Bernstein Polynomial

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.

There has been increasing interest in estimating a multivariate regression function subject to various shape restrictions, such as nonnegativity, isotonicity, convexity and concavity among many others. The estimation of such shape-restricted regression curves is more challenging for multivariate predictors, especially for functions with compact support. Most of the currently available statistical estimation methods for shape restricted regression functions are generally computationally very intensive. Some of the existing methods have perceptible boundary biases.

Every s×s matrix A yields a composition map acting on polynomials on R s . Specifically, for every polynomial p we define the mapping C A by the formula (C A p)(x):=p(Ax), x∈R s . For each nonnegative integer n, homogeneous polynomials of degree n form an invariant subspace for C A . We let A (n) be the matrix representation of C A relative to the monomial basis and call A (n) a binomial matrix. This paper studies the asymptotic behavior of A (n) as n→∞. The special case of 2×2 matrices A with the property that A 2=I corresponds to discrete Taylor series and motivated our original interest in binomial matrices.

The aim of this paper is to give main properties of the generating function of the Bernstein polynomials. We prove recurrence relations and derivative formula for Bernstein polynomials. Furthermore, some new results are obtained by using this generating function of these polynomials.

In geometric modelling and computer-aided design a polynomial is usually represented in Bernstein form. Forward error analysis for the computation of the derivative of a polynomial in Bernstein form is performed. The corresponding running error is also analyzed. We have compared our results with those corresponding to a polynomial represented in power basis and using Legendre and Chebyshev forms. Since the conversions between the Bernstein and the power bases and between Bernstein and Chebyshev polynomials were already studied in the literature, for the sake of completeness the conversion between Bernstein and Chebyshev polynomials of the first kind is analyzed in an appendix.

In some applications of survival analysis with covariates, the commonly used semiparametric assumptions (e.g., proportional hazards) may turn out to be stringent and unrealistic, particularly when there is scientific background to believe that survival curves under different covariate combinations will cross during the study period. We present a new nonparametric regression model for the conditional hazard rate using a suitable sieve of Bernstein polynomials. The proposed nonparametric methodology has three key features: (i) the smooth estimator of the conditional hazard rate is shown to be a unique solution of a strictly convex optimization problem for a wide range of applications; making it computationally attractive, (ii) the model is shown to encompass a proportional hazards structure, and (iii) large sample properties including consistency and convergence rates are established under a set of mild regularity conditions. Empirical results based on several simulated data scenarios indicate that the proposed model has reasonably robust performance compared to other semiparametric models particularly when such semiparametric modeling assumptions are violated. The proposed method is further illustrated on the gastric cancer data and the Veterans Administration lung cancer data.

In this paper, we present some efficient direct solvers for solving a system of high order linear Volterra-Fredholm integro-differential equations (VFIDEs). A new approach implementing a collocation method in combination with operational matrices of Bernstein polynomials for the numerical solution of VFIDEs is introduced. The main characteristic behind this approach is that it reduces such problems to ones of solving systems of algebraic equations. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving systems of high order linear VFIDEs.

This paper presents a method based on polynomial approximation using Bernstein polynomial basis to obtain approximate numerical solution of a singular integro-differential equation with Cauchy kernel. The numerical results obtained by the present method compares favorably with those obtained by various Galerkin methods earlier in the literature. Also the convergence of the method is established rigorously for the studied integro-differential equation.

We study the behaviour of the notion of "thema", introduced in our previous article [B.09b], by a change of variable. We show not only that the fundamental invariants of such a thema, corresponding to the Bernstein polynomial, are stable by a change of variable, but also other numerical invariants called principal parameters. \\ We show on a rank 3 example that nevertheless the isomorphism class of a thema is not stable in general by a change of variable. We conclude in proving that a change of variable transforms an holomorphic family of thema in an holomorphic family. This implies that non principal parameters change holomorphically.

In this work, we present a comparative study of meshless method, modified Bernstein polynomials (BP) and B-Spline finite element method (BS-FEM) for the numerical solution of two different models of Korteweg–de Vries (KdV) equation. The multiquadric (MQ) and Gaussian (GA) Radial basis functions (RBFs) are used due to exponential convergence rate. Excellent agreement is found between exact and RBFs solutions and same accuracy is obtained as Bernstein polynomials and B-Spline finite element method.

