Interval arithmetic is able to be applied when we include the ranges of various functions. When we include them applying the interval arithmetic, the serious problem that the widths of the range inclusions increase extremely exists. In range inclusion of polynomials particularly, Horner's method and Alefeld's method are well known as the conventional methods which mitigates this problem. The purpose of this paper is to propose the new methods which are able to mitigate this problem more efficiently than the conventional methods. And in this paper, we show and compare the efficiencies of the new methods by some numerical examples.

Affine arithmetic is a kind of interval arithmetic defined by Stolfi et al. In affine arithmetic, it is difficult to realize the efficient nonlinear binomial operations. The purpose of this letter is to propose a new dividing method which is able to supply more suitable evaluation than the old dividing method. And this letter also shows the efficiency of the new dividing method by numerical examples.