Robustness analysis of polynomials with polynomial parameterdependency using Bernstein expansion

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     

基于约束Jacobi基的多项式反函数逼近及应用

求解多项式反函数是CAGD中的一个基本问题.提出一种带端点Ck约束的反函数逼近算法.利用约束Jacobi基作为有效工具, 推导了它与Bernstein基的转换公式,采用Bernstein多项式的升阶、乘积、积分与组合运算, 给出了求解反函数系数的具体算法.该算法稳定、简易, 克服了以往计算反函数的系数时每次逼近系数需全部重新计算的缺陷.最后通过具体逼近实例验证了文中算法的正确性和有效性, 同时给出了它在PH曲线准弧长参数化中的应用.     

Optimal computation of the Bernstein algorithm for the bound of an interval polynomial

If a polynomial is expanded in terms of Bernstein polynomial over an interval then the coefficients of the expansion may be used to provide upper and lower bounds for the value of the polynomial over the interval. When applying this method to interval polynomials straightforwardly, the coefficients of the expansion are computed with an increase in width due to dependency intervals. In this paper we show that if the computations are rearranged suitably then the Bernstein coefficients can be computed with no increase in width due to dependency intervals.     

Inclusion Isotonicity of Convex–Concave Extensions for Polynomials Based on Bernstein Expansion

In this paper the expansion of a polynomial into Bernstein polynomials over an interval I is considered. The convex hull of the control points associated with the coefficients of this expansion encloses the graph of the polynomial over I. By a simple proof it is shown that this convex hull is inclusion isotonic, i.e. if one shrinks I then the convex hull of the control points on the smaller interval is contained in the convex hull of the control points on I. From this property it follows that the so-called Bernstein form is inclusion isotone, which was shown by a longish proof in 1995 in this journal by Hong and Stahl. Inclusion isotonicity also holds for multivariate polynomials on boxes. Examples are presented which document that two simpler enclosures based on only a few control points are in general not inclusion isotonic. Received September 12, 2002; revised February 5, 2003 Published online: April 7, 2003     

Integral and non-negativity preserving Bernstein-type polynomial approximations

In this paper, we consider the problem of approximating a function by Bernstein-type polynomials that preserve the integral and non-negativity of the original function on the interval [0, 1], obtaining the Kantorovich–Bernstein polynomials, but providing a novel approach with advantages in numerical analysis. We then develop a Markov finite approximation method based on piecewise Bernstein-type polynomials for the computation of stationary densities of Markov operators, providing numerical results for piecewise constant and piecewise linear algorithms.     

Solving nonlinear polynomial systems in the barycentric Bernstein basis

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.     

On the computing time of the continued fractions method

The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm.     

Approximate GCDs of polynomials and sparse SOS relaxations

The problem of computing approximate GCDs of several polynomials with real or complex coefficients can be formulated as computing the minimal perturbation such that the perturbed polynomials have an exact GCD of given degree. We present algorithms based on SOS (Sums Of Squares) relaxations for solving the involved polynomial or rational function optimization problems with or without constraints.     

Structural Interval Reliability Algorithm Based on Bernstein Polynomials andEvidence Theory

Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors. Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid. The uncertain parameters mainly exist in the form of intervals. This method requires a lot of calculation and is often difficult to achieve efficiently. In order to solve this problem, this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty. Based on the non-probabilistic reliability index method, the extreme value of the limit state function is obtained using the properties of Bernstein polynomials, thus avoiding the need for a lot of sampling to solve the reliability analysis problem. The method is applied to numerical examples and engineering applications such as experiments, and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method, especially for some reliability problems with higher nonlinearity. Moreover, this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.     

Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n?10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.     

The range of values of a complex polynomial over a complex interval

We discuss algorithms for the computation of the range of values of a complex interval polynomial over a complex interval. The mathematical results needed are based upon a result by Rivlin [7] valid for the range of values of a complex polynomial over the line segment [0, 1]. In the present work we extend his results to an arbitrary line segment in the complex plane. Based upon these results we then generate algorithms suitable for both line segments and rectangular regions in the complex plane executed in rectangular complex interval arithmetic. The algorithms are then tested on a variety of complex interval polynomials and compared both to the true range of values and to the values obtained by the Horner scheme.     

