Totally positive bases and progressive iteration approximation

Uniform approximation error estimates for weighted Shepard-type operators more flexible than the unweighted analogues are given. Error estimates for a linear combination of their iterates faster converging than previous ones are also showed. The results are applied in CAGD to construct Shepard-type curves useful in modeling and a weighted progressive iterative approximation technique exponentially converging.

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“…, n (cf. Amato and Della Vecchia 2016;Lin et al 2005). We have Corollary 1 Curve γ 1 1 satisfies the PIA property.…”

Section: Theorem 8 It Resultsmentioning

confidence: 84%

Weighting Shepard-type operators

Amato

Vecchia

2015

Comp. Appl. Math.

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“…Numerical results show that Algorithms 2.1 and 5.1 are better than B-spline method [17]. What's more, if we assume the coefficient matrix in [16] is B 1 , then B-spline method is convergent when the maximum eigenvalue of I − B 1 is less than 1, which will not always hold. But, our Algorithms 2.1 and 5.1 do not have this constraint.…”

Section: Numerical Resultsmentioning

confidence: 92%

“…In Table 3, we compare the relative error between the B-spline method in [16] and Algorithms 2.1 and 5.1. Numerical results show that Algorithms 2.1 and 5.1 are better than B-spline method [17].…”

Section: Numerical Resultsmentioning

confidence: 99%

Two efficient algorithms for surface construction

Tang

Jiang

2013

Applicable Analysis

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In this paper, we are concerned with two efficient algorithms for surface construction. Based on the Gauss equations, a discretized nonlinear equation of the form Ax + b(x) = g should be solved in the process of surface construction. We first consider a regularized fixed-point iterative algorithm for solving the discretized equation, in which we determine the actual regularization parameter by the Morozov discrepancy principle.Atwo-parameter algorithm is employed for solving the Morozov equation, and the convergence of the regularized fixed-point iterative algorithm is demonstrated. Secondly, we also propose the regularizing Levenberg-Marquardt scheme to solve the discretized equation, in which the regularization parameter is chosen to be a small constant. Numerical experiments are provided to demonstrate the robustness and efficiency of the two algorithms.

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“…Iterative fitting: The progressive-iterative approximation (PIA), proposed in Lin et al (2004Lin et al ( , 2005, is an iterative fitting algorithm with explicit geometric meaning for curves and surfaces with totally positive basis functions. In Shi and Wang (2006), the PIA algorithm was proven to be convergent for NURBS, and the convergence rate was accelerated in Lu (2010).…”

Section: Related Workmentioning

confidence: 99%