Computation of all solutions to a system of polynomial equations

This paper proposes a homotopy continuation method for approximating all solutions to a system of polynomial equations in several complex variables. The method is based on piecewise linear approximation and complementarity theory. It utilizes a skilful artificial map and two copies of the triangulationJ ₃ with continuous refinement of grid size to increase the computational efficiency and to avoid the necessity of determining the grid size a priori. Some computational results are also reported.     

A PL homotopy for finding all the roots of a polynomial

This paper presents a constructive method which gives, for any polynomialF(Z) of the degreen, approximate values of all the roots ofF(Z).. The point of the method is on the use of a piecewise linear function (Z, t) which approximates a homotopyH(Z, t) betweenF(Z) and a polynomialG(Z) of the degreen withn known simple roots. It is shown that the set of solutions to (Z, t) = 0 includesn distinct paths,m of which converges to a root ofF(Z) if and only if the root has the multiplicitym. Starting from givenn roots ofG(Z), a complementary pivot algorithm generates thosen paths.This work was supported by grants from Management Science Development Foundation and Takeda Science Foundation.     

The octahedral algorithm,a new simplicial fixed point algorithm

A new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented. It uses the restrart technique of Merrill to improve the accuracy of the solution. The algorithm is shown to converge quadratically under certain conditions. The algorithm should be efficient and relatively easy to implement.Partially supported by the Western Michigan University Sabbatical and Faculty Research Funds.     

Finding all solutions of nonlinearly constrained systems of equations

A new approach is proposed for finding all-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. All-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in BB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.     

On the computation of solutions of systems of interval polynomial equations

Systems of algebraic equations with interval coefficients are very common in several areas of engineering sciences. The computation of the solution of such systems is a central problem when the characterization of the variables related by such systems is desired.In this paper we characterize the solution of systems of algebraic equations with real interval coefficients. The characterization is obtained considering the approach introduced in J. Comput. Appl. Math. 136 (2001) 271.     

Existence and structure results on almost periodic solutions of difference equations

We study the almost periodic solutions of Euler equations and of some more general Difference Equations. We consider two different notions of almost periodic sequences, and we establish some relations between them. We build suitable sequences spaces and we prove some properties of these spaces. We also prove properties of Nemytskii operators on these spaces. We build a variational approach to establish existence of almost periodic solutions as critical points, We obtain existence theorems fornonautonomous linear equations and for an Euler equation with a concave and coercive Lagrangian. We also use a Fixed Point approach to obtain existence results for quasi-linear Difference Equations.     

ON THE EXISTENCE OF GLOBAL GENERAL SOLUTIONS OF POLYNOMIAL SYSTEMS

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Comparative computer results of a new complementary pivot algorithm for solving equality and inequality constrained optimization problems

This paper reports the development of a new algorithm for solving the general constrained optimization problem (that of optimizing an objective function subject to both equality and inequality constraints). The approach is based on the complementary pivoting algorithms which have been developed to solve certain classes of fixed point problems. The specific approach is to use the equality constraints to solve for some variables in terms of the remaining ones thus enabling one to eliminate the equality constraints altogether. The result, under certain circumstances, is an optimization problem which may be transformed into a fixed point problem in such a way that a complementary pivoting code may be used to search for a solution.Seventeen test problems have been solved by this method and the results are compared against those obtained from GRG (Generalized Reduced Gradient method). The results of the tests indicate that the fixed point approach is robust (all 17 problems were solved by this method where as GRG solved 16). As to the computer times, the fixed point code proved to be as fast or faster than GRG on the lower dimensional problems; however, as the dimension increased, the trend reversed and on a 40 dimensional problem GRG was approximately 11 times faster. The conclusion from these tests is that when the dimension of the original problem can be reduced sufficiently by the equality constraints, the fixed point approach appears to be more effective than GRG.     

Mean square polynomial stability of numerical solutions to a class of stochastic differential equations

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler–Maruyama method and the backward Euler–Maruyama method. The key technical contribution is based on various estimates involving the gamma function.