高次实系数代数方程的因子优化解

Optimal trinomial factor for nth degree algebra equation with real-coefficients

求解高次实系数代数方程的根,对于控制系统的分析和综合设计有着重要意义.计算给定高次代数方程的复根的方法很少.采用劈因子法和首次提出的因子优化方法能够解得实系数代数方程的全部根.这里提出的因子优化方法在收敛性和计算精度等方面优于劈因子法.因子优化方法的立足点是:高次实系数多项式总能够表达为多个三项式(二阶)因子和一个阶次为4阶或3阶的低次多项式的乘积,得到原代数方程的所有三项式因子和低次多项式,就等于得到了方程的根.文中提出的因子优化方法是高效的计算工具,计算精度满足工程实践需要,在迭代次数上优于劈因子法.文中给出的5个计算例子是从测试因子优化方法的有效性、计算精度和收敛性的众多计算例子中选出的典型,恰当地展示了因子优化方法的特性:有效地计算方程的全部复数根和实数根;计算结果有足够的精度.

Solving the algebra equation with real_coefficients of nth degree is of great importance for analysis and synthesis of a control system A few methods, such as Splitting Trinomial Factor method and the Optimal Trinomial Factor method proposed firstly in this paper, can be used to solve the algebra equation with complex roots of nth degree.This paper proposes that the Optimal Trinomial Factor method is superior to Splitting Trinomial Factor method in convergence,accuracy and other variables.The Optimal Trinomial Factor method is based on the factor that a real_coefficient polynomial of high degree can always be the product of several trinomials and a sub_polynomial whose degree is 4 or 3. Getting all trinomials and a sub_polynomial of the original algebra equation, the solution is obtained.OTF has satisfying accuracy for engineering practice and is superior to STF in iteration numbers. Five examples were selected from testing examples for the effectiveness, accuracy and convergence of OTF. They are suitable instances in illustration of properties of OTF: effectiveness to compute all of complex and/or real roots of an equation, satisfying accuracy of obtained roots.