A test, based on conversion to the Bernstein polynomial basis, for an interval to be free of zeros applicable to polynomials in Chebyshev form and to transcendental functions approximated by Chebyshev series

Abstract

Polynomials of high degree are much less vulnerable to roundoff error when expressed as truncated Chebyshev series rather than the usual power series form. Recent articles have developed subdivision methods in which all real roots on the canonical Chebyshev interval, x ∈ [−1, 1], are found by subdividing the interval and finding the roots of separate Chebyshev series of moderate degree on each subdomain. This strategy can be applied either to polynomials or to transcendental functions if the latter are analytic on the search interval and thus have rapidly convergent Chebyshev polynomial approximations. The last step is to compute the eigenvalues of the Chebyshev–Frobenius companion matrix for each local polynomial. Here, we propose a simple strategy for flagging some subdomains as “zero-free” so that eigensolving can be omitted. The test requires conversion of the polynomial from Chebyshev form to a Bernstein polynomial basis. The interval is zero-free if all coefficients in the Bernstein basis are of the same sign. We give the conversion matrices for various small and moderate N and quote Rababah’s formulas for general N. We show how to exploit parity so as to halve the cost of the conversion for large N to about N 2 floating point operations. We show that the conversion matrices have condition numbers that are approximately (5/8)2 N .