Mean Value and Taylor Forms in Interval Analysis
Louis B. Rall · SIAM Journal on Mathematical Analysis · 1983
Basic spaces for interval analysis are constructed as Cartesian products of the real line. The spaces obtained in this way include real finite- and infinite-dimensional real vector spaces, and have a number of important Hilbert and Banach spaces as subspaces in the sense of set inclusion. A Gâteaux-type derivative is defined in these spaces, and is used in the corresponding interval spaces, together with interval arithmetic, to obtain interval versions of the mean value theorem and Taylor’s theorem. These theorems provide ways to construct accurate interval inclusions of operators, called mean value and Taylor forms. The forms resulting from expansion about midpoints of intervals are shown to be inclusion monotone, and the effect of outward rounding on this class of forms is also considered. An application is made to show that interval iteration operators for the solution of operator equations can be constructed which have arbitrarily high order of convergence in width. Derivations of the fundamental theorems of less generality from results in real and functional analysis are also presented. As in the case of real and functional analysis, the interval Taylor’s theorem given here provides a powerful tool for applications of interval analysis to problems in applied mathematics.