A combined method for enclosing all solutions of nonlinear systems of polynomial equations

Abstract

We consider the problem of finding interval enclosures ot all zeros of a nonlinear system of polynomial equations. We presen~ a method which combines th e meth(g! of Grfibner: bases (used as a preprocessing step), some techniques frmn interval analysis, and a special version of the algorithm of E. Hansen for s~lving nonlinear equations in i~ne variable. The latter is applied to a tri.qngular fi)rm of the system ~d" eqnations, which is generated by. the preprocessing step: ~ Our" method is able to check if the given system has a finite mnnher of zeros and to r verifg'd enclosures for all these zeros. Several test resnhs detn~mstrate that ,our tneth~ is mpch faster than the application of Hansen's multidimensional algorithm (or similar methods) to the original nonlinear systems of polyt~nmial eqnations.

A Weighted Biobjective Transformation Technique for Locating Multiple Optimal Solutions of Nonlinear Equation Systems

“…As suggested in [16], ρ = 0.01 and β = 1000. From Equation (18), we can see that the repulsion strategy creates the repulsion areas around the found optimal solutions and that an individual lies within one of the repulsion areas will be penalized.…”

“…• Rep-SHADE and Rep-CLPSO: The repulsion strategy presented in [16] is combined with two powerful EAs, i.e., SHADE [31] and CLPSO [43], to solve NESs. …”

“…Therefore, AWeB gets great benefit from "DE/current/1" to find multiple optimal solutions of a NES simultaneously in a single run. 16 …”

“…Due to this feature, EAs may merely focus on one of the optimal solutions in one run when solving Equations (2)- (4). In order to locate multiple optimal solutions simultaneously, some extra diversity preservation strategies should be incorporated into EAs, such as the repulsion strategy [16], [17], [18]. However, such extra diversity preservation strategies would inevitably introduce some complicated operators and/or problem-dependent parameters, which have a side effect on the generalization.…”

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

“…Since this is a system of polynomial type, all real solutions of (2.26) could be found by a continuation method [15] or using a special interval method [6]. An alternative method for solving (2.26) will be suggested here that is much simpler for implementation and seems to be more effective than either method mentioned above.…”