The numerical condition of univariate and bivariate degree elevated Bernstein polynomials

The numerical condition of the degree elevation operation on Bernstein polynomials is considered and it is shown that it does not change the condition of the polynomial. In particular, several condition numbers for univariate and bivariate Bernstein polynomials, and their degree elevated forms, are developed and it is shown that the condition numbers of the degree elevated polynomials are identically equal to their forms prior to degree elevation. Computational experiments that verify this theoretical result are presented. The results in this paper differ from those in [Comput. Aided Geom. Design 4 (1987) 191–216] and [Comput. Aided Geom. Design 5 (1988) 215–252], where it is claimed that degree elevation causes a reduction in the numerical condition of a Bernstein polynomial. It is shown, however, that there is an error in the derivation of this result.     

A weak Galerkin finite element method for the Oseen equations

In this paper, a weak Galerkin finite element method for the Oseen equations of incompressible fluid flow is proposed and investigated. This method is based on weak gradient and divergence operators which are designed for the finite element discontinuous functions. Moreover, by choosing the usual polynomials of degree i ≥ 1 for the velocity and polynomials of degree i ? 1 for the pressure and enhancing the polynomials of degree i ? 1 on the interface of a finite element partition for the velocity, this new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained by defining a weak convection term. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the Oseen problem.     

On shifted Eisenstein polynomials

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree \(n\) polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.     

Chebyshev polynomials are not always optimal

We are concerned with the problem of finding the polynomial with minimal uniform norm on among all polynomials of degree at most n and normalized to be 1 at c. Here, is a given ellipse with both foci on the real axis and c is a given real point not contained in . Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. In this work, we show that this is not true in general. Moreover, we derive sufficient conditions which guarantee that Chebyshev polynomials are optimal. Also, some numerical examples are presented.     

Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients

In this paper, we explore the effect of numerical integration on the Galerkin meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered Galerkin meshless methods with shape functions that reproduce polynomials of degree k?≥?1. We have obtained an estimate for the energy norm of the error in the approximate solution under the presence of numerical integration. This result has been established under the assumption that the numerical integration rule satisfies a certain discrete Green’s formula, which is not problem dependent, i.e., does not depend on the non-constant coefficients of the problem. We have also derived numerical integration rules satisfying the discrete Green’s formula.     

Permutation polynomials over algebraic number fields

Let K be an algebraic number field. It is known that any polynomial which induces a permutation on infinitely many residue class fields of K is a composition of cyclic and Chebyshev polynomials. This paper deals with the problem of deciding, for a given K, which compositions of cyclic or Chebyshev polynomials have this property. The problem is reduced to the case where K is an Abelian extension of $\text{Q}$. Then the question is settled for polynomials of prime degree, and the Abelian case for composite degree polynomials is considered. Finally, various special cases are dealt with.     

基于高斯伪谱的最优控制求解及其应用

研究一种基于高斯伪谱法的具有约束受限的最优控制数值计算问题.方法将状态演化和控制规律用多项式参数化近似,微分方程用正交多项式近似.将最优控制问题求解问题转化为一组有约束的非线性规划求解.详细论述了该种近似方法的有效性.作为该种方法的应用,讨论了一个障碍物环境下的机器人最优路径生成问题.将机器人路径规划问题转化为具有约束条件最优控制问题,然后用基于高斯伪谱的方法求解,并给出了仿真结果.     

Do Orthogonal Polynomials Dream of Symmetric Curves?

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.     

Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nyström’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems.     

五次键合多项式P-不可约的两个定理

在生物化学研究领域,对键合多项式P-不可约性的判定是一个重要问题.已有结果主要考虑四次或四次以下的多项式,而对五次或五次以上键合多项式的讨论尚未见报道.文章在这方面作了一定的探索,给出了五次键合多项式P-不可约的两组充分条件,这两组条件均是用多项式的系数构成的等式或不等式组显式表示的.     

Direct and inverse polynomial perturbations of hermitian linear functionals

This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.     

Analysis of the combined finite element and Fourier interpolation

Summary We consider Lagrange interpolation involving trigonometric polynomials of degree N in one space direction, and piecewise polynomials over a finite element decomposition of mesh size h in the other space directions. We provide error estimates in non-isotropic Sobolev norms, depending additively on the parametersh andN. An application to the convergence analysis of an elliptic problem, with some numerical results, is given.     

On Local Smoothness Classes of Periodic Functions

We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. We construct a sequence of operators whose values are (global) trigonometric polynomials, and yet their behavior at different points reflects the behavior of the target function near each of these points. In addition to being localized, our operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Our constructions are “universal;” i.e., they do not require an a priori knowledge about the smoothness of the target functions. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order.     

A weak Galerkin finite element method for the stokes equations

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k≥1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.     

A note on stability results for scattered data interpolation on Euclidean spheres

The present work considers the interpolation of the scattered data on the d-sphere by spherical polynomials. We prove bounds on the conditioning of the problem which rely only on the separation distance of the sampling nodes and on the degree of polynomials being used. To this end, we establish a packing argument for well separated sampling nodes and construct strongly localized polynomials on spheres. Numerical results illustrate our theoretical findings. Dedicated to Professor Manfred Tasche on the occasion of his 65th birthday.     

Numerical Solution of Riemann–Hilbert Problems: Random Matrix Theory and Orthogonal Polynomials

Recently, a general approach to solving Riemann–Hilbert problems numerically has been developed. We review this numerical framework and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with the approach of Bornemann to compute Fredholm determinants, we are able to calculate spectral densities and gap statistics for a broad class of finite-dimensional unitary invariant ensembles. We show that the accuracy of the numerical algorithm for approximating orthogonal polynomials is uniform as the degree grows, extending the existing theory to handle g-functions. As another example, we compute the Hastings–McLeod solution of the homogeneous Painlevé II equation.     

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.     

Some results on numerical ranges and factorizations of matrix polynomials

Some new factorization theorems for monic matrix polynomials are obtained. These theorems are based on the numerical range having the number of connected components equal to the degree of the polynomial. For second degree polynomials, sufficient conditions are given for the numerical range to have two connected components.