Progressive Iterative Approximation for Extended Cubic Uniform B-Splines with Shape Parameters
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Progressive Iterative Approximation for Extended Cubic Uniform B-Splines with Shape Parameters Yeqing Yi1 · Lijuan Hu1 · Chengzhi Liu2
· Shen Liu2 · Fangyu Luo2
Received: 3 February 2020 / Revised: 20 August 2020 / Accepted: 5 October 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we concern with the data interpolation by using extended cubic uniform B-splines with shape parameters. Two iterative formats, namely the progressive iterative approximation (PIA) and the weighted progressive iterative approximation (WPIA), are proposed to interpolate given data points. We study the optimal shape parameter and the optimal weight for the proposed methods by solving the eigenvalues of the collocation matrix. The optimal shape parameter can make the iterative methods not only have the fastest convergence speed but also have smallest initial interpolation error. Numerical experiments are given to illustrate the effectiveness of the proposed methods. Keywords Progressive iterative approximation · Extended cubic uniform B-spline · Shape parameter · Convergence rate · Spectral radius Mathematics Subject Classification 65D05 · 65D10 · 65D17 · 65F10
1 Introduction We consider the classical interpolation problem by using spline. Given blending basis n to be interpolated whose interpolation functions {b j (t)}nj=0 and a set of data { pi }i=0
Communicated by Theodore E. Simos.
B
Chengzhi Liu [email protected] Yeqing Yi [email protected]
1
School of Information, Hunan University of Humanities, Science and Technology, Loudi 417000, China
2
School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi 417000, China
123
Y. Yi et al. n . Very often, these interpolation nodes satisfy t < t < . . . < t . We nodes are {ti }i=0 0 1 n n , i.e., seek a curve γ (t) that interpolates { pi }i=0
γ (ti ) =
n
q j b j (ti ) = pi ,
j=0
where {q j }nj=0 are control points need to be determined, which can be computed by solving the matrix Eq. B Q = P,
(1)
where Q = [q 0 , q 1 , . . . , q n ]T , P = [ p0 , p1 , . . . , pn ]T and B is the so-called collocation matrix given by B = b j (ti ) , i, j = 0, 1, . . . , n. In linear algebra, B is also called generalized Vandermonde matrix. If the coefficient matrix B is nonsingular, the matrix Eq. (1) can be solved uniquely by direct solvers such as Neville elimination [3,15,16] etc., or iteration methods such as Jacobi iteration, Gauss-Sield iteration, GMRES [3,15,16], etc. Recently, a geometric iterative method named progressive iterative approximation (PIA) are frequently used in data interpolation/approximation due to its clear geometry meaning, stable convergence and simple iteration format. The PIA starts with an initial approximate interpolant whose control points are the ones needed to be interpolated, and then approximates the corresponding exact interpolant iteratively. For more details about PIA, readers can refer to two recent survey papers [6,8], in which the authors summar
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