Polynomials In Bernstein Form Research Articles

Approximation with One-Bit Polynomials in Bernstein Form

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from \(\{ \pm 1\}\) only. A basic case of our results states that for any Lipschitz function \(f:[0,1] \rightarrow [-1,1]\) and for any positive integer n, there are signs \(\sigma _0,\dots ,\sigma _n \in \{\pm 1\}\) such that $$\begin{aligned} \left| f(x) - \sum _{k=0}^n \sigma _k \, \left( {\begin{array}{c}n\\ k\end{array}}\right) x^k (1-x)^{n-k} \right| \le \frac{C (1+|f|_{\textrm{Lip}})}{1+\sqrt{nx(1-x)}} ~ \text{ for } \text{ all } x \in [0,1]. \end{aligned}$$More generally, we show that higher accuracy is achievable for smoother functions: For any integer \(s\ge 1\), if f has a Lipschitz \((s{-}1)\)st derivative, then approximation accuracy of order \(O(n^{-s/2})\) is achievable with coefficients in \(\{\pm 1\}\) provided \(\Vert f \Vert _\infty < 1\), and of order \(O(n^{-s})\) with unrestricted integer coefficients, both uniformly on closed subintervals of (0, 1) as above. Hence these polynomial approximations are not constrained by the saturation of classical Bernstein polynomials. Our approximations are constructive and can be implemented using feedforward neural networks whose weights are chosen from \(\{\pm 1\}\) only.

Parameter-Space Stability Analysis of LTI Time-Delay Systems With Parametric Uncertainties

A novel approach to stability analysis of linear time-invariant (LTI) time-delay systems of retarded type with incommensurate time delays and polynomial dependence on parametric uncertainties is presented. Using a branch and bound algorithm, which relies on Taylor models and polynomials in Bernstein form, we first determine the stability-crossing set in the delay/parameter space and then evaluate stability for each disjoint region. The novel approach can be used to either nonconservatively check stability of time-delay systems with interval parameters or to map stable regions to a low-dimensional parameter space while taking additional interval parameters into account. The algorithm is applied to several examples with parametric uncertainties and/or incommensurate time delays.

The symmetric equilibria of symmetric voter participation games with complete information

We characterize the symmetric Nash equilibria of the symmetric voter participation game with complete information from Palfrey and Rosenthal (1983). To do so, we use methods based on polynomials in Bernstein form to determine how the probability that a voter is pivotal depends on the participation probability and the number of players in the game.

Algorithm 960

The design and implementation of a Matlab object-oriented software library for working with polynomials is presented. The construction and evaluation of polynomials in Bernstein form are motivated and justified. Efficient constructions for the coefficients of a polynomial in Bernstein form when the polynomial is not given with this representation are provided. The presented adaptive evaluation algorithm uses the VS (Volk and Schumaker) algorithm, the de Casteljau algorithm, and a compensated VS algorithm. In addition, we have completed the library with other algorithms to perform other usual operations with polynomials in Bernstein form.

The Symmetric Equilibria of Symmetric Voter Participation Games with Complete Information

We characterize the symmetric Nash equilibria of the symmetric voter participation game with complete information introduced by Palfrey and Rosenthal (1983). To do so, we use methods based on polynomials in Bernstein form to determine how the probability that a voter is pivotal depends on the participation probability and the number of players in the game.

Gains from switching and evolutionary stability in multi-player matrix games

In this paper we unify, simplify, and extend previous work on the evolutionary dynamics of symmetric N-player matrix games with two pure strategies. In such games, gains from switching strategies depend, in general, on how many other individuals in the group play a given strategy. As a consequence, the gain function determining the gradient of selection can be a polynomial of degree N−1. In order to deal with the intricacy of the resulting evolutionary dynamics, we make use of the theory of polynomials in Bernstein form. This theory implies a tight link between the sign pattern of the gains from switching on the one hand and the number and stability of the rest points of the replicator dynamics on the other hand. While this relationship is a general one, it is most informative if gains from switching have at most two sign changes, as is the case for most multi-player matrix games considered in the literature. We demonstrate that previous results for public goods games are easily recovered and extended using this observation. Further examples illustrate how focusing on the sign pattern of the gains from switching obviates the need for a more involved analysis.

