Research on Efficient Numerical Analysis Method for Nonlinear Resistive Circuits and on the Properties of the Solutions of the above Circuits

我们研究了由非线性抵抗性单端口或非线性放大器组成的非线性抵抗性电路，以及线性抵抗性元件，包括线性活性元件。考虑到这些电路的基本问题，例如，有效的数值分析方法、存在和唯一性、解决方案的数目和解决方案的稳定性分析。许多研究人员都在研究这些问题，但大多数问题都是如此的遥远，无法完全解决。在这项研究中，我们研究了一些线性方程的有效数值方法和非线性方程的重要类的解决方案的数量。我们提出了一种有效的方法，以找到碎片线性方程的所有解决方案。此方法需要O （Mn）的计算量（multiplications），资本字母O代表计算复杂性的顺序，而在最坏情况下，先前最好的结果是如此已知的O （Mn^2）计算量（Mn^2）。我们提供了一个摊位 ... More 1989年由IEEE CAS学会赞助。我们认为，我们的结果在计算复杂性的意义上是最好的，但还没有证明。在先前关于非线性抵抗性电路的理论研究中，非线性抵抗性单一端口的v-i特征i=f （v）和放大函数v_2=f （v_1）的非线性放大器是单方面增加的。在假设1下，如何，我们只能获得唯一解决方案的条件。为了在本研究中得出更多的实际结果，我们认为：假设2 ：与非线性函数f （.）的第一个衍生物一样好，假设3 ：放大器具有理想饱和度特征。在这些假设下，我们调查了解决方案的数量，并给出了一些必要和适当的条件来衡量它们。它们中的一些是等式的条件，以便有有限的解决方案数目，以及解决方案数目的最高限额。Less（低）

We studied on the nonlinear resistive circuits composed of nonlinear resistive one-ports or nonlinear amplifiers and linear resistive elements including linear active elements. Fundamental problems concerning these circuits are, for example, efficient numerical analysis methods, existence and uniqueness of a solution, the number of solutions, and stability analysis of solutions. Many researchers have studied on these problems, but most problems have so far been unsolved completely. In this research we studied on efficient numerical method of piecewise-linear equations and the number of solutions of an important class of nonlinear equations.1. We proposed an efficient method to find all solutions of piecewise-linear equations. This method requires the amount of computation (multiplications) of O (Mn), where the capital letter O means the order of computational complexity, while previous best results so far known required the computatio of O (Mn^2) in the worst case. We presented the res … More ult at the 1989 ISCAS sponsored by IEEE CAS Society. We conjecture that our result is the best in the sense of computational complexity, but have not proven it yet.2. In the previous theoretical research on the nonlinear resistive circuits it has exclusively been assumed thatAssumption 1 : The v-i characteristics i=f (v) of nonlinear resistive one-ports and the amplification functions v_2=f (v_1) of nonlinear amplifiers are monotonically increasing.Under Assumption 1, however, we can obtain only the condition for the unique solution. To obtain more practical results we assume in this research that :Assumption 2 : The second derivative as well as the first derivative of the nonlinear function f (・) are always positive.Assumption 3 : Amplifiers have ideal saturation characteristics.Under these assumptions we investigated the number of solutions and gave some necessary and sufficient conditions concerning them. Some of them are the conditions for equations to have a finite number of solutions and the upper limit of the number of solutions. Less