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Progressive Iterative Approximation for Extended Cubic Uniform B-Splines with Shape Parameters

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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.

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Notes

  1. A matrix is called totally positive if all its minors are positive. A basis \(\{b_{i}(t)\}_{i=0}^{n}\) is called normalized if \(b_{i}(t)\ge 0\) and \(\sum \limits _{i=0}^{n}b_{i}(t)=1\); totally positive if its collocation matrix at any increasing sequence is a totally positive matrix.

  2. A matrix is called stochastic if it is nonnegative and the sum of the entries of each row equals to 1.

  3. A matrix is called Toeplitz if its entries along the diagonal are constant.

References

  1. Carnicer, J., Delgado, J.: On the progressive iterative approximation property and alternative iterations. Comput. Aided Geom. Des. 28, 523–526 (2011)

    Article  Google Scholar 

  2. Carnicer, J., Delgado, J., Peńa, J.: Richardson method and totally nonnegative linear systems. Linear Algebra Appl. 11, 2010–2017 (2010)

    Article  MathSciNet  Google Scholar 

  3. Demmel, J., Koev, P.: The accurate and efficient solution of a totally positive generalized Vandermonde. Linear Syst. SIAM J. Matrix Anal. Appl. 27, 142–152 (2005)

    Article  MathSciNet  Google Scholar 

  4. Deng, S., Wang, G.: Numerical analysis of the progressive iterative approximation method. Comput. Aided Des. Graph. 7, 879–884 (2012)

    Google Scholar 

  5. Ebrahimi, A., Loghmani, G.: A composite iterative procedure with fast convergence rate for the progressive-iteration approximation of curves. J. Comput. Appl. Math. 359, 1–15 (2019)

    Article  MathSciNet  Google Scholar 

  6. Lin, H.: Survey on geometric iterative methods with applications. J. Comput.-Aided Des. Comput. Graph. 27, 583–589 (2015)

    Google Scholar 

  7. Lin, H., Bao, H., Wang, G.: Totally positive bases and progressive iterative approximation. Comput. Math. Appl. 50, 575–586 (2005)

    Article  MathSciNet  Google Scholar 

  8. Lin, H., Maekawa, T., Deng, C.: Survey on geometric iterative methods and their applications. Comput.-Aided Des. 95, 40–51 (2018)

    Article  Google Scholar 

  9. Liu, C., Han, X., Li, J.: The Chebyshev accelerating method for progressive iterative approximation. Commun. Inf. Syst. 17, 25–43 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Liu, C., Han, X., Li, J.: Progressive-iterative approximation by extension of cubic uniform B-spline curves. J. Comput.-Aided Des. Comput. Graph. 31, 899–910 (2019)

    Google Scholar 

  11. Liu, C., Xin, S., Shu, Z., Chen, S., Zhang, R., Zhao, J.: Accelerated geometric iteration method. J. Comput.-Aided Des. Comput. Graph. 28, 1838–1843 (2016)

    Google Scholar 

  12. Liu, X., Deng, C.: Jacobi-PIA algorithm for non-uniform cubic B-spline curve interpolation. J. Comput.-Aided Des. Comput. Graph. 27, 485–491 (2015a)

    Google Scholar 

  13. Liu, X., Deng, C.: Non-uniform cubic B-spline curve interpolation algorithm of GS-PIA. J Hangzhou Dianzi Univ Nat Sci 35, 79–82 (2015b)

    Google Scholar 

  14. Lu, L.: Weighted progressive iterative approximation and convergence analysis. Comput. Aided Geom. Des. 2, 129–137 (2010)

    Article  Google Scholar 

  15. Marco, A., Martínez, J.: A fast and accurate algorithm for solving Bernstein–Vandermonde linear systems. Linear Algebra Appl. 2, 616–628 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Marco, A., Martínez, J.: Accurate computations with totally positive Bernstein–Vandermonde matrices. Electron. J. Linear Algebra 26, 357–380 (2013)

    Article  MathSciNet  Google Scholar 

  17. Varga, R.S.: Matrix iterative analysis. Springer, Berlin (2000)

    Book  Google Scholar 

  18. Wang, Z., Li, J., Deng, C.: Convergence proof of GS-PIA algorithm. J. Comput.-Aided Des. Comput. Graph. 30, 60–66 (2018)

    Google Scholar 

  19. Yan, L., Han, X.: The extended cubic uniform B-spline curve based on totally positive basis. J. Graph. 37, 329–336 (2016)

    Google Scholar 

  20. Zhang, L., Tan, J., Ge, X., Guo, Z.: Generalized B-splines’ geometric iterative fitting method with mutually different weights. J. Comput. Appl. Math. 329, 331–343 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The research is supported by National Natural Science Foundation of China (Grant No. 11771453), Natural Science Foundation of Hunan Province (Grant No. 2020JJ5267), Scientific Research Funds of Hunan Provincial Education Department (Grant No. 18C877).

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Authors and Affiliations

  1. School of Information, Hunan University of Humanities, Science and Technology, Loudi, 417000, China

    Yeqing Yi & Lijuan Hu

  2. School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi, 417000, China

    Chengzhi Liu, Shen Liu & Fangyu Luo

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  1. Yeqing Yi
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  2. Lijuan Hu
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  3. Chengzhi Liu
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  4. Shen Liu
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  5. Fangyu Luo
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Correspondence to Chengzhi Liu.

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Communicated by Theodore E. Simos.

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Yi, Y., Hu, L., Liu, C. et al. Progressive Iterative Approximation for Extended Cubic Uniform B-Splines with Shape Parameters. Bull. Malays. Math. Sci. Soc. 44, 1813–1836 (2021). https://doi.org/10.1007/s40840-020-01034-2

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  • DOI: https://doi.org/10.1007/s40840-020-01034-2

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