Theorems on the Computation of the Transient Response of Nonlinear Networks Containing Transistors and Diodes

The set P0 of all real square matrices each with all principal minors nonnegative plays a key role in the study 1 - 3 of nonlinear equations of the form F(x) + Ax = B, and more generally4 of equations of the form CF(x) + Ax = B, in which F(-) is a "diagonal monotone-nondecreasing mapping" of real Euclidean n-space E" into itself, A and C are real 1730 1742 T H E BELL SYSTEM T E C H N I C A L J O U R N A L , OCTOBER 197(1 n X n matrices and B is an element of En. Such equations arise in the dc analysis of transistor networks, the computation of the transient response of transistor networks, and the numerical solution of certain nonlinear partial-differential equations. In Ref. 3 a nonuniqueness theorem is proved which focuses attention on a simple special property of transistor-type nonlinearities. It shows that for any transistor-type exponential F(-) the equation F(x) + Ax = B has at least two solutions x for some B t En whenever A if P0. The theorem shows that some earlier conditions1,2 for the existence of a unique solution cannot be improved by taking into account more information concerning the nonlinearities, and therefore makes more clear that the set of matrices P0 plays a basic role in the theory of nonlinear transistor networks. Ref. 3 also contains material concerned with the convergence of algorithms for computing the solution of F(x) + Ax = B as well as of more general equations, and some related problems concerning the numerical integration of the ordinary differential equations which govern the transient response of nonlinear transistor networks are considered briefly.