高次实系数代数方程最优解

Optimal solution 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.采用劈因子法和因子优化方法能够解得实系数代数方程的全部根.但是,这些方法在优化1个三项式因子时会有计算残差,必然影响后续三项式因子的计算精度.为尽可能地减小计算残差,提出最优解方法,该方法使系数拟合误差的评价函数值最小.最优解方法分为2个主要步骤:先使用因子优化方法计算出所有三项式因子的系数,再同时优化代数方程所有三项式因子.最优解方法就是使优化后因子乘积多项式系数与原多项式系数之间的差为0.联合使用因子优化方法和最优解方法,能够有效地求解n阶代数方程,且计算精度高.文中给出最优解方法的数学表达式,并推导出相应的计算步骤.文中给出的5个计算例子是从测试最优解方法的有效性、计算精度和收敛性的众多计算例子中选出的典型,它们恰当地展示了最优解方法的特性: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 the Splitting Trinomial Factor method and the Optimal Trinomial Factor method, can be used to solve the algebra equation with complex roots of nth degree. As they have computing residuals optimizing a trinomi factor of the algebra equation, it has immediately influence on accuracies of consequent trinomials. To minimize the computing residuals, an Optimal Solution is proposed, which has a minimum value of criterion function of fitting errors. The Optimal Solution method has two main steps: first,calculating coefficients of trinomials with the Optimal Trinomial Factor method; second,optimizing all trinomials of the algebra equation at the same time. As a result, the differences of coefficients between the product of all optimized trinomials and the original polynomial are optimized to zero. Combining the OTF method and Optimal Solution, one will solve a given nth degree algebra equation efficiently and accurately.After preseting the formulation,and derivation of calculation steps,five examples were selected for the effectiveness, accuracy and convergence of the Optimal solution computing all complex and/or real roots of an equation, satisfying accuracy of obtained roots.