The concept of an Ω-matrix was introduced by Nishi in order to estimate the number of solutions of a resistive circuit containing active elements. He gave a finite characterization by means of four conditions which are all satisfied if and only if the matrix under investigation is an Ω-matrix. In this note we show that none of the four conditions can be omitted.

The author once defined the Ω-matrix and showed that it played an important role for estimating the number of solutions of a resistive circuit containing active elements such as CCCS's. The Ω-matlix is a generalization of the wellknown P-matrix. This paper gives the necessary and sufficient conditions for the Ω-matrix.

An efficient algorithm is proposed for finding all solutions of piecewise-linear resistive circuits The algorithm is based on the idea of "contraction" of the solution domain using a sign test. The proposed algorithm is efficient because many large super-regions containing no solution are eliminated in early steps.

In this paper, we discuss computational methods for obtaining the bifurcation points and the branch directions at branching points of solution curves for the nonlinear resistive circuits. There are many kinds of the bifurcation points such as limit point, branch point and isolated point. At these points, the Jacobian matrix of circuit equation becomes singular so that we cannot directly apply the usual numerical techniques such as Newton-Raphson method. Therefore, we propose a simple modification technique such that the Newton-Raphson method can be also applied to the modified equations. On the other hand, a curve tracing algorithm can continuously trace the solution curves having the limit points and/or branching points. In this case, we can see whether the curve has passed through a bifurcation point or not by checking the sign of determinant of the Jacobian matrix. We also propose two different methods for calculating the directions of branches at branching point. Combining these algorithms, complicated solution curves will be easily traced by the curve tracing method. We show the example of a Hopfield network in Sect.5.

An efficient algorithm is given for finding all solutions of piecewise-linear resistive circuits containing nonseparable transistor models such as the Gummel-Poon model or the Shichman-Hodges model. The proposed algorithm is simple and can be easily programmed using recursive functions.

An efficient algorithm is presented for finding all solutions of piecewise-linear resistive circuits containing sophisticated transistor models such as the Gummel-Poon model or the Shichman-Hodges model. When a circuit contains these nonseparable models, the hybrid equation describing the circuit takes a special structure termed pairwise-separability (or tuplewise-separability). This structure is effectively exploited in the new algorithm. A numerical example is given, and it is shown that all solutions are computed very rapidly.

Recently, efficient algorithms that exploit the separability of nonlinear mappings have been proposed for finding all solutions of piecewise-linear resistive circuits. In this letter, it is shown that these algorithms can be extended to circuits containing piecewise-linear resistors that are neither voltage nor current controlled. Using the parametric representation for these resistors, the circuits can be described by systems of nonlinear equations with separable mappings. This separability is effectively exploited in finding all solutions. A numerical example is given, and it is demonstrated that all solutions are computed very rapidly by the new algorithm.

An efficient algorithm is presented for finding all solutions of piecewise-linear resistive circuits. In this algorithm, a simple sign test is performed to eliminate many linear regions that do not contain a solution. This makes the number of simultaneous linear equations to be solved much smaller. This test, in its original form, is applied to each linear region; but this is time-consuming because the number of linear regions is generally very large. In this paper, it is shown that the sign test can be applied to super-regions consisting of adjacent linear regions. Therefore, many linear regions are discarded at the same time, and the computational efficiency of the algorithm is substantially improved. The branch-and-bound method is used in applying the sign test to super-regions. Some numerical examples are given, and it is shown that all solutions are computed very rapidly. The proposed algorithm is simple, efficient, and can be easily programmed.

An efficient algorithm is presented for finding all solutions of piecewise-linear resistive circuits. In this algorithm, a simple sign test is performed to eliminate many linear regions that do not contain a solution. Therefore, the number of simultaneous linear equations to be solved is substantially decreased. This test, in its original form, requires O(Ln2) additions and comparisons in the worst case, where n is the number of variables and L is the number of linear regions. In this paper, an effective technique is proposed that reduces the computational complexity of the sign test to O(Ln). Some numerical examples are given, and it is shown that all solutions can be computed very efficiently. The proposed algorithm is simple and can be easily programmed by using recursive functions.

This paper surveyed the research topics and results on nonlinear circuits and systems which have been achieved in Japan or by Japanese researchers (sometimes as co-authors) during the last 20 years. The particular emphasis is placed on the analysis of nonlinear resistive circuits and periodic dynamic circuits.

This paper deals with the uniqueness of a solution of the basic equation obtained from the analysis of resistive circuits including ideal diodes. The equation in consideration is of the type of (A-)X=b, where A is a constant matrix, b a constant vector, X an unknown vector satisfying X 0, and a diagonal matrix whose diagonal elements take the value 0 or 1 arbitrarily. The necessary and sufficient conditions for the equation to have a unique solution X 0 for an arbitrary vector b are shown. Some numerical examples are given for the illustration of the result.