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New differential-recurrence relations for B-spline basis functions are given. Using these relations, a recursive method for finding the Bernstein-Bézier coefficients of B-spline basis functions over a single knot span is proposed. The algorithm works for any knot sequence and has an asymptotically optimal computational complexity. Numerical experiments show that the new method gives results which preserve a high number of digits when compared to an approach which uses the well-known de Boor-Cox formula. • New differential-recurrence relations for B-spline basis functions are given. • A method for finding the Bézier forms of B-splines over one knot span is given. • The algorithm works for any knot sequence and has an optimal complexity. • Numerical experiments show that the new method preserves a high number of digits. [ABSTRACT FROM AUTHOR]

Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis, and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of these polynomials and their linear combinations. New simple recurrence relations of low order satisfied by dual Bernstein polynomials are given. In particular, a first-order non-homogeneous recurrence relation linking dual Bernstein and shifted Jacobi orthogonal polynomials has been obtained. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all n + 1 dual Bernstein polynomials of degree n with O(n) computational complexity. [ABSTRACT FROM AUTHOR]

New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees. • New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. • They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. • Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. • The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees. [ABSTRACT FROM AUTHOR]

New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation satisfied by dual Bernstein polynomials has been constructed. Also, a fourth-order recurrence relation for these polynomials has been obtained; this result may be useful in the efficient solution of some computational problems. [ABSTRACT FROM AUTHOR]

In a recent paper (Appl. Math. Comput. 215:1622-1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel approach to the problem of merging of Bézier curves with respect to the L 2 -norm. We give illustrative examples to show that the solution of the conventional merging problem may not be suitable for further modification and applications. As in the case of the degree reduction problem, we apply the so-called restricted area approach–proposed recently in Gospodarczyk (2015)–to avoid certain defects and make the resulting curve more useful. A method of solving the new problem is based on box-constrained quadratic programming approach. [ABSTRACT FROM AUTHOR]

A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein–Bézier form is derived. Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. In other cases, the complexity is increased by at most O (m 3). When the Bernstein–Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. This algorithm scales well when evaluating the B-spline curve at multiple points, e.g., in order to render it, since one only needs to find the coefficients for each knot span once. When evaluating many B-spline curves at multiple points (as is the case when rendering tensor product B-spline surfaces), such approach has lower computational complexity than using the de Boor–Cox algorithm. The numerical tests show that the new method is efficient. The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly. [Display omitted] • A new differential-recurrence relation for the B-spline functions of the same degree • A fast method of finding the Bernstein–Bézier coefficients of B-spline functions • The evaluation of B-spline curves and tensor product surfaces is accelerated • The method is numerically stable • A simplified case of uniform B-splines is included [ABSTRACT FROM AUTHOR]

This paper deals with the merging problem of segments of a composite Bézier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (Woźny and Lewanowicz, 2009) [12]to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods. [ABSTRACT FROM AUTHOR]

Abstract: Using new properties of dual functions, we show that the method of constructing the dual bases recently presented in [P. Woźny, Construction of dual bases, Journal of Computational and Applied Mathematics 245 (2013) 75–85] can be significantly simplified. Next, we propose a simple algorithm of constructing the dual -spline functions, which can be useful in solving some approximation problems related to numerical analysis or computer graphics. [Copyright &y& Elsevier]

Abstract: Fast and efficient methods of evaluation of the connection coefficients between shifted Jacobi and Bernstein polynomials are proposed. The complexity of the algorithms is , where n denotes the degree of the Bernstein basis. Given results can be helpful in a computer aided geometric design, e.g., in the optimization of some methods of the degree reduction of Bézier curves. [Copyright &y& Elsevier]

Abstract: Let be the sets of linearly independent functions. We give a simple method of construction, the dual functions satisfying the following conditions: and , where , for , and is a given inner product. The proposed algorithm allows us to construct all the sets of the dual functions in the time , where is a natural number. Four illustrative examples presenting the possible applications of obtained results are given. [Copyright &y& Elsevier]

Abstract: We give structure relations for the orthogonal polynomials in two variables, defined by Lewanowicz and Woźny [S. Lewanowicz, P. Woźny, J. Comput. Appl. Math. 233 (2010) 1554–1561] where , and are univariate big q-Jacobi polynomials. We discuss in full detail the case of the polynomials , which are closely related to Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, SIAM J. Algebra. Discr. Methods 1 (1980) 137–151]. [Copyright &y& Elsevier]

Abstract: Simple recurrence relations for the Bézier coefficients of the dual Bernstein polynomials are given. Using the recurrence relations, we propose fast methods for computing these coefficients. The complexity of the proposed algorithms is , where n is the degree of the considered dual Bernstein polynomial basis. [Copyright &y& Elsevier]

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed. [Copyright &y& Elsevier]