A surface reconstruction strategy based on deformable template for repairing damaged turbine blades

Recently, rapid repair of damaged blade has become the focus of considerable interest for extending its service life. However, due to the defects caused by high temperature and pressure of operations as well as foreign object impact, the turbine blades often undergo the deviations of the actual part profile from its design model, such that this nominal Computer Aided Design (CAD) model cannot be directly used in the process of repair for tool path generation of laser cladding and Numerical Control (NC) machining, thus to nicely repair the damaged or worn blades, it is necessary to reconstruct the surface model of the actual blade. This paper develops a deformable template-based approach to recovering the surface of blade from the cross-sectional profiles. The mathematical model for cross-sectional profile reconstruction is first established and is then solved by an alternate iteration optimization strategy consisting of registration and deformation of the template curve. Since the proposed method can automatically transform and deform the template curve to best fit the cross-sectional points, the compatibility conditions between different sections are automatically satisfied and there is no need for the data preprocessing such as data sorting, parameterization, etc. which are necessary for the traditional surface fitting methods. Undoubtedly, this considerably simplifies the reconstruction problem of the damaged blade and nicely adapts to blade part-to-part variation. Moreover, a method of closest point computation that combines the arithmetic for Bernstein-form polynomials and Bézier curve subdivision is also given based on bintree decomposition to improve the iteration processes of 2D profile reconstruction. Then, according to these reconstructed sectional profiles, the actual blade surface is reconstructed by surface skinning operations. Finally, the proposed method is tested on a sample blade, and the experimental results show that our method can precisely reconstruct the surface of the damaged blade, especially for the blades with area defects.

Automatic accurate surface reconstruction of a class of wrap-around models

For a wrap-around surface, many issues, such as difficulty in parameterisation of data points, incapability of automatic modelling, and high-accuracy surface reconstruction, need to be resolved. Therefore, an automatic accurate non-uniform rational B-spline (NURBS) surface reconstruction method is proposed in the present paper. First, after a method that recognises the central axis from data points is given, the data points in physical space are projected to a plane domain. Consequently, the boundary curve is extracted. Second, by local surface fitting of the data points on sector-distributed cross-sectional plane, the unorganised scatter points become ordered array points. A base surface is created using skinning technology. Finally, using robust arithmetic of the multivariate Bernstein-form polynomials, an algorithm for calculating the closest point is proposed. Using base-surface parameterisation, an accurate surface fitting is finally implemented based on least square iterative procedure. Examples verify the feasibility and validity of the proposed surface reconstruction method.

High-speed cornering by CNC machines under prescribed bounds on axis accelerations and toolpath contour error

To exactly execute a sharp corner in the toolpath, the feedrate of a CNC machine must instantaneously drop to zero at that point. This constraint is problematic in the context of high-speed machining, since it incurs very high deceleration/acceleration rates near sharp corners, which increase the total machining time, and may incur significant path deviations (contour errors) at these points. A strategy for negotiating sharp corners in high-speed machining is proposed herein, based upon a priori toolpath/feedrate modifications in their vicinity. Each corner is smoothed by replacing a subset of the path that contains it with a conic “splice” segment, deviating from the exact corner by no more than a prescribed tolerance ϵ, along which the square of the feedrate is specified as a Bernstein-form polynomial. The problem of determining the fastest traversal of the conic segments under known axis acceleration bounds can then be formulated as a constrained optimization problem, and by exploiting some well-known properties of Bernstein-form polynomials this can be approximated by a simple linear programming task. Some computed examples are presented to illustrate the implementation and performance of the high-speed cornering strategy.

A unified localization approach for machining allowance optimization of complex curved surfaces

The goal of workpiece localization is of interest to find the optimal Euclidean transformation that aligns the sampled points to the nominal CAD model to ensure sufficient stock allowance during the machining process. In this paper, a unified localization technique is developed for sculptured surface machining. This technique concerns an alignment process to satisfy a user-defined set of constraints for some specific surfaces where the machining allowance is preferentially guaranteed. The mathematical model of the constrained optimization alignment is firstly established, and is efficiently solved by a combination of the multipliers method and the BFGS algorithm to handle the large number of constraints in allowance optimization. To efficiently calculate the Euclidean oriented distance, a novel approach, which combines the robust arithmetic for multivariate Bernstein-form polynomials and Bezier surface segmentation algorithm, is presented based on recursive quadtree decomposition. Two typical sculptured surfaces are used to test the developed algorithm and comparisons between the proposed algorithm and the existing algorithms are given. Experiment results show that the proposed method is appropriate and feasible to distribute the stock allowance for proper sculptured surface machining.

Optimal localization of free-form shaped parts in precision inspection

Optimal localization of free-form shaped parts is a key issue in precision inspection. Aiming at the problems on estimate of the initial transformation and the calculation of closest points, an effective and exact localization method is presented. First, this method establishes the corresponding rela-tionships between two free-form surfaces with constraints of angle and distance by means of the surface features of the Gaussian curvature and the mean curvature. Then the rough localization is realized. Using the robust arithmetic for multi-variate Bernstein-form polynomials, a novel algorithm to cal-culate closest point is proposed. An improved iterative algo-rithm is used in the exact adjustment of localization to ensure the global optimization. Experiment results demonstrate the feasibility of the proposed method.

Jacobi polynomials in Bernstein form

The paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so-called Bernstein–Bézier form of a polynomial.

Bernstein–Bezoutian matrices

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 coefficients 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.

Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The orthogonal polynomials reduce to the usual Legendre polynomials along one edge of the domain triangle, and within each fixed degree are characterized by vanishing Bernstein coefficients on successive rows parallel to that edge. Closed-form expressions and recursive algorithms for computing the Bernstein coefficients of these orthogonal bivariate polynomials are derived, and their application to surface smoothing problems is sketched. Finally, an extension of the scheme to the construction of orthogonal bases for polynomials over higher-dimensional simplexes is also presented.

Algebraic manipulation in the Bernstein form made simple via convolutions

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.

Algorithm 812: BPOLY

The design, implementation, and testing of a C++ software library for univriate polynomials in Bernstein form is described. By invoking the class environment and operator overloading, each polynomial in an expression is interpreted as an object compatible with the arithmetic operations and other common functions (subdivision, degree, elevation, differentiation and integration, compoistion, greatest common divisor, real-root solving, etc.) for polynomials in Bernstein form. The library allows compact and intuitive implementation of lengthy manipulation of Bernstein-form polynomials, which often arise in computer graphics and computer-aided design and manufacturing applications. A series of empirical tests indicates that the library functions are typically very accurate and reliable, even for polynomials of surprisingly high degree.

Robust arithmetic for multivariate Bernstein-form polynomials

There are several ways to represent, to handle and to display curved surfaces in computer-aided geometric design that involve the use of polynomials. This paper deals with polynomials in the Bernstein form. Other work has shown that these polynomials are more numerically stable and robust than power-form polynomials. However, these advantages are lost if conversions to and from the customary power form are made. To avoid this, algebraic manipulations have to be done in the Bernstein basis. Farouki and Rajan (R.T. Farouki, V.T. Rajan, Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design 5 (1988) 1–26) present methods for doing arithmetic on univariate Bernstein-basis polynomials. This paper extends all polynomial arithmetic operations to multivariate Bernstein-form polynomials.

Convergent inversion approximations for polynomials in Bernstein form

Given a monotone polynomial function λ= f( t) in Bernstein form on t∈[ 0,1 ] , we formulate an algorithm that computes a sequence of rapidly convergent Bernstein-form polynomial approximations P n ( λ) to the inverse t= f −1( λ) of this function. The method is based upon a least-squares minimization of the error e n ( λ)= f −1( λ)− P n ( λ), using the orthogonal Legendre polynomials to yield an uncoupled sequential computation of the least-squares coefficients, and it requires only the standard arithmetic, degree elevation, and composition algorithms for polynomials in Bernstein form. An extension of the method to error minimization under the constraint of fixed end values is also presented. As an example application, we show that the algorithm can be used to derive representations for polynomial PH curves that come arbitrarily close to the ideal of arc-length parameterization.

Algorithms for polynomials in Bernstein form

Aspects of the formulation and numerical stability of geometric modeling algorithms in the Bernstein polynomial basis are further explored. Bernstein forms for various basic polynomial procedures (the arithmetic operations, substitution of polynomials, elimination of variables, determination of greatest common divisors, etc.) required in such algorithms are developed, and are found to be of similar complexity to their customary power forms. This establishes the viability of systematic computation with the Bernstein form, avoiding the need for (numerically unstable) basis conversions. Procedures which nominally improve root condition numbers—the power-to-Bernstein conversion and Bernstein degree elevation and subdivision techniques—are examined in greater detail. These procedures are found to have a relatively poor inherent conditioning, which essentially cancels the expected improvement, and their explicit use does not therefore provide a means of enhancing numerical stability. We also examine the condition of computation in floating point arithmetic of the power and Bernstein formulations for the basic polynomial procedures. However, the detailed dependence of the computational errors on the input parameters, the algorithm structure, and the floating point system preclude comparisons as definitive and general as those pertaining to the root condition numbers in the power and Bernstein bases. Empirical evaluation of carefully selected test cases may provide firmer conclusions in this regard.

On the numerical condition of polynomials in Bernstein form

The Bernstein-Bézier curve and surface forms have enjoyed considerable popularity in computer aided design applications, due to their elegant geometric properties and the simple recursive algorithms available for processing them. In this paper we describe a different and largely unexplored favorable aspect of Bernstein-Bézier methods: their inherent numerical stability under the influence of imprecise computer arithmetic or perturbed input data. We show that the condition numbers of the simple real roots of a polynomial on the unit interval are always smaller in the Bernstein basis than in the power basis. These arguments are then generalized to accommodate multiple roots and roots on an arbitrary interval. We further show that the standard Bernstein polynomial subdivision and degree elevation procedures induce a strictly monotonic decrease in root condition numbers. Finally, we prove that among a large family of polynomial bases characterized by certain simple properties, the Bernstein basis exhibits optimal root conditioning. Some well-known examples are employed to illustrate these results, and their practical implications for floating-point implementation of curve and surface intersection algorithms in a geometric modeling system are discussed. Important issues requiring further attention in the processing of polynomials in Bernstein form are also briefly indicated.