of the interval integral is given for the case that the integrand and interval of integration are finite the interval integral is shown to be the intersection of the interval Darboux sums corresponding to the partitions of the interval of integration into subintervals of equal length. A rate of convergence of these interval Darboux sums to the interval integral is given for Lipschitz continuous integrands. An alternate approach to interval integration in the unbounded case via finite interval integrals is presented. The results give theoretical support to interval methods for the solution of integral equations and finding extreme values of functionals defined in terms of integrals. Author
- Numerical Mathematics