For univariate polynomials f ( x 1 ), Horner's scheme provides the fastest way to compute a value. For multivariate polynomials, several different version of Horner's scheme are possible; it is not clear which of them is optimal. In this paper, we propose a greedy algorithm, which it is hoped will lead to good computation times.The univariate Horner scheme has another advantage: if the value x 1 is known with uncertainty, and we are interested in the resulting uncertainty in f ( x 1 ), then Horner scheme leads to a better estimate for this uncertainty that many other ways of computing f ( x 1 ). The second greedy algorithm that we propose tries to find the multivariate Horner scheme that leads to the best estimate for the uncertainty in f ( x 1 ,..., x n ).

Abstract. In many engineering applications, we have to combine probabilistic and interval uncertainty. For example, in environmental analysis, we observe a pollution level x(t) in a lake at different moments of time t, and we would like to estimate standard statistical characteristics such as mean, variance, autocorrelation, correlation with other measurements. In environmental measurements, we often only measure the values with interval uncertainty. We must therefore modify the existing statistical algorithms to process such interval data.In this paper, we provide a survey of algorithms for computing various statistics under interval uncertainty and their computational complexity. The survey includes both known and new algorithms.

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The paper is a continuation of our previous work towards the use of probability information in interval computations. While in the previous work, bounds on the first order moments are taken into account, the contribution of this article is to deal with correlations. Specifically, in this paper, we develop a new method that takes into account both correlation among measured parameters and bounds on their expected values when doing interval computation.

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A reliable symbolic-numeric algorithm for solving nonlinear systems over the reals is designed. The symbolic step generates a new system, where the formulas are different but the solutions are preserved, through partial factorizations of polynomial expressions and constraint inversion. The numeric step is a branch-and-prune algorithm based on interval constraint propagation to compute a set of outer approximations of the solutions. The processing of the inverted constraints by interval arithmetic provides a fast and efficient method to contract the variables' domains. A set of experiments for comparing several constraint solvers is reported.