Problem Of Polynomial Approximation Research Articles

USE OF POLYNOMIAL APPROXIMATION OF FORCES IN THE METHOD OF CORRECTION FUNCTIONS

The paper considers the problem of polynomial approximation of the "force - time" dependencies in the elements of corroding hinged-rod structures from the point of view of the influence of the degree of the polynomial on the error in calculating their durability. A method for determining the coefficients of approximating polynomials is proposed, which is based on the use of a numerical-analytical algorithm for solving a system of differential equations describing the corrosion process. The results of numerical experiments are presented, illustrating for various constructions the dependence of the error in solving the problem on the degree of approximating polynomials.

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

Approximation by polynomials composed of Weierstrass doubly periodic functions

The problem of describing classes of functions in terms of the rate of approximation of these functions by polynomials, rational functions, splines entered in the theory of approximation more than 100 years ago and still retains its relevance. Among a large number of problems related to approximation, we considered the problem of polynomial approximation in two variables of a function defined on the continuum of an elliptic curve in C2 and holomorphic in its interior. The formulation of such a question led to the need to study the approximation of a function that is continuous on the continuum of the complex plane and analytic in its interior, using polynomials in doubly periodic Weierstrass functions and their derivatives. This work is devoted to the development of this topic.

Markov Parameter Identification via Chebyshev Approximation

This paper proposes an identification algorithm for Single Input Single Output (SISO) Linear Time-Invariant (LTI) systems. In the noise-free setting, where the first T Markov parameters can be precisely estimated, all Markov parameters can be inferred by the linear combination of the known T Markov parameters, of which the coefficients are obtained by solving the uniform polynomial approximation problem, and the upper bound of the asymptotic identification bias is provided. For the finite-time identification scenario, we cast the system identification problem with noisy Markov parameters into a regularized uniform approximation problem. Numerical results demonstrate that the proposed algorithm outperforms the conventional Ho-Kalman Algorithm for the finite-time identification scenario while the asymptotic bias remains negligible.

RESEARCH OF POLYNOMIAL APPROXIMATION OF FORCES IN ROD ELEMENTS OF CORRODING STRUCTURES

The paper considers the problem of polynomial approximation of the "force - time" depend-encies in the elements of corroding hinged-rod structures from the point of view of the in-fluence of the degree of the polynomial on the error in calculating their durability. A method for determining the coefficients of approximating polynomials is proposed, which is based on the use of a numerical-analytical algorithm for solving a system of differential equations describing the corrosion process. The results of numerical experiments are presented, illus-trating for various constructions the dependence of the error in solving the problem on the degree of approximating polynomials.

Tikhonov regularization for polynomial approximation problems in Gauss quadrature points

This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on [−1, 1] by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing only a multiplicative correction factor into both classical barycentric formulae. An L 2 error bound and a uniform error bound are derived, providing similar information that Tikhonov regularization is able to reduce the operator norm (Lebesgue constant) and the error term related to the level of noise, both by multiplying a correction factor which is less than one. Numerical examples show the benefits of Tikhonov regularization when data is noisy or data size is relatively small.

Finite Alternation Theorems and a Constructive Approach to Piecewise Polynomial Approximation in Chebyshev Norm

One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment.

Approximation of Infinitely Differentiable Functions in Unbounded Convex Domains by Polynomials in Weighted Spaces

We examine the problem of polynomial approximation in the space of infinitely differentiable functions in an unbounded convex domain in ℝn that have a prescribed growth rate near the boundary of the domain and at infinity.

Weierstrass type approximation by weighted polynomials in [formula omitted

In this paper we consider weighted polynomial approximation on unbounded multidimensional domains in the spirit of the weighted version of the Weierstrass trigonometric theorem according to which every continuous function on the real line with equal finite limits at ±∞ is a uniform limit on R of weighted algebraic polynomials of degree 2n with varying weights (1+t2)−n. We will verify a similar statement in the multivariate setting for a general class of convex weights.We also consider the similar problem of multivariate polynomial approximation with varying weights for some typical non convex weights. In case of non convex weights of the form wα(x)≔(1+|x1|α+...+|xd|α)1α,0<α<1 in order for weighted polynomial approximation to hold for a given continuous function it is necessary that the function vanishes on a certain exceptional set consisting of all coordinate hyperplanes and ∞. Moreover, in case of rational α this condition is also sufficient for weighted polynomial approximation to hold.

Determination of amplitude levels of the piecewise constant signal by using polynomial approximation

A new approach to determining the amplitude levels of piecewise constant signal has been proposed that is based on using its multiplicative model and solving the problem of polynomial approximation. In case of the absence of noise, the statement of polynomial approximation problem is based on the requirement of exact match of the current signal value with the amplitude value of one of its levels. In case of the presence of ordinary additive noise, the problem statement is based on the least squares criterion, while the solution of problem is presented in the analytical form. For the case of the presence of pulse-type noise, the problem statement is based on the minimum duration criterion, while the problem solution is achieved numerically by an appropriate functional minimization in unknown amplitudes of levels. The case of binary piecewise constant signal is considered in detail. The results of numerical simulation are presented for the cases, where the binary signal is distorted by ordinary additive noise with Gaussian distribution law and the pulse-type noise with the Cauchy distribution law.

