Polynomials Of Arbitrary Degree Research Articles

A New Discretization for $m$th-Laplace Equations with Arbitrary Polynomial Degrees

This paper introduces new mixed formulations and discretizations for $m$th-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary $m=1,2,3,\dots$ based on novel Helmholtz-type decompositions for tensor-valued functions. The new discretizations allow for ansatz spaces of arbitrary polynomial degree and the lowest-order choice coincides with the nonconforming FEMs of Crouzeix and Raviart for $m=1$ and of Morley for $m=2$. Since the derivatives are directly approximated, the lowest-order discretizations consist of piecewise affine and piecewise constant functions for any $m=1,2,\dots.$ Moreover, a uniform implementation for arbitrary $m$ is possible. Besides the a priori and a posteriori analysis, this paper proves optimal convergence rates for adaptive algorithms for the new discretizations.

Squarefree parts of polynomial values

Given a separable nonconstant polynomial f(x) with integer coefficients, we consider the set S consisting of the squarefree parts of all the rational values of f(x), and study its behavior modulo primes. Fixing a prime p, we determine necessary and sufficient conditions for S to contain an element divisible by p. We conjecture that if p is large enough, then S contains infinitely many representatives from every nonzero residue class modulo p. The conjecture is proved by elementary means assuming f(x) has degree 1 or 2. If f(x) has degree 3, or if it has degree 4 and has a rational root, the conjecture is shown to follow from the parity conjecture for elliptic curves. For polynomials of arbitrary degree, a local analogue of the conjecture is proved using standard results from class field theory, and empirical evidence is given to support the global version of the conjecture.

Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters

Proofs of convergence of adaptive finite element methods (AFEMs) for approximating eigenvalues and eigenfunctions of linear elliptic problems have been given in several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai, He, and Zhou in [IMA J. Numer. Anal., 35 (2015), pp. 1934--1977], who proved convergence and optimality of AFEMs for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (noncomputable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. In [Numer. Math., 130 (2015), pp. 467--496], Gallistl used a similar tool to prove that a standard AFEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or nonconstant diffusion coefficients, however, the arguments of Dai, He, and Zhou and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this article we provide this missing key step, thus showing that standard adaptive FEMs for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

SummaryA framework to validate and generate curved nodal high‐order meshes on Computer‐Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin‐shell and 3D finite element analysis with unstructured high‐order methods. First, we define a distortion (quality) measure for high‐order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high‐order), and handles with low‐quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high‐order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process. Copyright © 2015 John Wiley & Sons, Ltd.

Generation of high order geometry representations in Octree meshes

We propose a robust method to convert triangulated surface data into polynomial volume data. Such polynomial representations are required for high-order partial differential solvers, as low-order surface representations would diminish the accuracy of their solution. Our proposed method deploys a first order spatial bisection algorithm to find robustly an approximation of given geometries. The resulting voxelization is then used to generate Legendre polynomials of arbitrary degree. By embedding the locally defined polynomials in cubical elements of a coarser mesh, this method can reliably approximate even complex structures, like porous media. It thereby is possible to provide appropriate material definitions for high order discontinuous Galerkin schemes. We describe the method to construct the polynomial and how it fits into the overall mesh generation. Our discussion includes numerical properties of the method and we show some results from applying it to various geometries. We have implemented the described method in our mesh generatorSeeder, which is publically available under a permissive open-source license.

Lower estimates of measure of deviation of polynomials from mathematical expectations

Inequalities that are in a sense reverse to the Carbery–Wright inequality are investigated. Inequalities of this type are proved in the case of a Gaussian measure and a polynomial of arbitrary degree and also in the case of an arbitrary convex measure and a polynomial degree of 2.

Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming Approximations

This paper derives a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue. They are valid without restrictions on the computational mesh or on the approximate eigenvector; we only need to assume that the approximate eigenvalue is separated from the surrounding smaller and larger exact ones, which can be checked in practice. Guaranteed, fully computable, optimally convergent, and polynomial-degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well, under the same hypotheses. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory, and no convexity/regularity assumption on the computational domain/exact eigenvector(s) is needed. Two improvements of the multiplicative constant appearing in our estimates are presented. First, it is reduced by a fixed factor under an explicit, a posteriori calculable condition on the mesh and on the approximate eigenvector--eigenvalue pair. Second, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, the multiplicative factor can be brought to the optimal value of one. Inexact algebraic solvers are taken into account; the estimates are valid on each iteration and can serve for the design of adaptive stopping criteria. The application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided, along with a numerical illustration on a set of test problems.

