The basic problem of interval computations is: given a function f(x 1,..., x n) and n intervals [x i, x i], find the (interval) range yof the given function on the given intervals. It is known that even for quadratic polynomials f(x 1,..., x n), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound ε on the width of the input intervals tends to 0. We show that for small c > 0, if we want to compute the range with an accuracy c ⋅ ε2, then the problem is still NP-hard; on the other hand, for every δ > 0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c ⋅ ε2−δ.
Reliable Computing – Springer Journals
Published: Oct 13, 2004
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