Bernstein form of a polynomial

Bounds on a polynomial

“…The problem of computing the range of values of a polynomial over an interval has received a good deal of attention in the past [1][2][3][4][5][6][7][8][9][10]. In this paper a new algorithm is presented for isolating the maxima, minima, and real roots of a polynomial with some advantages over methods previously published.…”

“…In this paper a new algorithm is presented for isolating the maxima, minima, and real roots of a polynomial with some advantages over methods previously published. As in Cargo and Shisha [1] and Rivlin [2], the algorithm makes use of the Bernstein form of the polynomial, and like Collins and Akritas [10], the method utilizes the variation-diminishing properties of polynomials combined with a bisection technique. A bound on the complexity is derived and analysis of the algorithm performed as suggested in [8].…”

Bounds for an interval polynomial

“…Analysis Since the formulas by Rivlin [4] both are improvements over those by Cargo and Shiska [2], we will only discuss those by Rivlin.…”

“…This problem has been dealt with in a paper by Dussel and Schmitt [1] and, disregarding the computational cost of their algorithm, solved in a satisfactory manner. In this paper we will discuss two further approaches to the problem by ] (see also Cargo and Shiska [2]) where the accuracy of the bounds depend on the amount of work one is willing to do.…”

“…Several algorithms are proposed and tested on numerical examples. The algorithms are based on ideas by Cargo and Shiska [2] and Rivlin [4]. The one basic algorithm uses Bernstein polynomials.…”