Finding all solutions of a class of nonlinear equations using an improved LP test

Abstract

Recently, a new computational test has been proposed for nonexistence of a solution to a system of nonlinear equations using linear programming. This test is termed the LP test. It has been shown that the LP test is much more powerful than the conventional nonexistence test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. By introducing the LP test to interval analysis, all solutions of nonlinear equations can be found very efficiently. In this paper, we propose some techniques for improving the computational efficiency of the LP test in some special cases. Using the proposed techniques, all solutions of a special class of nonlinear equations (including circuit equations) can be found very efficiently.

New interval methodologies for reliable chemical process modeling

“…Ratscheck and Rokne, 1984;Makino and Berz, 1999;Jansson, 2000); 2. Techniques that involve changes to the methodology at the level of the nonlinear equation system (e.g., Alefeld et al, 1998;Granvilliers and Hains, 2000;van Hentenryck et al, 1997;Ratz, 1994;Herbort and Ratz, 1997;Yamamura et al, 1998;Yamamura and Nishizawa, 1999); 3. Methods that seek to make improvements in solving the linear interval system defined by Eq.…”

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].…”

Finding all solutions of weakly nonlinear equations using the dual simplex method

“…Example 1: We first consider a system of n nonlinear equations: which is often used as a test problem [3][4][5]. The initial box is D = ([-10, 10], .…”

“…4, an efficient variation of the LP test algorithm in Ref. 3 is proposed which can be used when the component nonlinear functions are monotone and convex (as seen in the equations describing transistor circuits).…”