Isolating all the real roots of a mixed trigonometric-polynomial

Abstract

Mixed trigonometric-polynomials (MTPs) are functions of the form $f(x,\sin x,\cos x)$ where f is a trivariate polynomial with rational coefficients, and the argument x ranges 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 $I_i\subset [-\pi ,\pi ]$, integers $c_i>0$ and multisets of root multiplicity ${\left\{m_{j,i}\right\}}_{j=1}^{c_i}$ such that any root $>{\mu }_{+}$ is in the set $\left({\cup }_i{\cup }_{k\in N}\right(I_i+2k\pi \left)\right)$ and any interval $I_i+2k\pi \subset ({\mu }_{+},\infty )$ contains exactly $c_i$ distinct roots with multiplicities $m_{1,i},...,m_{c_i,i}$, 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 there is any root, or whether there are infinitely many roots in unbounded intervals of the form $(-\infty ,a)$ or $(a,\infty )$ with $a\in Q$.