A note on the Bernstein algorithm for bounds for interval polynomials

In computing the range of values of a polynomial over an intervala≤x≤b one may use polynomials of the form $$\left( {\begin{array}{*{20}c} k \\ j \\ \end{array} } \right)\left( {x - a} \right)^j \left( {b - x} \right)^{k - j} $$ called Bernstein polynomials of the degreek. An arbitrary polynomial of degreen may be written as a linear combination of Bernstein polynomials of degreek≥n. The coefficients of this linear combination furnish an upper/lower bound for the range of the polynomial. In this paper a finite differencelike scheme is investigated for this computation. The scheme is then generalized to interval polynomials.