Subdivision methods for solving polynomial equations

This paper presents a new algorithm for solving a system of polynomials, in a domain of Rⁿ${\mathbb{R}}^n$. It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis [Sherbrooke, E.C., Patrikalakis, N.M., 1993. Computation of the solutions of nonlinear polynomial systems. Comput. Aided Geom. Design 10 (5), 379–405]. It uses a powerful reduction strategy based on univariate root finder using Bernstein basis representation and Descarte’s rule  . We analyse the behavior of the method, from a theoretical point of view, shows that for simple roots, it has a local quadratic convergence speed and gives new bounds for the complexity of approximating real roots in a box of Rⁿ${\mathbb{R}}^n$. The improvement of our approach, compared with classical subdivision methods, is illustrated on geometric modeling applications such as computing intersection points of implicit curves, self-intersection points of rational curves, and on the classical parallel robot benchmark problem.     

Topologically guaranteed bivariate solutions of under-constrained multivariate piecewise polynomial systems

We present a subdivision based algorithm to compute the solution of an under-constrained piecewise polynomial system of $n-2$ equations with $n$ unknowns, exploiting properties of B-spline basis functions. The solution of such systems is, typically, a two-manifold in ${\mathbb{R}}^n$. To guarantee the topology of the approximated solution in each sub-domain, we provide subdivision termination criteria, based on the (known) topology of the univariate solution on the domain’s boundary, and the existence of a one-to-one projection of the unknown solution on a two dimensional plane, in ${\mathbb{R}}^n$. We assume the equation solving problem is regular, while sub-domains containing points that violate the regularity assumption are detected, bounded, and returned as singular locations of small (subdivision tolerance) size. This work extends (and makes extensive use of) topological guarantee results for systems with zero and one dimensional solution sets. Test results in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ are also demonstrated, using error-bounded piecewise linear approximations of the two-manifolds.     

Bessel polynomial solutions of high-order linear Volterra integro-differential equations

In this study, a practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on the computer using a program written in MATLAB v7.6.0 (R2008a).     

广义预测控制中Diophantine矩阵多项式方程的显式解

直接利用被控对象的离散差分方程推导出多变量广义预测控制中Diophantine矩阵多项式方程的显式解,从而避免了其递推求解或迭代求解,使广义预测控制的应用更加方便．     

Numerical solution of a fuzzy system of linear equations with polynomial parametric form

A systems of linear equations are used in many fields of science and industry, such as control theory and image processing, and solving a fuzzy linear system of equations is now a necessity. In this work we try to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers.     

Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.     

Global strong solutions to the 3D incompressible MHD equations with density-dependent viscosity

This paper is concerned with the 3D incompressible MHD equations with density-dependent viscosity in smooth bounded domains. Through time-weighted a priori estimates, the global existence of strong solutions is established under the assumption that the initial energy is suitably small. This generalizes previous results for the 3D Navier–Stokes equations in Huang and Wang (2015) and Zhang (2015), which need $6?u_0{6}_{L^2}$ to be small. The initial vacuum is allowed.     

Asymptotic properties of solutions to discrete Coulomb equations

We construct two finite-difference models for the Coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles. These difference equations correspond to the standard and Mickens-Ramadhani schemes for the Coulomb equation. Our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the Coulomb differential equation. In particular, the form of the anomalous phase term is examined.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Particular solutions of products of Helmholtz-type equations using the Matern function

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation (LOOCV) algorithm is employed to select an appropriate shape parameter for the Matern basis functions. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations

In this article an iterative method to compute the maximal solution and the stabilising solution, respectively, of a wide class of discrete-time nonlinear equations on the linear space of symmetric matrices is proposed. The class of discrete-time nonlinear equations under consideration contains, as special cases, different types of discrete-time Riccati equations involved in various control problems for discrete-time stochastic systems. This article may be viewed as an addendum of the work of Dragan and Morozan (Dragan, V. and Morozan, T. (2009), ‘A Class of Discrete Time Generalized Riccati Equations’, Journal of Difference Equations and Applications, first published on 11 December 2009 (iFirst), doi: 10.1080/10236190802389381) where necessary and sufficient conditions for the existence of the maximal solution and stabilising solution of this kind of discrete-time nonlinear equations are given. The aim of this article is to provide a procedure for numerical computation of the maximal solution and the stabilising solution, respectively, simpler than the method based on the Newton–Kantorovich algorithm.     

Algorithms for quadratic forms

We present algorithms for square classes, quadratic forms and Witt classes of quadratic forms over the field of rational functions of one variable over the reals. The algorithms are capable of: finding the unique representative of a square class, deciding if a given function is a square or a sum of squares and deciding if a quadratic form is isotropic or hyperbolic. Moreover we propose a representation for Witt classes of quadratic forms. With this representation one can manipulate Witt classes without operating directly on their coefficients. We present algorithms both for computing this representation and manipulating Witt classes.     

Relationship between polynomial and state-space solutions of the optimal regulator problem

A relationship is established between the state-space and polynomial matrix solutions of the LQG state-feedback optimal regulator problem. The solution of polynomial matrix Diophantine equations is related to the solution of the steady-state Riccati equation.     

Numerical solution of system of first-order delay differential equations using polynomial spline functions

The use of two constructed polynomial spline functions to approximate the solution of a system of first-order delay differential equations is described. The first spline function is a polynomial with an undetermined constant coefficient in the last term. The other has a polynomial spline form. The error analysis and stability of the second function are theoretically investigated and a test example is given. A comparison of the two forms is carried out to illustrate the pertinent features of the proposed techniques.     

Non-trivial solutions for nonlinear fourth-order elastic beam equations

The existence of at least one non-trivial solution to a boundary value problem for fourth-order elastic beam equations, under a non-standard growth condition of the nonlinear term, is established. Our approach is based on a local minimum theorem for differentiable functionals.     

Algorithms for brush movement

Animators frequently usepaint systems to create and refine their images. In such a program, an artist creates a brush that is moved across a frame buffer, providing a simplified simulation of a physical brush moving across an actual canvas. Movement of the brush often requires modification of a large number of pixels in a small amount of time. Existing algorithms for brush movement are discussed, and two new algorithms are presented that reduce the amount of i/o needed to move a brush, but at the expense of increased computational complexity.This paper is an extension and revision of a paper which appeared in Proceedings of Graphics Interface '84 (Fishkin and Barsky 1984). Temporary address (until summer 1986): Laboratoire Image, Ecole Nationale Supérieure des Telecommunications, 46, Rue Barrault, F-75634 Paris Cedex 13, FranceThis work was supported in part by the Semiconductor Research Corporation under grant number 82-11-008 and the National Science Foundation under grant number ECS-8204381     

A note on Gao’s algorithm for polynomial factorization

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f.     

Gradient based iterative solutions for general linear matrix equations

In this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.