Interval Arithmetic and Standardization

Interval arithmetic is arithmetic for continuous sets. Floating-point intervals are intervals of real numbers with floating-point bounds. Operations for intervals can be efficiently implemented. Hence, the time is ripe for standardization. In this paper we present an interval model that is mathematically sound and closed for the 4 basic operations. The model allows for exception free interval arithmetic, if we carefully distinguish between clean and reliable interval arithmetic on one side and rounded floating-point arithmetic on the other side. Elementary functions for intervals can be defined. In some application areas loose evaluation of functions, i.e. evaluation over an interval which is not completely contained in the function domain, is recommended, In this case, however, a discontinuity flag has to be set to inform that Brouwer’s fixed point theorem is no longer applicable. 1 Real Interval Arithmetic 1.1 Real Interval Arithmetic Real interval arithmetic is defined as arithmetic on continuous (in the sense of complete, not discrete) sets. Definition 1 For intervals A = [a1, a2] and B = [b1, b2] ∈ IR, arithmetic operations are defined as set operations A ◦B := {a ◦ b | a ∈ A, b ∈ B} for operations ◦ ∈ {+,−, ·, /}, 0 / ∈ B in case of division. Remark 1 Since the operations are continuous, the result is an interval, A ◦B ∈ IR This definition can be extended to elementary functions. Definition 2 f(X) = {f(x) | x ∈ X} denotes the range of values of the function f : Df ⊆ R→ R over the interval X ⊆ Df . Remark 2 If f is continuous, the range is an interval.