Abstract: We propose an efficient approach to the problem of multi-degree reduction of rectangular Bézier patches, with prescribed boundary control points. We observe that the solution can be given in terms of constrained bivariate dual Bernstein polynomials. The complexity of the method is with m ≔min(m 1, m 2), where (n 1, n 2) and (m 1, m 2) is the degree of the input and output Bézier surface, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined rectangular Bézier surfaces, the result is a composite surface of global C r continuity with a prescribed r ⩾0. In the detailed discussion, we restrict ourselves to r ∈{0,1}, which is the most important case in practical application. Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

Abstract: The transformation, introduced recently by Woźny and Nowak, may serve as a good tool for summation of slowly convergence series. This approach can be easily applied to the case of generalized or basic hypergeometric series, as well as some orthogonal polynomial expansions. It is closely related to the famous Wynn''s epsilon algorithm, Weniger''s or Homeier''s transformations, and the method introduced by Čížek, Zamastil and Skála. However, it is difficult to use the algorithm proposed by Woźny and Nowak in the general case, because of its high complexity, and some other restrictions. We propose another realization of the transformation, which results in obtaining a simpler and faster algorithm. Four illustrative numerical examples are given. [Copyright &y& Elsevier]

Abstract: We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is , and being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global continuity with a prescribed order . Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

Abstract: We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller . To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates. [Copyright &y& Elsevier]

Abstract: A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type. The polynomials, which are expressed in terms of univariate big q-Jacobi polynomials, form an extension of Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, Orthogonal polynomials in two variables of -Hahn and -Jacobi type, SIAM J. Algebr. Discrete Methods 1 (1980) 137–151]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation. [Copyright &y& Elsevier]

Abstract: A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189–371] or the technique recently introduced by Čížek et al. [J. Čížek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962–968]. In the case of trigonometric series, our method is very similar to the Homeier’s transformation, while in the case of orthogonal series — to the transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given. [Copyright &y& Elsevier]

Abstract: We present a novel approach to the problem of multi-degree reduction of Bézier curves with constraints, using the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product. We give properties of these polynomials, including the explicit orthogonal representations, and the degree elevation formula. We show that the coefficients of the latter formula can be expressed in terms of dual discrete Bernstein polynomials. This result plays a crucial role in the presented algorithm for multi-degree reduction of Bézier curves with constraints. If the input and output curves are of degree n and m, respectively, the complexity of the method is , which seems to be significantly less than complexity of most known algorithms. Examples are given, showing the effectiveness of the algorithm. [Copyright &y& Elsevier]

Abstract: Connection coefficients between the two-variable Bernstein and Jacobi polynomial families on the triangle are given explicitly as evaluations of two-variable Hahn polynomials. Dual two-variable Bernstein polynomials are introduced. Explicit formula in terms of two-variable Jacobi polynomials and a recurrence relation are given. [Copyright &y& Elsevier]

Abstract: The generalized Bernstein basis in the space of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78], We give explicitly the dual basis functions for the polynomials , in terms of big q-Jacobi polynomials , a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial as a linear combination of big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by , as well as an identity which may be seen as an analogue of the extended Marsden''s identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346]. [Copyright &y& Elsevier]

A new algorithm for computing a point on a polynomial or rational curve in Bézier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational complexity is linear with respect to the number of control points and its memory complexity is O (1). Some remarks on similar methods for surfaces in rectangular and triangular Bézier form are also given. • A new algorithm for computing a point on a polynomial or rational curve in Bézier form is proposed. • The method has a geometric interpretation and uses only convex combinations of control points. • The new algorithm's computational complexity is linear with respect to the number of control points and its memory complexity is O (1). • Some remarks on similar methods for surfaces in rectangular and triangular Bézier are also given. [ABSTRACT FROM AUTHOR]

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L -solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier 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. [ABSTRACT FROM AUTHOR]

We propose a novel approach to the problem of polynomial approximation of rational Bézier 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. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of multi-degree reduction of a composite Bézier curve with the parametric continuity constraints at the endpoints of the segments. We present a novel method which is based on the idea of using constrained dual Bernstein polynomials to compute the control points of the reduced composite curve. In contrast to other methods, ours minimizes the L 2 -error for the whole composite curve instead of minimizing the L 2 -errors for each segment separately. As a result, an additional optimization is possible. Examples show that the new method gives much better results than multiple application of the degree reduction of a single Bézier curve. [ABSTRACT FROM AUTHOR]

We present a new approach to the problem of G-constrained ( k, l ≤ 3) multi-degree reduction of Bézier curves with respect to the least squares norm. First, to minimize the least squares error, we consider two methods of determining the values of geometric continuity parameters. One of them is based on quadratic and nonlinear programming, while the other uses some simplifying assumptions and solves a system of linear equations. Next, for prescribed values of these parameters, we obtain control points of the multi-degree reduced curve, using the properties of constrained dual Bernstein basis polynomials. Assuming that the input and output curves are of degree n and m, respectively, we determine these points with the complexity O( m n), which is significantly less than the cost of other known methods. Finally, we give several examples to demonstrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]