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1.
An efficient, robust algorithm is presented for converting a bivariate polynomial expressed in the power basis directly to its equivalent Bernstein (Bézier) form over an arbitrary triangular region. It represents an improvement over existing algorithms which, in the course of transforming the polynomial, may encounter degeneracy problems that must be handled with additional logic. 相似文献
2.
We prove an identity for multivariate Bernstein polynomials on simplices, which may be considered a pointwise orthogonality relation. Its integrated version provides a new representation for the polynomial dual basis of Bernstein polynomials. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order. 相似文献
3.
提出了一个基于CODIC的计算Bernstein多项式的移位-加算法.该算法可以在存在于许多领域的基本计算系统中实现.证明了算法的收敛性,给出了误差分析,做了数值实验,验证了算法的有效性和效率. 相似文献
4.
It is well-known that in two or more variables Bernstein polynomials do not preserve convexity. Here we present two variations, one stronger than the classical notion, the other one weaker, which are preserved and do coincide with classical convexity in the univariate case. Moreover, it will be shown that even the weaker notion is sufficient for the monotonicity of successive Bernstein polynomials, strengthening the well-known result that monotonicity holds for classically convex functions. 相似文献
6.
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. 相似文献
7.
In computing the range of values of a polynomial over an interval a≤x≤b one may use polynomials of the form $$\left( {\begin{array}{*{20}c} k \\ j \\ \end{array} } \right)\left( {x - a} \right)^j \left( {b - x} \right)^{k - j} $$ called Bernstein polynomials of the degree k. An arbitrary polynomial of degree n may be written as a linear combination of Bernstein polynomials of degree k≥n. The coefficients of this linear combination furnish an upper/lower bound for the range of the polynomial. In this paper a finite differencelike scheme is investigated for this computation. The scheme is then generalized to interval polynomials. 相似文献
8.
It is well known that the range of polynomial f over an interval is bounded by the smallest and the largest coefficient of f with respect to the Bernstein basis over the interval. This defines an interval extension F of f, which is called Bernstein form. In this paper we show that the bernstein form is inclusion monotone, i.e. X? Y implies F(X)?F(Y). 相似文献
9.
The Bernstein polynomials (B-polynomials) operational matrices of integration P, differentiation D and product ? are derived. A general procedure of forming these matrices are given. These matrices can be used to solve problems such as calculus of variations, differential equations, optimal control and integral equations. Illustrative examples are included to demonstrate the validity and applicability of the operational matrices. 相似文献
10.
In this work we find numerical integration of a function by some approximations of the function using Bernstein polynomials, and we introduce a new fast method to approximate the value of a definite integral and we compare them. 相似文献
11.
Centered forms for computing the range of values of a real interval polynomial over an interval are considered. A number of methods are proposed and tested on randomly generated interval polynomials. Theoretical and numerical results show that none of the suggested methods produce the optimal results in all cases. The methods are also compared to evaluation using the Horner scheme 相似文献
13.
A new form of personalized quantifier is developed in this paper, with which to investigate and formalize people's personalities or behavior intentions that have to be considered in increasingly complex situations. As we show in the article, the developed quantifier is realized by generalized Bernstein polynomials combined with interpolation spline, and finally expressed as a sequence of piecewise nonlinear polynomials with an adjustable degree. It is characterized by many excellent properties exemplified by such terms as sufficient smoothness, shape‐preserving interpolation, and a high rate of convergence. In particular, the consistency of the ordered weighted averaging (OWA) aggregation under the guidance of the developed quantifier is addressed and proved. This actually provides a sound theoretical basis for practical use. We also experimentally show that the developed quantifier significantly outperforms, in all respects of geometrical characteristics, the other ones presented in previous work, whether from a viewpoint of global approximation of functions or local one. As such, the developed quantifier could be considered as an effective analytical tool for decision making under uncertainty in which different personality traits have to be taken into account. 相似文献
15.
This paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials. The first one is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set. The second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the Bernstein polynomials. Numerical results to real-world control problems are presented showing the efficiency of both algorithms 相似文献
16.
Running error analysis for the bivariate de Casteljau algorithm and the VS algorithm is performed. Theoretical results joint
with numerical experiments show the better stability properties of the de Casteljau algorithm for the evaluation of bivariate
polynomials defined on a triangle in spite of the lower complexity of the VS algorithm. The sharpness of our running error
bounds is shown. 相似文献
17.
Traditional methods for algebraic manipulation of polynomials in Bernstein form try to obtain an explicit formula for each coefficient of the result of a given procedure, such us multiplication, arbitrarily high degree elevation, composition, or differentiation of rational functions. Whereas this strategy often furnishes involved expressions, these operations become trivial in terms of convolutions between coefficient lists if we employ the scaled Bernstein basis, which does not include binomial coefficients. We also carry over this scheme from the univariate case to multivariate polynomials, Bézier simplexes of any dimension and B-bases of other functional spaces. Examples of applications in geometry processing are provided, such as conversions between the triangular and tensor-product Bézier forms. 相似文献
18.
A class of adaptation algorithms for adaptive control of nonlinear dynamic objects of a class is introduced. Implementation of algorithms without the need for direct measurements of the first time derivatives is investigated and sufficient conditions for the existence of such algorithms are formulated. For a class of nonlinear objects and formulated adaptive control aim, examples are given to illustrate the realization of adaptation algorithms. These algorithms are compared with the traditional gradient algorithms. 相似文献
19.
In practice, computer simulations cannot be perfectly controlled because of the inherent uncertainty caused by variability in the environment (e.g., demand rate in the inventory management). Ignoring this source of variability may result in sub-optimality or infeasibility of optimal solutions. This paper aims at proposing a new method for simulation–optimization when limited knowledge on the probability distribution of uncertain variables is available and also limited budget for computation is allowed. The proposed method uses the Taguchi robust terminology and the crossed array design when its statistical techniques are replaced by design and analysis of computer experiments and Kriging. This method offers a new approach for weighting uncertainty scenarios for such a case when probability distributions of uncertain variables are unknown without available historical data. We apply a particular bootstrapping technique when the number of simulation runs is much less compared to the common bootstrapping techniques. In this case, bootstrapping is undertaken by employing original (i.e., non-bootstrapped) data, and thus, it does not result in a computationally expensive task. The applicability of the proposed method is illustrated through the Economic Order Quantity (EOQ) inventory problem, according to uncertainty in the demand rate and holding cost. 相似文献
20.
This article presents a robust adaptive controller for electrically driven robots using Bernstein polynomials as universal approximator. The lumped uncertainties including unmodeled dynamics, external disturbances, and nonimplemented control signals (they assumed as a function of time, instead a function of several variables) are represented with this powerful mathematical tool. The polynomial coefficients are then tuned based on the adaptation law obtained in the stability analysis. A comprehensive approach is adopted to include the saturated and unsaturated areas and also the transition between these areas in the stability analysis. As a result, the stability and the performance of the proposed controller have been improved considerably in dealing with actuator saturation. Also, in comparison with a recent paper based on uncertainty estimation using Taylor series, the proposed controller is less computational due to reducing the size of the matrix of convergence rate. A performance evaluation has been carried out to verify satisfactory performance of transient response of the controller. Simulation results on a Puma560 manipulator actuated by geared permanent magnet dc motors have been presented to guarantee its satisfactory performance. 相似文献
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