A Comparison of Interval and Ellipsoidal Error Bounds for Vector Operations

摘要

At present, interval analysis is the most widespread approach to guaranteed error estimation in computations [1]. This approach deals with intervals that enclose quantities with their errors. For example, instead of a scalar a, an interval [a~, a+] that is guaranteed to contain a is considered. Similarly, instead of an n-dimensional vector x with components xi, i = 1,2,...,n, we consider the rectangular parallelepiped given by the inequalities x_i~- ≤ x_i ≤ x_i~=, i = 1,2,...,n, (1) with the parallelepiped's sides being parallel to the coordinate axes used. In what follows, for brevity, such parallelepipeds are called rectangular. Being sets that enclose unknown vectors, parallelepipeds (1) have a number of properties useful for computations. Specifically, these parallelepipeds are characterized by the relatively small number of parameters required for specifying a set (namely, 2n for x ∈ Rn), have boundaries of simple geometry, and provide clear results. However, this class of sets is not invariant under affine transformations; therefore, after every such transformation, the bound obtained has to be degraded by approximating the resulting parallelepiped with a minimal rectangular one. This leads to an undesirable accumulation of errors, which may ultimately yield a result of no practical importance.