Bézier form of dual bivariate Bernstein polynomials

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bezier form of the L 2-solution of the problem of best polynomial approximation of Bezier curve or surface. In this connection, the Bezier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bezier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.

A Christoffel function weighted least squares algorithm for collocation approximations

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

Constrained approximation of rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

We propose a novel approach to the problem of polynomial approximation of rational Bezier triangular patches with prescribed boundary control points. The method is very efficient thanks to using recursive properties of the bivariate dual Bernstein polynomials and applying a smart algorithm for evaluating a collection of two-dimensional integrals. Some illustrative examples are given.

A Robust and Scalable Implementation of the Parks-McClellan Algorithm for Designing FIR Filters

With a long history dating back to the beginning of the 1970s, the Parks-McClellan algorithm is probably the most well known approach for designing finite impulse response filters. Despite being a standard routine in many signal processing packages, it is possible to find practical design specifications where existing codes fail to work. Our goal is twofold. We first examine and present solutions for the practical difficulties related to weighted minimax polynomial approximation problems on multi-interval domains (i.e., the general setting under which the Parks-McClellan algorithm operates). Using these ideas, we then describe a robust implementation of this algorithm. It routinely outperforms existing minimax filter design routines.

On the zero-free polynomial approximation problem

Let E be a compact set in C with connected complement, and let A(E) be the class of all complex continuous function on E that are analytic in the interior E0 of E. Let f∈A(E) be zero free on E0. By Mergelyan’s theorem f can be uniformly approximated on E by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on E? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets E. The present paper describes classes of functions for which zero free approximation is possible on an arbitrary E.

Weighted discrete least-squares polynomial approximation using randomized quadratures

We discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation. The problem stems from uncertainty quantification (UQ), where the independent variables of the functions are random variables with specified probability measure. We propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product Gaussian quadrature points. We analyze the stability properties of this method and prove that the method is asymptotically stable, provided that the number of points scales linearly (up to a logarithmic factor) with the cardinality of the polynomial space. Specific results in both bounded and unbounded domains are obtained, along with a convergence result for Chebyshev measure. Numerical examples are provided to verify the theoretical results.

Polynomials and lemniscates of indefiniteness

For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in $$\,\mathbb {C}^{n \times n}$$Cn×n. A natural matrix analytic extension for lemniscates is introduced. Operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse. For the speed of convergence and lemniscate asymptotics, the notion of quasilemniscate arises. In the $$L^2$$L2-norm algorithms for solving the problem are devised.

Approximation of sgn $$(x)$$ ( x ) on Two Symmetric Intervals by Rational Functions with Fixed Poles

Recently, A. Eremenko and P. Yuditskii found explicit solutions of the best polynomial approximation problems of sgn\((x)\) over two intervals in terms of conformal mappings onto special comb domains. We give analogous solutions for the best approximation problems of sgn\((x)\) over two symmetric intervals by odd rational functions with fixed poles. Here, the existence of the related conformal mapping is proved using convexity of the comb domains along the imaginary axis.

Multivariate Approximation Theorems in Weighted Lorentz Spaces

We consider trigonometric polynomial approximation problems in the multivariate weighted Lorentz spaces. In particular, we prove direct and converse theorems for trigonometric approximation in the multivariate weighted Lorentz spaces with \({\mathcal{A}_{p}}\) -weights.

Optimality Conditions in Terms of Alternance: Two Approaches

In Optimization Theory, necessary and sufficient optimality conditions play an essential role. They allow, first of all, checking whether a point under study satisfies the conditions, and, secondly, if it does not, finding a "better" point. For the class of directionally differentiable functions, a necessary condition for an unconstrained minimum requires the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form of inclusion. The problem of verifying this condition is reduced to that of finding the point of some convex and compact set C which is nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction, and are able to construct a numerical method. In the classical Chebyshev polynomial approximation problem, necessary optimality conditions are expressed in the form of alternation of signs of some values. In the present paper, a generalization of the alternance approach to a general optimization problem is given. Two equivalent forms of the alternance condition (the so-called inside form and the outside one) are discussed in detail. In some cases, it may be more convenient to use the conditions in the form of inclusion, in some other--the condition in the alternance form as in the Chebyshev approximation problem. Numerical methods based on the condition in the form of inclusion usually are "gradient-type" methods, while methods employing the alternance form are often "Newton-type". It is hoped that in some cases it will be possible to enrich the existing armory of optimization algorithms by a new family of efficient tools. In the paper, we discuss only unconstrained optimization problems in the finite-dimensional setting. In many cases, a constrained optimization problem can be reduced (via Exact Penalization Techniques) to an unconstrained one.