Solving piecewise‐linear equations for resistive networks
Ming‐Jeh Chien, E.S. Kuh · International Journal of Circuit Theory and Applications · 1976
Abstract Nonlinear resistive networks can be characterized by the equation f(x)= y where f(x) is a continuous piecewise‐linear mapping of Rn into itself. The n‐dimensional Euclidean space is divided into a finite number of regions, and, in each region say region Rm, we can express f by J(m)x + w(m) where J(m) is a constant n × n Jacobian matrix and w(m) is a constant n‐vector. In this paper we obtain the following results: If all the Jacobian determinants in the unbounded regions have the same sign, the equation f(x)= y has at least one solution and an algorithm is developed, which obtains one or more solutions in a finite number of steps. The work represents a generalization of early work by Fujisawa, Kuh and Ohtsuki and relaxes the condition imposed on the function. For example, in the bounded regions, the Jacobian matrices can be singular and the sign of Jacobian determinants can be arbitrary.