On the Solutions of Equations for Nonlinear Resistive Networks
01 October 1968
In this paper we consider the solution of the equation F(x) + Ax = B .r, where x = is a point in the w-dimensional Euclidean space En, (1) F(x) = /.(*.) U(Xn) is a nonlinear function mapping En into En, A is an 1755 1756 T H E BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1968 6. n X n matrix of real numbers, and B = is an arbitrary point [b,, in E n . We prove (Theorem 1) that there is a unique solution of (1) if: (i) Each /,· is a strictly monotone increasing function mapping E1 onto E1 , and (ii) The elements a,, of the matrix A satisfy the inequality fl« > 2 I o,-f I. for i = 1, ··· , n. i-i jVt We then demonstrate a straightforward method of computing bounds on the location of this solution. Finally, we present two iterative techniques for computing the solution; and prove (Theorem 3) that the two additional assumptions: (iii) Either all of the functions /,· are convex, or else all /, are concave, and (iv) aif ^ 0 if i j, are sufficient to guarantee that the iterations converge to the solution. Equations of type (1) occur often in the study of nonlinear electrical networks. For example, if a linear n-port containing resistors, independent sources, and dependent sources has a two-terminal device whose V vs I curve is specified by 7, = /, (F t ), for i = 1, · · · , n, connected across each port, then the port voltages may often be expressed as the solution of an equation of type (1). In this case the matrix A will be the ^/-parameter matrix of the w-port, the constant vector B will account for the presence of the independent sources, and the components of the vector x will be the desired port voltages.