Eliciting dependability requirements: a control cases based approach

At present,great demands are posed on software dependability.But how to elicit the dependability requirements is still a challenging task.This paper proposes a novel approach to address this issue.The essential idea is to model a dependable software system as a feedforward-feedback control system,and presents the use cases+control cases model to express the requirements of the dependable software systems.In this model,while the use cases are adopted to model the functional requirements,two kinds of control cases(namely the feedforward control cases and the feedback control cases)are designed to model the dependability requirements.The use cases+control cases model provides a unified framework to integrate the modeling of the functional requirements and the dependability requirements at a high abstract level.To guide the elicitation of the dependability requirements,a HAZOP based process is also designed.A case study is conducted to illustrate the feasibility of the proposed approach.     

Dual bases of a Bernstein polynomial basis on simplices

This paper extends the theory of dual bases of a one-variate B-spline basis to the case of Bernstein polynomials and presents methods of constructing dual bases of a Bernstein polynomial basis on a simplex from that of a one-variate polynomial basis, s{xⁱs}ⁿ_i=0 and s{B^k_i,k-i(x) = (^k_i)xⁱ(l-x)^k-i}s^k_i=0 on the interval [0,1].     

Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) point of view. An original method for estimating distribution functions and densities with Bernstein polynomials is proposed, which takes advantage of results about the eigenstructure of the Bernstein operator to refine a convergence acceleration method. Furthermore, an original data-driven method for choosing the degree of the polynomial is worked out. The method is successfully applied to two data-sets which are important benchmarks in the field of Density Estimation.     

Bernstein多项式的快速复合算法

在计算机辅助几何设计中,Bernstein多项式的复合是一个重要的研究课题。目前,实现复合的方法主要有Blossoming算法和优化的Blossoming算法。这类方法虽然是数值稳定的,但是计算量很大,存储空间和程序复杂性方面也要求较高,文中基于多项式插值和符号运算,提出了一种新的复合算法。理论分析表明,新算法不但保持了数值稳定性,而且在计算量,存储空间和程序复杂性方面明显优于已有算法。     

Vergröberung von Intervallpolynomen

This paper deals with two new methods for reducing the degree of interval polynomials, which are based on Tchebychev approximation and the telescoping procedure. Compared with the well-known algorithms of Krückenberg [5] and Rokne [10], in some cases these methods give considerably better inclusions for the interval polynomials the degree of which is to be reduced.     

Analysis of nonlinear electrical circuits using bernstein polynomials

In electrical circuit analysis, it is often necessary to find the set of all direct current (d.c.) operating points (either voltages or currents) of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.     

Necessary conditions for Hadamard factorizations of Hurwitz polynomials

A property of Hurwitz polynomials is related with the Hadamard product. Garloff and Wagner proved that Hadamard products of Hurwitz polynomials are Hurwitz polynomials, and Garloff and Shrinivasan shown that there are Hurwitz polynomials of degree 4 which do not have a Hadamard factorization into two Hurwitz polynomials of the same degree 4. In this paper, we give necessary conditions for an even-degree Hurwitz polynomial to have a Hadamard factorization into two even-degree Hurwitz polynomials; such conditions are given in terms of the coefficients of the given polynomial alone. Furthermore, we show that if an odd-degree Hurwitz polynomial has a Hadamard factorization then a system of nonlinear inequalities has at least one solution.     

Idempotent and tropical mathematics; complexity of algorithms and interval analysis

A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics, constructions and algorithms are usually simpler than their traditional analogs. We examine in particular algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other “good” operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent semirings are examined. In this case, basic algorithms, like their interval versions, are polynomial. This situation is very different from the traditional linear algebra case, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.     

Globally convergent autocalibration using interval analysis

We address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters. Existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed, in general. To address this problem, we have developed a method where an interval branch-and-bound method is employed for numerical minimization. Thanks to the properties of interval analysis this method converges to the global solution with mathematical certainty and arbitrary accuracy and the only input information it requires from the user are a set of point correspondences and a search interval. The cost function is based on the Huang-Faugeras constraint of the essential matrix. A recently proposed interval extension based on Bernstein polynomial forms has been investigated to speed up the search for the solution. Finally, experimental results are presented.