Solving Nonlinear Network Equations Using Optimization Techniques
A. Gersho · Bell System Technical Journal · 1969
A class of nonlinear equations arising in transistor network analysis, as well as in other areas, has the form$f_{i}(x_{i})+ \sim_{i=1}^{n}a_{i j}x_{i}- b_{i}=0 \quad i=1,2,\ldots, n (1)$or in matrix notation$F(x) \ + \ Ax \ -\ b = 0, \quad (2)$where the nonlinearities fi(·) are continuously differentiable, strictly monotone increasing functions. Results by Willson1and Sandberg and Willson2,3on nonlinear networks have included broad conditions for the existence and uniqueness of a solution to equation (2). However, convergent computational algorithms for finding the solution have been given only for restricted subclasses of the class of equations that have unique solutions.1,2,4,5These subclasses are characterized by a variety of restrictions on the matrix A and on the type of nonlinearities. In this brief we show that a single convergent algorithm exists for solving these equations under conditions virtually as broad as the known existence and uniqueness conditions. Peripherally, we obtain under these conditions a conceptually simple proof of the existence of a solution.