Explicit [formula omitted] approximation of conic sections using Bézier curves of arbitrary degree

In this paper, we propose an explicit and effective method for G1 approximation of conic sections with arbitrary degree polynomial curves. The geometric constraints can be expressed by two variables. Then we construct an objective function as the alternative approximation error function in the L2-norm, which is a quadratic function in these two variables. To minimize the objective function is equivalent to solving a system of linear equations with two variables. Since its coefficient matrix is invertible, we can explicitly obtain the unique solution and then derive the explicit G1 approximation of conic sections by Bézier curves of arbitrary degree. The proposed method has an optimal approximation in the L2-norm, i.e., the L2-distance error reaches minimum, and also can process the conic section with center angle larger than π without subdivision scheme. Finally, numerical examples demonstrate the effectiveness of our method.

BMI-Based Stabilization of Linear Uncertain Plants With Polynomially Time Varying Parameters

This technical note considers the robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics. Each mode is characterized by a theoretically unbounded dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. Using a parameter dependent Lyapunov function polynomially depending on time, the stabilizing controller for each single mode is directly obtained by the solution of some BMI's, which become LMI's by fixing two positive scalars. The stability conditions of the switching closed loop system are derived defining a switched Lyapunov function and involving the permanence time interval of the switching plant over each single mode. A salient feature of the technical note is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

On the speed of convergence of Newton’s method for complex polynomials

We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\mathcal{S}_d$ of $3.33d\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting points to reach all roots with precision $\varepsilon$ is $O(d^2\log^4 d + d\log|\log \varepsilon|)$. This is an improvement of an earlier result in \cite{Schleicher}, where the number of iterations is shown to be $O(d^4\log^2 d + d^3\log^2d|\log \varepsilon|)$ in the worst case (allowing multiple roots) and $O(d^3\log^2 d(\log d + \log \delta) + d\log|\log \varepsilon|)$ for well-separated (so-called $\delta$-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than $O(d^2)$ for fixed $\varepsilon$.

A Hybrid Scheme of Public-Key Encryption and Somewhat Homomorphic Encryption

We introduce a hybrid homomorphic encryption that combines public-key encryption (PKE) and somewhat homomorphic encryption (SHE) to reduce the storage requirements of most somewhat or fully homomorphic encryption (FHE) applications. In this model, messages are encrypted with a PKE and computations on encrypted data are carried out using SHE or FHE after homomorphic decryption. To obtain efficient homomorphic decryption, our hybrid scheme combines IND-CPA PKE without complicated message padding with SHE with a large integer message space. Furthermore, if the underlying PKE is multiplicative, the proposed scheme has the advantage that polynomials of arbitrary degree can be evaluated without bootstrapping. We construct this scheme by concatenating the ElGamal and Goldwasser-Micali schemes over a ring ℤ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> for a composite integer N whose message space is ℤ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">×</sup> . To accelerate the homomorphic evaluation of the PKE decryption, we introduce a method to reduce the degree of the exponentiation circuit at the cost of additional public keys. Using the same technique, we present an efficient partial solution to an open problem which is to evaluate mod q mod p arithmetic homomorphically for large p. As an independent interest, we also obtain a generic method for converting from private-key SHE to public-key SHE. Unlike the method described by Rothblum, we are free to choose the SHE message space.

A New Stabilization Method for the Elasticity Problem

In this paper, we consider the elasticity problem based on the Hellinger---Reissner variational principle. We use the $$C^0$$C0 continuous arbitrary degree polynomial mixed finite element spaces to approximate the stress and displacement, and develop a new stabilization method for the finite element space pairs to overcome the lack of the inf-sup condition, then we give the corresponding error estimates of the stabilization approximation scheme. At last, we implement a numerical example to test the stability and effectiveness of the proposed method.

Algorithms for Hessenberg-Triangular Reduction of Fiedler Linearization of Matrix Polynomials

Small- to medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, requires an initial reduction of a matrix pair to Hessenberg-triangular (HT) form. In this paper, we discuss the design and evaluation of high-performance parallel algorithms and software for HT reduction of a specific linearization of matrix polynomials of arbitrary degree. The proposed algorithm exploits the sparsity structure of the linearization to reduce the number of operations and improve the cache reuse compared to existing algorithms for unstructured inputs. Experiments on both a workstation and a high-performance computing system demonstrate that our structure-exploiting parallel implementation can outperform both the general LAPACK routine \tt DGGHRD and the prototype implementation \tt DGGHR3 of a general blocked algorithm.

