指数函数多项式的实根分离算法

针对超越函数多项式的实根分离问题，提出了一种指数函数多项式的区间分离算法exRoot，将非多项式型实函数的实根分离问题转化为多项式正负性判定问题进而对其求解。首先，利用泰勒替换法构造目标函数的多项式区间套；然后，将指数函数的求根问题转化为多项式在区间内正负性的判定问题；最后，给出综合算法，并且试探性地应用于实特征值线性系统的可达性判定问题。所提算法在Maple中实现，输出的结果可读，且高效易行。区别于HSOLVER和数值计算方法fsolve，exRoot回避了直接讨论根的存在性问题，理论上具有终止性和完备性，且可达到任意精度，应用于最优化问题时可避免数值解带来的系统误差。

Real root isolation algorithm for exponential function polynomials

For addressing real root isolation problem of transcendental function polynomials， an interval isolation algorithm for exponential function polynomials named exRoot was proposed. In the algorithm， the real root isolation problem of non-polynomial real functions was transformed into sign determination problem of polynomial， then was solved. Firstly， the Taylor substitution method was used to construct the polynomial nested interval of the objective function. Then， the problem of finding the root of the exponential function was transformed into the problem of determining the positivity and negativity of the polynomial in the intervals. Finally， a comprehensive algorithm was given and applied to determine the reachability of rational eigenvalue linear system tentatively. The proposed algorithm was implemented in Maple efficiently and easily with readable output results. Different from HSOLVERand numerical calculation method fsolve， exRoot avoids discussing the existence of roots directly， and theoretically has termination and completeness. It can reach any precision and can avoid the systematic error brought by numerical solution when being applied into the optimization problem.