The Bernstein Algorithm
Jurgen Garlo · 1994
We solve the problem of finding an enclosure for the range of a multivariate polynomial over a rectangular region by expanding the given polynomial into Bernstein polynomials. Then the coecients of the expansion provide lower and upper bounds for the range and these bounds converge monotonically if the degree of the Bernstein polynomials is elevated. To obtain a faster improvement of the bounds we use subdivision and present an economical procedure for computing the bounds on subboxes. Then we apply the results to a problem of robust control, viz. checking the (Hurwitz) stability of a polynomial with coecients depending polynomially on parameters varying inside given intervals. Numerical examples are also presented.