In this paper, we develop a Bernstein dual-Petrov-Galerkin method for the numerical simulation of a two-dimensional fractional diffusion equation. A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polyno-mials as the trial ones. We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix based approach for the spatial discretization. It is shown that the method leads to banded linear systems that can be solved efficiently. The stability and convergence of the proposed method is discussed. Finally, some numerical examples are provided to support the theoretical claims and to show the accuracy and efficiency of the method.

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.

Let U n ⊂ C n [a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f 0 ∈ U n is strictly positive and f 1 ∈ U n has the property that f 1 /f 0 is strictly increasing. We search for conditions ensuring the existence of points t 0 , ..., t n ∈ [a, b] and positive coefficients α 0 , ..., α n such that for all f ∈ C [a, b], the operator B n :

We present an elementary proof of a conjecture by I. Raşa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szász operators and Baskakov operators.

Given a real function f c C2 ' [ 0, 1], k > 1 and the corresponding Bernstein polynomials {B, (f)}, we derive an asymptotic expansion formula for & (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence than Bn (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique .

In this short note, we establish the uniform integrability and pointwise convergence of an (unbounded) family of polynomials on the unit interval that arises in work on statistical density estimation using Bernstein polynomials. These results are proved by first establishing/generalizing some combinatorial and probability inequalities that rely on a new family of completely monotonic functions.

We present an open source toolbox in Scilab for multivariate polynomial optimization based on the Bernstein form. The developed toolbox finds the global minimum of unconstrained polynomial optimization problems. We first describe the method used by the toolbox, and then demonstrate its performance on several standard optimization examples. We also compare the quality of the results obtained using the developed toolbox with those of popular global optimization methods, such as genetic algorithms, simulated annealing, pattern search, and global search interior point methods. The tests show the superior performance of the presented toolbox.

We introduce polynomials B n i (x; ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B n i (x; ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.

We prove that the real roots of normal random homogeneous polynomial systems with n + 1 variables and given degrees are, in some sense, equidistributed in the projective space P R n+1. From this fact we compute the average number of real roots of normal random polynomial systems given in the Bernstein basis.

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One of the standard problems in statistics consists of determining the relationship between a response variable and a single predictor variable through a regression function. Background scientific knowledge is often available that suggests the regression function should have a certain shape (e.g., monotonically increasing or concave) but not necessarily a specific parametric form. Bernstein polynomials have been used to impose certain shape restrictions on regression functions. The Bernstein polynomials are known to provide a smooth estimate over equidistant knots. Bernstein polynomials are used in this paper due to their ease of implementation, continuous differentiability, and theoretical properties. In this work, we demonstrate a connection between the monotonic regression problem and the variable selection problem in the linear model. We develop a Bayesian procedure for fitting the monotonic regression model by adapting currently available variable selection procedures. We demonstrate the effectiveness of our method through simulations and the analysis of real data.

The results in single beam gamma transmission tomography are presented by an image processing based on the Algebraic Reconstruction Technique -ART. The mathematical reconstruction was carried out by Bezier triangles with Bernstein polynomials, and also by spline functions, and as a matter of comparison the Filtered Backprojection -FBP method was used. The image processing reconstructs the FCC (fluidized catalytic cracking) catalyst density distribution in an experimental riser. Typical problems are characterized as reconstruction models involve the inverse problem, basis function and solution of linear system of equations. A least squares estimator is required for the numerical solution, the model parameters are evaluated, and a comparison with literature values is carried out.

In this paper, using the concept of B-statistical convergence for sequence of infinite matrices B=(Bi) with Bi=(bnk(i)) we investigate various approximation results concerning the classical Korovkin theorem. Then we present two examples of sequences of positive linear operators. The first one shows that the statistical Korovkin type theorem does not work but our approximation theorem works. The second one gives that our approximation theorem does not work but the statistical Korovkin type theorem works. Also, we study the rates of B-statistical convergence of approximating positive linear operators and give a Voronovskaya-type theorem.

