Isolating all the real roots of a mixed trigonometric-polynomial

Mixed trigonometric-polynomials (MTPs) are functions of the form f(x, sinx, cosx) where fis a trivariate polynomial with rational coefficients, and the argument xranges over the reals. In this paper, an algorithm "isolating" all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [ mu(-), mu(+)] while the other consists of countably many roots in R\[ mu(-), mu(+)]. For the roots in [ mu(-), mu(+)], the algorithm returns isolating intervals and corresponding multiplicities while for those greater than mu+, it returns finitely many mutually disjoint small intervals I-i subset of[- pi, pi], integers c(i)> 0and multisets of root multiplicity {mj,i}c(j- 1)(i)such that any root > mu(+) is in the set (boolean OR i boolean OR(k subset of N)( I-i+ 2k pi)) and any interval I-i+ 2k pi subset of ( mu+, infinity) contains exactly cidistinct roots with multiplicities m(1,i),..., m(ci,i), respectively. The efficiency of the algorithm is shown by experiments. The method used to isolate the roots in [ mu(-), mu(+)] is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether unbounded intervals of the form (-infinity, a) or (a, infinity) with a is an element of Q. (c) 2023 Elsevier Ltd. All rights reserved.