Representations of intervals and optimal error bounds
Louis B. Rall · Mathematics of Computation · 1983
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 y ∗ {y^\ast } in an interval I and the corresponding minimum error bound ε ∗ {\varepsilon ^\ast } . Conversely, it is shown that y ∗ {y^\ast } , ε ∗ {\varepsilon ^\ast } 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.