In this article we study holomorphic deformations of the filtered Gauss-Manin systems associated to a vanishing period integral. For that purpose we introduce a new sub-class of the class of monogenic (a,b)-modules (Brieskorn modules) which was studied in our previous article [B. 09]. We show that these new objects, called ?themes?, have good functorial properties and that there exists a canonical order on the roots of the corresponding Bernstein polynomial. We construct, for given fundamental invariants, a finite dimensional versal holomorphic family and we show that, when all themes with these fundamental invariants are ?stable?, this versal family is in fact universal. We also give a sufficient condition on the roots of the Bernstein polynomial in order that the previous condition is satisfied. We show with an example that a universal family may not exist for some values of the fundamental invariants.

Several computational and structural properties of Bezoutian matrices expressed with respect to the Bernstein polynomial basis are shown. The exploitation of such properties allows the design of fast algorithms for the solution of Bernstein-Bezoutian linear systems without never making use of potentially ill-conditioned reductions to the monomial basis. In particular, we devise an algorithm for the computation of the greatest common divisor (GCD) of two polynomials in Bernstein form. A series of numerical tests are reported and discussed, which indicate that Bernstein-Bezoutian matrices are much less sensitive to perturbations of the coe cients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the Bernstein and the power basis.

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.

The main aim of this paper is to provide a unified approach to deriving identities for the Bernstein polynomials using a novel generating function. We derive various functional equations and differential equations using this generating function. Using these equations, we give new proofs both for a recursive definition of the Bernstein basis functions and for derivatives of the nth degree Bernstein polynomials. We also find some new identities and properties for the Bernstein basis functions. Furthermore, we discuss analytic representations for the generalized Bernstein polynomials through the binomial or Newton distribution and Poisson distribution with mean and variance. Using this novel generating function, we also derive an identity which represents a pointwise orthogonality relation for the Bernstein basis functions. Finally, by using the mean and the variance, we generalize Szasz-Mirakjan type basis functions.

Approximation on an unbounded interval is studied in this work by means of a newlydefined two-parameter polynomial operator based on Chlodowsky polynomials. The operator's properties including convergence rate are investigated using the weighted modulus of continuity. An example is given to illustrate the effect of the parameters. Some basic results are presented.

A new system of multivariate distributions with fixed marginal distributions is introduced via the consideration of random variates that are randomly chosen pairs of order statistics of the marginal distributions. The distributions allow arbitrary positive or negative Pearson correlations between pairs of random variates and generalise the Farlie-Gumbel-Morgenstern distribution. It is shown that the copulas of these distributions are special cases of the Bernstein copula. Generation of random numbers from the distributions is described, and formulas for the Kendall and grade (Spearman) correlations are given. Procedures for data fitting are described and illustrated with examples.

This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD.

Polynomials of high degree are much less vulnerable to roundoff error when expressed as truncated Chebyshev series rather than the usual power series form. Recent articles have developed subdivision methods in which all real roots on the canonical Chebyshev interval, x 2 [À1, 1], are found by subdividing the interval and finding the roots of separate Chebyshev series of moderate degree on each subdomain. This strategy can be applied either to polynomials or to transcendental functions if the latter are analytic on the search interval and thus have rapidly convergent Chebyshev polynomial approximations. The last step is to compute the eigenvalues of the Chebyshev-Frobenius companion matrix for each local polynomial. Here, we propose a simple strategy for flagging some subdomains as ''zero-free'' so that eigensolving can be omitted. The test requires conversion of the polynomial from Chebyshev form to a Bernstein polynomial basis. The interval is zero-free if all coefficients in the Bernstein basis are of the same sign. We give the conversion matrices for various small and moderate N and quote Rababah's formulas for general N. We show how to exploit parity so as to halve the cost of the conversion for large N to about N 2 floating point operations. We show that the conversion matrices have condition numbers that are approximately (5/8)2 N .

The objective of this article is to develop a computationally efficient estimator of the regression function subject to various shape constraints. In particular, nonparametric estimators of monotone and/or convex (concave) regression functions are obtained by using a nested sequence of Bernstein polynomials. One of the key distinguishing features of the proposed estimator is that a given shape constraint (e.g., monotonicity and/or convexity) is maintained for any finite sample size and satisfied over the entire support of the predictor space. Moreover, it is shown that the Bernstein polynomial based regression estimator can be obtained as a solution of a constrained least squares method and hence the estimator can be computed efficiently using a quadratic programming algorithm. Finally, the asymptotic properties (e.g., strong uniform consistency) of the estimator are established under very mild conditions, and finite sample properties are explored using several simulation studies and real data analysis. The predictive performances are compared with some of the existing methods.

