Existence Theorems and a Solution Algorithm for Piecewise-Linear Resistor Networks
Existence Theorems and a Solution Algorithm for Piecewise-Linear Resistor Networks
复制标题
分段线性电阻网络的存在定理和求解算法
DOI：
10.1137/0508005
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发表时间：
1977
影响因子：
2
通讯作者：
S. Kumagai
中科院分区：
文献类型：
--
作者：
T. Ohtsuki;T. Fujisawa;S. Kumagai
本文研究了可由方程${\bf f}（{\bf x}）= {\bf y}$表征的非线性网络，其中${\bf f}$是从$R^n $到自身的连续分段线性映射。主要定理断言，对于任意给定的${\bf y} \in R^n $，${\bf f}（{\bf x}）= {\bf y}$的解${\bf x} \in R ^n $的存在性由基于映射度理论的相当一般的条件保证。然后，它表明，迭代算法（广义Katzenelson算法）导致在有限数量的迭代步骤的解决方案。最后，对物理非线性元件的综合研究表明，该理论可以应用于目前使用的大多数非线性网络。
This paper deals with nonlinear networks which can be characterized by the equation ${\bf f}({\bf x}) = {\bf y}$, where ${\bf f}$ is a continuous piecewise-linear mapping from $R^n $ into itself. The main theorem asserts that the existence of solutions ${\bf x} \in R^n $ of ${\bf f}({\bf x}) = {\bf y}$ for an arbitrary given ${\bf y} \in R^n $ is guaranteed by fairly general conditions based on the theory of the degree of mapping. Then it is shown that an iterative algorithm (generalized Katzenelson algorithm) leads to a solution in a finite number of iteration steps. Finally, a comprehensive study of physical nonlinear elements demonstrates that the theory can be applied to most of the currently used nonlinear networks.