Philosophy and practicalities of interval arithmetic
Reliability in computing: the role of interval methods in scientific computing
Precise Numerical Analysis with Disk
Precise Numerical Analysis with Disk
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
Intelligent control in space exploration: interval computations are needed
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For many measuring devices, the only information that we have about them is their biggest possible error ε > 0. In other words, we know that the error Δx = x - x (i.e., the difference between the measured value x and the actual values x) is random, that this error can sometimes become as big as ε or - ε, but we do not have any information about the probabilities of different values of error.Methods of statistics enable us to generate a better estimate for x by making several measurements x1, ..., xn. For example, if the average error is 0 (E(Δx) = 0), then after n measurements, we can take an average x = (x1 + ... + xn)/n, and get an estimate whose standard deviation (and the corresponding confidence intervals) are √n times smaller.Another estimate comes from interval analysis: for every measurement xi, we know that the actual value x belongs to an interval [xi-ε, xi+ε]. So, x belongs to the intersection of all these intervals. In one sense, this estimate is better than the one based on traditional engineering statistics (i.e., averaging): interval estimation is guaranteed. In this paper, we show that for many cases, this intersection is also better in the sense that it gives a more accurate estimate for x than averaging: namely, under certain reasonable conditions, the error of this interval estimate decreases faster (as 1/n) than the error of the average (that only decreases as 1/ √n).A similar result is proved for a multi-dimensional case, when we measure several auxiliary quantities, and use the measurement results to estimate the value of the desired quantity y.