The problem of simultaneous approximations of a given function and its derivatives, has been addressed frequently in pure and applied mathematics. In pure mathematics, Bernstein polynomials get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of an unknown function and its derivatives. In this paper, we consider fuzzy logic systems with the membership functions of each input variables are chosen as the translations and dilations of one appropriately fixed function. We prove, by a constructive proof based on discretization of the convolution operator, that under certain conditions made on the input variables membership functions, fuzzy logic systems of Sugeno type are universal approximators of a given function and its derivatives.

We study the existence and shape preserving properties of a generalized Bernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and a second function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, within the framework of extended Chebyshev spaces $U_{n}$. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator $B_{n}:C[a,b]\to U_{n}$ with strictly increasing

The aim of this paper is to use symmetric properties of cir-cles and Bernstein polynomials to define a series of interesting properties of rational biquadric Bézier patches, called barycentric properties. A ro-bust algorithm based on these properties is proposed for the conrversion of revolution quadrics to rational biquadric Bézier surfaces. A set of con-version examples is given to illustrate the contribution of this algorithm.

We provide sufficient conditions for a sequence of positive linear approximation operators, L n ( f, x), converging to f (x) from above to imply the convexity of f. We show that, for the convolution operators of Feller type, K n ( f, x), generated by a sequence of iid random variables taking values in an interval I, having a finite moment generating function, the inequalities K n ( f, x) f (x) (x # I, n 1) are necessary and sufficient conditions for f to be convex. This provides a converse of a well-known result of R. A. Khan (Acta. Math. Acad. Sci. Hungar. 39 (1980), 193 203). It contains, as a special case, the corresponding result for the Bernstein polynomials and extends two results obtained for bounded continuous functions by Horova for Sza sz and Baskakov operators. As examples, similar results are also provided for the beta, Meyer-Ko nig Zeller, Picard, and Bleiman, Butzer, and Hahn operators. 1998 Academic Press 1. INTRODUCTION Let I be an interval and, for each x # I, let [+ n, x , n=1, 2, ...] be a sequence of finite measures concentrating on I. Let g(t) be a non-negative continuous function increasing to infinity as t Ä \ . We define D g (I ) to Article No. AT963113 22

In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite differences, or particular techniques, such as finite elements. Also, the method doesn't require the use of matrices, as in resolution of linear algebraic systems, nor the use of like-Newton algorithms, as in resolution of non linear sets of equations. An initial equation is resolved only once, then the method is based on iterated evaluations of appropriate polynomials.

This paper compares the performance and efficiency of different function range interval methods for plotting f (x, y)= 0 on a rectangular region based on a subdivision scheme, where f (x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Horner form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubin's method, Rivlin's method, Gopalsamy's method, and related methods ...

We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K. E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the case of locally conical divisors along a stratification, generalizing a formula of Mustaţǎ in the case of hyperplane arrangements. We also give another proof of a formula of U. Walther on the b-function of a generic hyperplane arrangement, including the multiplicity of −1.

We consider the problem of nonparametric estimation of unknown smooth functions in the presence of restrictions on the shape of the estimator and on its support, using polynomial splines. We provide a general computational framework that treats these estimation problems in a unified manner, without the limitations of the existing methods. Applications of our approach include computing optimal spline estimators for regression, density estimation, and arrival rate estimation problems in the presence of various shape constraints. Our approach can also handle multiple simultaneous shape constraints. The approach is based on a characterization of nonnegative polynomials that leads to semidefinite programming (SDP) and second order cone programming (SOCP) formulations of the problems. These formulations extend and generalize a number of previous approaches in the literature, including those with piecewise linear and B-spline estimators. We also consider a simpler approach, in which nonnegative splines are approximated by splines whose pieces are polynomials with nonnegative coefficients in a nonnegative basis. A condition is presented to test whether a given nonnegative basis gives rise to a spline cone that is dense in the space of nonnegative continuous functions. The optimization models formulated in the paper are solvable with minimal running time using off-the-shelf software. We provide numerical illustrations for density estimation and regression problems. These examples show that the proposed approach requires minimal computational time, and that the estimators obtained using our approach often match and frequently outperform kernel methods and spline smoothing without shape constraints.