Piecewise-linear theory and computation of solutions of homeomorphic resistive networks

Piecewise-linear resistive networks can be characterized by the equation
f(x)=J^{(m)}x + w^{(m)} = y, m = 0, l, \cdots,l,
where
l
is a finite positive number. The domain (
n
-dimensional Euclidean space) is divided into
l+1
regions (closed convex polyhedrons). In each region
j(m)
is a constant
n \times n
matrix and
w^{(m)}
is a constant
n
-vector. In this paper, we derive necessary and sufficient conditions for the function
f(x)
to be a homeomorphism. Different formulations of network equations are investigated, and results in terms of the matrices
J^{(m)}
's are obtained. An algorithm with a new perturbation method is also developed which is capable of locating the unique solution in a finite number of steps. The work is different from the early work by Kuh and Fujisawa in many ways; comparisons are presented.