Application Of Lupa(?) Q-bernstein Operators In Approximation And Geometric Computation

Barycentric rational interpolation is a research hotspot for high efficiency and good numerical stability.Lupa(?) q-Bernstein operator is a generalization of Bernstein operator involving q-integers with well approximation and conformal.The operator is directly used to construct the interpolation nodes of barycentric rational interpolation,and the basis functions from the operator can also be used to construct Lupa(?) q-B(?)zier curves.In this paper,new interpolation nodes are constructed based on Lupa(?) q-Bernstein operator and Lupa(?) q-Bernstein operator reparameterized.The approximation property for the Berrut's rational at the nodes is studied.Simultaneously,a new de Casteljau algorithm with explicit matrix representation of Lupa(?) q-B(?)zier curves is construted by means of Pascal-type formula.The main research results include the f'ollowing aspects:Firstly,regular distribution function sequence is defined.The upper bound of' the Lebesgue constant of Berrut's rational interpolation at the interpolation nodes based on the regular distribution function sequence is discussed.It is proved that the Lebesgue constants of the Brrrmt's rational interpolation at the two sets of interpolation nodes satisfying the inverse symmetry are equal.By using the relationship between equidistant nodes and q-equidistant nodes,q-logarithmic regular distribution function sequence is constructed.It is proved that the q-logarithmic regular distribution nodes based on the distribution function sequence are well-spaced nodes.The upper bound of the Lebesgue constant of Berrut's rational interpolation at the nodes is obtained.By the numerical experiments,we compare the Lebesgue constant of Berrut's rational interpolation at q-logarithmic regular distribution nodes and logarithmic nodes,and there exsit q and z such that the Lebesgue constant of Berrut's rational interpolation at q-logarithmic regular distribution nodes is smaller.Then,the Lupa(?) q-Bernstein operator is combined with the theory of regular dis-tribution function sequence,and the operator is applied to constructing interpolation nodes.The three classes of interpolation nodes,Lupa(?) regular distribution nodes,Lupa(?) symmetric regular distribution nodes and Lupa(?) q-symmetric regular distribution nodes,are based on the Lupa(?) q-Bernstein operator and the Lupa(?) q-Bernstein operator repa-rameterized.It is proved that three kinds of nodes are well-spaced nodes.From the point of the Lebesgue constant,we proved the Lebesgue constants of Berrut's rational interpolation at the three kinds of nodes are logarithmic about the number of nodes.By the numerical experiments,we compare the Lebesgue constant of Berrut's rational interpolation at the classes of nodes and equidistant points.Under certain condition-s,the Lebesgue constant of Berrut's rational interpolation at Lupa(?) symmetric regular distribution nodes and Lupa(?) q-symmetric regular distribution nodes are smaller.Finally,to obtain Lupa(?) q-B(?)zier curves by recursive evaluation algorithms with bet?ter properties,new de Casteljau algorithms and Lupa(?) q-B(?)zier curves with symmetry are constructed by means of Pascal-type formula and reparameterization.A new de Castel-jau algorithm with explicit matrix representation is constructed by applying Pascal-type formula,and the algorithm shares three properties with de Casteljau algorithm of clas-sical B(?)zier curves.Lupa(?) q-Bernstein basis functions and Lupa(?) q-B(?)zier curves with symmetry are gained from reparameterization.Moreover,Lupa(?) q-B(?)zier curves repa-rameterized can be generated by multiply bidiagonal matrices successively on control polygon.In addition,numerical examples of using one Lupa(?) q-B(?)zier curve to approx-imate two blending B(?)zier curves are presented as a simple application of de Casteljau algorithm with explicit matrix representation and the effectiveness of the algorithm is verified.