Finding all solutions of piecewise‐linear circuits by using linear programming

Abstract

SUMMARYAn efficient algorithm is proposed for finding all solutions of piecewise-linear (PWL) resistive circuits using linear programming (LP). This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of PWL equations of the form Pg(x) + Qx s = 0 in a given region. In the LP test, the system of PWL equations is transformed into an LP problem, to which the simplex method is applied. Such an LP problem is obtained by surrounding the PWL functions by rectangles. In this paper, it is shown that the LP test can be performed very efficiently mostly by Phase II of the simplex method. Using the proposed techniques, the number of pivotings in the LP test becomes much smaller, which makes the algorithm very efficient. By numerical examples, it is shown that the proposed algorithm could find all solutions of relatively large scale circuits (

Finding all characteristic curves of nonlinear resistive circuits using linear programming

“…In that case, it is more efficient to surround it by a right-angled triangle as shown in Fig. 3(b) that is smaller than a rectangle [29,30]. Since a right-angled triangle can be represented by one inequality constraint and the two nonnegativity constraints, the LP test becomes more powerful and more efficient than using a rectangle (represented by two inequality constraints and the two nonnegativity constraints).…”

“…This algorithm is much more efficient than the conventional algorithms if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. It was also applied to transistor circuits and succeeded in finding all solutions of systems of 100 nonlinear equations in practical computation time [29,30].…”

“…However, such a triangle requires three inequality constraints (other than the nonnegativity constraints). Hence, the number of constraints increases, which often makes the LP test less efficient [29,30]. * As is well known, the simplex method consists of Phase I and Phase II.…”