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Some New Results Concerning the Classical Bernstein Cubature Formula S S symmetry Article Some New Results Concerning the Classical Bernstein Cubature Formula Dan Micl˘au¸s Department of Mathematics and Computer Science, Technical University of Cluj-Napoca, North University Center at Baia Mare, Victoriei 76, 430122 Baia Mare, Romania; [email protected] or [email protected] Abstract: In this article, we present a solution to the approximation problem of the volume obtained by the integration of a bivariate function on any finite interval [a, b] × [c, d], as well as on any symmetrical finite interval [−a, a] × [−a, a] when a double integral cannot be computed exactly. The approximation of various double integrals is done by cubature formulas. We propose a cubature formula constructed on the base of the classical bivariate Bernstein operator. As a valuable tool to approximate any volume resulted by integration of a bivariate function, we use the classical Bernstein cubature formula. Numerical examples are given to increase the validity of the theoretical aspects. Keywords: Bernstein operator; quadrature formula; cubature formula; upper bound estimation MSC: 41A36; 41A80; 65D32 1. Introduction Citation: Micl˘au¸s,D. Some New Let N be the set of positive integers and N0 = N [ f0g. Recent studies concerning the Results Concerning the Classical classical Bernstein operators associated with any real-valued function F : [a, b] ! R, where Bernstein Cubature Formula. a, b are two real and finite numbers, such that a < b can be found in a series of research Symmetry 2021, 13, 1068. https:// papers, see [1–12], which shows that there is sustained interest in the study of the classical doi.org/10.3390/sym13061068 linear positive operators. These operators are given by Academic Editors: Ana-Maria Acu, 1 n n k(b − a) Carmen Violeta Muraru, Marius B (F; x) = (x − a)k(b − x)n−k F a + (1) n ( − )n ∑ Birou and Radu Voichita Adriana b a k=0 k n Received: 5 May 2021 for any x 2 [a, b] and any n 2 N. If a := 0 and b := 1 in (1), then Accepted: 10 June 2021 n n Published: 15 June 2021 k n k n−k k Bn( f ; x) = ∑ pn,k(x) f n = ∑ (k)x (1 − x) f n , (2) k=0 k=0 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in which are the well-known Bernstein operators [13] associated with any real-valued function published maps and institutional affil- f : [0, 1] ! R, any x 2 [0, 1], and any n 2 N. The Bernstein polynomials (2) opened a new iations. era in the approximation theory beginning with Sergei Natanovich Bernstein’s famous proof of the Weierstrass approximation theorem in 1912, and continuing until today with thousands of interesting papers. Some generalizations (approximation of integrable func- tions, approximation of measurable functions, Bernstein polynomials on an unbounded interval, degenerate Bernstein polynomials), as well as many other applications of the Bern- Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. stein polynomials, can be consulted in the excellent book [14]. An exceptional historical This article is an open access article perspective is provided in [6] on the introduction and evolution of the Bernstein polynomi- distributed under the terms and als. Constructed as an important tool in approximation of functions by polynomials, they conditions of the Creative Commons became in time more useful in exploiting computers to interactively design polynomial Attribution (CC BY) license (https:// functions. In this context, the Bernstein polynomial basis provides valuable insight into its creativecommons.org/licenses/by/ behavior over a certain finite interval, yielding many useful properties and elegant algo- 4.0/). rithms that are now being increasingly adopted in other application domains. In the real Symmetry 2021, 13, 1068. https://doi.org/10.3390/sym13061068 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1068 2 of 16 daily applications, different situations arise where it is necessary to solve certain definite integrals, much more complicated than those presented in the courses of mathematical analysis. In order to solve such integrals in an efficient way, numerous numerical methods called quadrature formulas have been developed. The most known method of numerical integration is obtained by integrating the Lagrange interpolation formula resulting in the class of Newton–Cotés quadrature formulas. Engineers and researchers use numerical integration as a primary tool to get approximate values for definite integrals that cannot be solved analytically. The quadrature formulas based on the linear positive Bernstein type operators represented a theme of intensive study for the distinguished mathematician Dimitrie D. Stancu. His contributions in this area were collected in the second volume of the monograph “Numerical Analysis and Approximation Theory ”[15]. Looking deep at this problem of numerical integration based on the Bernstein polynomial, we remarked that it is not fully researched. In [10], we managed to assign a place on the map of the closed Newton–Cotés quadrature formulas to a new approximation formula based on the classical Bernstein polynomials. In order to increase the validity of the theoretical aspects, we tested a series of numerical examples. By approximation of definite integrals on some intervals [a, b], we observed that the resulting error is proportional with the interval length. Therefore, if the interval [a, b] is large, then it is advisable to divide [a, b] into smaller inter- vals and use the classical Bernstein quadrature formula on each subinterval. By adding the results, we get the approximation of the definite integral on the whole interval [a, b]. Such a scheme is called the classical composite Bernstein quadrature formula, and it can be found in [11]. Although effective in most situations, there are instances when the composite quadrature formulas cannot be applied, as they use equally-spaced nodes. To avoid this fact, we constructed an adaptive method in [10]. All the mentioned results were established in the univariate case. In natural logic, we can continue our study in the bivariate case. Our aim is to compute the definite integral (on any finite bivariate interval [a, b] × [c, d] as well as on any symmetrical finite bivariate interval [−a, a] × [−a, a]) of a function depending on two variables. For instance, let us consider the following two functions: 2 2 F1 : [−5, 5] × [−5, 5] ! R, F1(x, y) = x + y + cos x + sin y, −(x2+y2) F2 : [−10, 10] × [−10, 10] ! R, F2(x, y) = e , having the graphs (see Figure1 for F1 and Figure2 for F2) given below. Figure 1. The graph of F1. Symmetry 2021, 13, 1068 3 of 16 Figure 2. The graph of F2. In general, the graph of a bivariate function F is a surface having the equation z = F(x, y) as we can see in Figure3. Therefore, the definite integral of a positive function with two variables is a double integral, which represents the volume of the solid between the surface of the function and the plane that contains its domain (see Figure3). Figure 3. The graph of a bidimensional function F. In this article, we present a solution to the approximation problem of the volume ob- tained by the integration of a bivariate function when a double integral cannot be computed exactly. The approximation of various double integrals can be done by a few cubature formulas (for instance, the Newton–Cotés cubature formulas) according to the specialty literature. Constructed by means of the bivariate Lagrange polynomial, trapezoidal and Simpson cubature formulas use a fixed number of nodes, resulting in a single possible approximation for a double integral. In order to be more flexible with this fact, we bring to light a cubature formula constructed on the base of the classical bivariate Bernstein operator. As a valuable tool to approximate any volume resulting by integration of a bivariate function, we use the classical Bernstein cubature formula. If the bivariate interval [a, b] × [c, d] (the bivariate symmetrical interval [−a, a] × [−a, a]) is large, then the classical composite Bernstein cubature formula is suitable for the approximation of a double integral. Numerical examples are given to increase the validity of the theoretical aspects. Symmetry 2021, 13, 1068 4 of 16 2. Auxiliary Results For any n 2 N, let a ≤ x0 < x1 < ··· < xn ≤ b be some distinct nodes, and let F : [a, b] ! R be a real-valued function. In many books on numerical analysis, divided differences are defined recursively 1 [x0; F] = F(x0),..., [x0, x1, ··· , xn; F] = [x1, ··· , xn; F] − [x0, ··· , xn−1; F] . xn − x0 If I, J are certain real intervals, F : I × J ! R is a real-valued function and (x0, y0), (x1, y1) 2 I × J (x0 6= x1, y0 6= y1), then the bivariate divided differences of F with respect to the nodes (x0, y0), (x1, y1) are defined in [16], using the method of parametric extensions by x , x F(x , y ) − F(x , y ) − F(x , y ) + F(x , y ) 0 1 ; F = 1 1 0 1 1 0 0 0 . y0, y1 (x1 − x0)(y1 − y0) Other equivalent definitions for univariate and bivariate divided differences can be found in the excellent monograph [17]. In the definition of the divided differences, the number of abscissas in general is not equal to the number of ordinates. If x0, x1, ... , xp 2 I and y0, y1,..., yq 2 J are distinct nodes, then the following recurrence formula x , x ,..., x 1 x ,..., x x ,..., x 0 1 p ; F = 1 p ; F − 0 p−1 ; F y0, y1,..., yq (xp − x0)(yq − y0) y1,..., yq y1,..., yq x ,..., x x ,..., x − 1 p ; F + 0 p−1 ; F y0,..., yq−1 y0,..., yq−1 holds (see [16]), for p, q 2 N, p, q ≥ 2 and " # x , x ,..., x x , x ,..., x 0 1 p ; F = i0 i1 ip ; F , (3) y0, y1,..., yq yj0 , yj1 ,..., yjq where (i0, i1, ..
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