Divergence in right-angled Coxeter groups

Let W W be a 2 2 -dimensional right-angled Coxeter group. We characterise such W W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.

A Matrix Recurrence for Systems of Clifford Algebra-Valued Orthogonal Polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory (Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Gürlebeck, Bock, Lávička, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.

Extended Gaussian quadratures for functions with an end-point singularity of logarithmic type

The extended Gaussian quadrature rules are shown to be an efficient tool for numerical integration of wide class of functions with singularities of logarithmic type. The quadratures are exact for the functions pol1n−1(x)+lnxpol2n−1(x), where pol1n−1(x) and pol2n−1(x) are two arbitrary polynomials of degree n−1 and n is the order of the quadrature formula. We present an implementation of numerical algorithm that calculates the nodes and the weights of the quadrature formulas, provide a Fortran code for numerical integration, and test the performance of different kinds of Gaussian quadratures for functions with logarithmic singularities. Program summaryProgram title: GAUSEXTCatalogue identifier: AETP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2535No. of bytes in distributed program, including test data, etc.: 39963Distribution format: tar.gzProgramming language: Mathematica, Fortran.Computer: PCs or higher performance computers.Operating system: Linux, Windows, MacOS.RAM: Kilobytes.Classification: 4.11.Nature of problem:Quadrature formulas for numerical integration, effective for a wide class of functions with end-point singularities of logarithmic type.Solution method:The method of solution is based on the algorithm developed in Ref. [1] with some modifications.Running time:Milliseconds to minutes.

Generalized Schultz iterative methods for the computation of outer inverses

We consider a general matrix iterative method of the type Xk+1=Xkp(AXk) for computing an outer inverse AR(G),N(G)(2), for given matrices A∈Cm×n and G∈Cn×m such that AR(G)⊕N(G)=Cm. Here p(x) is an arbitrary polynomial of degree d. The convergence of the method is proven under certain necessary conditions and the characterization of all methods having order r is given. The obtained results provide a direct generalization of all known iterative methods of the same type. Moreover, we introduce one new method and show a procedure how to improve the convergence order of existing methods. This procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples.

An asymptotically exact LMI solution to the robust discretisation of LTI systems with polytopic uncertainties and its application to sampled-data control

In this paper, the problem of robust discretisation of linear time-invariant (LTI) systems with polytopic uncertainties is introduced. More specifically, the main objective is to provide a systematic way to find an approximate discrete-time (DT) model of a continuous-time (CT) plant with uncertainties in polytopic domain. The system matrices of polytopic DT model to be found are expressed as parameter-dependent matrices which are homogeneous polynomials of arbitrary degree with respect to the uncertain variables in the simplex, and is obtained in such a way that the norm between the system matrices and the truncated power series of the exact DT model is minimised while preserving the polytopic structure of the original CT plant. The solution procedures proposed are presented in terms of single-parameter minimisation problems subject to linear matrix inequality (LMI) constraints which are numerically tractable via LMI solvers. Finally, examples are given to show the validity and effectiveness of the proposed techniques.

Polynomial problems of the Casas-Alvero type

We establish necessary and sufficient conditions for an arbitrary polynomial of degree n , especially with only real roots, to be trivial, i.e. to have the form a(x− b)n . To do this, we derive new properties of polynomials and their roots. In particular, it concerns new bounds and genetic sum representations of the Abel-Goncharov interpolation polynomials. Moreover, we prove the Sz.-Nagy type identities, the Laguerre and Obreshkov-Chebotarev type inequalities for roots of polynomials and their derivatives. As applications these results are associated with the known problem, conjectured by Casas-Alvero in 2001, which says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. We investigate particular cases of the problem, when the conjecture holds true or, possibly, is false.

Robust stabilization of linear uncertain plants with polynomially time varying parameters

The robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics is considered. Each mode is characterized by a dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. The stability conditions of the switching closed-loop system are derived defining a switched Lyapunov function and involving the dwell time of the system over each single mode. An important feature of the paper is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.