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.     

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).     

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.     

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.     

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.     

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     

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.     

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.     

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.     

Integer programming with 2-variable equations and 1-variable inequalities

We present an efficient algorithm to find an optimal integer solution of a given system of 2-variable equalities and 1-variable inequalities with respect to a given linear objective function. Our algorithm has worst-case running time in O(N²) where N is the number of bits in the input.     

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.     

Sparse polynomial division using a heap

In 1974, Johnson showed how to multiply and divide sparse polynomials using a binary heap. This paper introduces a new algorithm that uses a heap to divide with the same complexity as multiplication. It is a fraction-free method that also reduces the number of integer operations for divisions of polynomials with integer coefficients over the rationals. Heap-based algorithms use very little memory and do not generate garbage. They can run in the CPU cache and achieve high performance. We compare our C implementation of sparse polynomial multiplication and division with integer coefficients to the routines of the Magma, Maple, Pari, Singular and Trip computer algebra systems.     

A new proximity test for polynomial zeros

We combine some known techniques and results of Turan and Schönhage to improve substantially numerical performance of the computation of the minimum and the maximum distances from a fixed complex point to roots (zeros) of a fixed univariate polynomial.     

Parametric solutions to Sylvester-conjugate matrix equations

By two recently proposed operations with respect to complex matrices, a simple explicit solution to the Sylvester-conjugate matrix equation is given in a finite series form. The obtained solution can also be equivalently expressed in terms of the so-called controllability-like matrix and observability-like matrix. The proposed solution can provide all the degrees of freedom which is represented by a free parameter matrix. An illustrative example is employed to show the effectiveness of the proposed method.