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Optimization of complex functions and the algorithm for exact geometric search for complex roots of a polynomial

机译：复杂函数的优化与多项式复杂根的精确几何搜索算法

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The paper describes an application for visualization of four-dimensional graphs of a complex variable function. This application allowed us to construct an exact geometric algorithm for finding the real and complex roots of a polynomial on the same plane. A graph of ann-th order polynomial on the real plane allows us to define geometrically all the real roots. Number of real roots varies from 0 to n. The rest of the roots are complex and not determined by the graph. In the article, in addition to the graph of the basic polynomial, two auxiliary graphs are constructed, which allow us to represent all complex roots on the same real plane. Realization of this method is considered in detail for the solution of a cubic polynomial. In this case the method has exceptional features in comparison with polynomials of other degrees. We also propose an algorithm for constructing auxiliary functions for the general case of a polynomial of order n which have exact formulas for polynomials with order n ≤ 10. The algorithm for the first time builds the exact hodograph of poles for the control systems with feedback. We generalize the concepts of stationary and extremal points to the case of a complex function. The absence of the possibility of comparing the complex values of the objective function is compensated by an analysis of the behavior of the stationary point under small perturbations of the polynomial by linear functions. Optimality criteria are proposed using complex trajectories of stationary points.
机译：本文介绍了一种用于可视化复杂变量函数的四维图的应用。此应用程序允许我们构建一个精确的几何算法，用于在同一平面上找到多项式的真实和复杂根。真正平面上的Ann-Tround多项式图允许我们在几何上定义所有真实根部。实际根数因0而异。其余的根部是复杂的，而不是由图表确定。在本文中，除了基本多项式的图之外，构造了两个辅助图，这使我们能够在同一真实平面上表示所有复杂根。对立方多项式的溶液详细考虑了该方法的实现。在这种情况下，该方法与其他度的多项式相比具有卓越的特征。我们还提出了一种用于构建辅助功能的算法，用于构造具有顺序N≤10的多项式的多项式的多项式的辅助功能的算法。第一次算法为具有反馈的控制系统构建极点的精确唱片。我们概括了静止和极值点的概念到复杂功能的情况。通过通过线性函数对多项式的小扰动下的静止点的行为进行分析来补偿比较目标函数的复数的可能性。使用静止点的复杂轨迹提出了最佳标准。

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