Solving nonlinear resistive networks using piecewise-linear analysis and simplicial subdivision

In recent years numerous results of piecewise-linear analysis of nonlinear resistive networks have been derived. The applicability of the method relies on the fact that every nonlinear device is modeled by a piecewise-linear continuous function. In order to extend the applicability of piecewise-linear analysis to treat more general nonlinear networks, three steps need to be carried out: i) the subdivision of the domain of the multi-dimensional nonlinear network function; ii) the interpolation of a piecewise-linear continuous function on the subdivided domain; and iii) the application of piecewise-linear analysis. It turns out that the above three steps can be accomplished effectively by using simplicial subdivision. In addition, the difficulties encountered in the conventional piecewise-linear analysis are simplified. The memory space needed for the analysis is also greatly reduced. The complete analysis has been implemented in a program on CDC 6400.

[1]  Leon O. Chua,et al.  Efficient Computer Algorithms for Piecewise-Linear Analysis of Resistive Nonlinear Networks , 1971 .

[2]  Thomas E. Stern,et al.  Piecewise-linear network theory , 1956 .

[3]  Harold W. Kuhn,et al.  Some Combinatorial Lemmas in Topology , 1960, IBM J. Res. Dev..

[4]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[5]  F. Wu.  Existence of an operating point for a nonlinear circuit using the degree of mapping , 1974 .

[6]  I. Sandberg,et al.  Existence of solutions for the equations of transistor-resistor-voltage source networks , 1971 .

[7]  J. M. Bennett.  Triangular factors of modified matrices , 1965 .

[8]  J. Katzenelson.  An algorithm for solving nonlinear resistor networks , 1965 .

[9]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[10]  E. Kuh,et al.  Solving piecewise-linear equations for resistive networks , 1976 .

[11]  Ibrahim N. Hajj,et al.  Nonlinear circuit theory: Resistive networks , 1971 .

[12]  Ernest S. Kuh,et al.  A sparse matrix method for analysis of piecewise-linear resistive networks , 1972 .

[13]  E. S. Kuh,et al.  Piecewise-Linear Theory of Nonlinear Networks , 1972 .

[14]  Herbert E. Scarf,et al.  The Computation of Economic Equilibria , 1974 .

[15]  Jr. A. Willson.  The no-gain property for networks containing three-terminal elements , 1975 .

[16]  B. Eaves.  Computing Kakutani Fixed Points , 1971 .

[17]  H. Kuhn.  Simplicial approximation of fixed points. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[18]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[19]  T. Ohtsuki,et al.  DC Analysis of Nonlinear Networks Based on Generalized Piecewise-Linear Characterization , 1971 .