Kiyotaka Yamamura

A fixed-point homotopy method for solving modified nodal equations

Recently, the application of homotopy methods to practical circuit simulation has been remarkably developed, and bipolar analog integrated circuits with more than 10 000 elements are now solved efficiently by the homotopy methods. There are several approaches to applying the homotopy methods to large-scale circuit simulation. One of them is combining the publicly available software package of the homotopy methods (such as HOMPACK) with the general-purpose circuit simulators such as SPICE. However, the homotopy method using the fixed-point (FP) homotopy (that is provided as a default in HOMPACK) is not guaranteed to converge for the modified nodal (MN) equations that are used in SPICE. In this paper, we propose a modified algorithm of the homotopy method using the FP homotopy and prove that this algorithm is globally convergent for the MN equations. We also show that the proposed algorithm converges to a stable operating point with high possibility from any initial point.

An efficient algorithm for finding all solutions of piecewise-linear resistive circuits

An efficient algorithm for finding all solutions of piecewise-linear resistive circuits is presented. First, a technique that substantially reduces the number of function evaluations needed in the piecewise-linear modeling process is proposed. Then a simple and efficient sign test is proposed that remarkably reduces the number of linear simultaneous equations to be solved for finding all solutions. An effective technique that makes the sign test even more efficient is introduced. All of the techniques exploit the separability of nonlinear mappings. Some numerical examples are given, and it is shown that all solutions are computed rapidly. The algorithm is simple and efficient, and can be easily programmed. >

Finding all solutions of piecewise-linear resistive circuits using linear programming

An efficient algorithm is proposed for finding all solutions of piecewise-linear resistive circuits. This algorithm is based on a new test for nonexistence of a solution to a system of piecewise-linear equations f/sub i/(x)=0(i=1.2,/spl middot//spl middot//spl middot/,n) in a super-region. Unlike the conventional sign test, which checks whether the solution surfaces of the single piecewise-linear equations exist or not in a super-region, the new test checks whether they intersect or not in the super-region. Such a test can be performed by using linear programming. It is shown that the simplex method can be performed very efficiently by exploiting the adjacency of super-regions in each step. The proposed algorithm is much more efficient than the conventional sign test algorithms and can find all solutions of large scale circuits very efficiently. Moreover, it can find all characteristic curves of piecewise-linear resistive circuits.

INTERVAL SOLUTION OF NONLINEAR EQUATIONS USING LINEAR PROGRAMMING

A new computational test is proposed for nonexistence of a solution to a system of nonlinear equations in a convex polyhedral regionX. The basic idea proposed here is to formulate a linear programming problem whose feasible region contains all solutions inX. Therefore, if the feasible region is empty (which can be easily checked by Phase I of the simplex method), then the system of nonlinear equations has no solution inX. The linear programming problem is formulated by surrounding the component nonlinear functions by rectangles using interval extensions. This test is much more powerful than the conventional test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. By introducing the proposed test to interval analysis, all solutions of nonlinear equations can be found very efficently.

A globally and quadratically convergent algorithm for solving nonlinear resistive networks

A globally convergent algorithm that is also quadratically convergent for solving bipolar transistor networks is proposed. The algorithm is based on the homotopy method using a rectangular subdivision. Since the algorithm uses rectangles, it is much more efficient than the conventional simplicial-type algorithms. It is shown that the algorithm is globally convergent for a general class of nonlinear resistive networks. Here, the term globally convergent means that a starting point which leads to the solution can be obtained easily. An efficient acceleration technique which improves the local convergence speed of the rectangular algorithm is proposed. By this technique, the sequence of the approximate solutions generated by the algorithm converges to the exact solution quadratically. Also, in this case the computational work involved in each iteration is almost identical to that of Newtons method. Therefore, the algorithm becomes as efficient as Newtons method when it arrives sufficiently close to the solution. It is also shown that sparse-matrix techniques can be introduced to the rectangular algorithm, and the partial linearity of the system of equations can be exploited to improve the computational efficiency. Some numerical examples are also given in order to demonstrate the effectiveness of the proposed algorithm. >

Finding all solutions of piecewise-linear resistive circuits using simple sign tests

This paper presents an efficient algorithm for finding all solutions of piecewise-linear resistive circuits. The algorithm uses two types of sign tests; one is a new test that is proposed in this paper, and the other is the test proposed by Yamamura and Ochiai (1992). The computational complexity of the new test is much smaller than that of Yamamura and Ochiais test. These tests eliminate many linear regions that do not contain a solution. Therefore, the number of simultaneous linear equations to be solved is substantially reduced. The proposed algorithm is very simple and efficient. >

Finding all solutions of piecewise-linear resistive circuits using the simplex method

Recently, an efficient algorithm was proposed for finding all solutions of piecewise-linear (PWL) resistive circuits using the simplex method, which could solve a problem where the number of PWL resistors is 200 and the number of linear regions is 10/sup 200/. In this work, an improved version of this algorithm is proposed, which can be applied to a broader class of PWL resistive circuits and could solve problems where the number of PWL resistors is 500 and the number of linear regions is 10/sup 500/ in practical computation time.

Find all solutions of piecewise-linear resistive circuits using an LP test

An 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 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. It is shown that the LP test can deal with nonseparable functions of more than one variable by using more than two-dimensional rectangles. It is also shown that, for bipolar transistor circuits, the LP test becomes more efficient and more powerful by surrounding the exponential functions by right-angled triangles. The proposed algorithm is simple, efficient, and can be easily implemented.

Finding all solutions of nonlinear equations using the dual simplex method

Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.

An algorithm for representing functions of many variables by superpositions of functions of one variable and addition

A computer algorithm is given for representing functions of many variables by superpositions of functions of one variable and addition. By this algorithm, nonseparable functions are represented in separable forms automatically by computer.