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Weighted Fifth Degree Polynomial Spline

Received: 9 November 2015     Accepted: 20 November 2015     Published: 22 December 2015
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Abstract

Global fifth degree polynomial spline is developed. Ideas applied in the field of high order WENO (Weighted Essentially non Oscillating) methods for numerical solving compressible flow equations are used to construct interpolation which has accuracy closed to accuracy of classical cubic spline for smooth interpolated functions, and which reduces undesirable oscillations often appearing in the case of data with break points. Fifth degree polynomial spline is constructed in two steps. Third degree spline is developed in first step by usage of additional stencils above three point central stencil, dealt in classical cubic splines. The Procedure of weights calculation provides choice of preferable stencils. Compensating terms are introduced to left side of governing equations for calculation of spline derivative knot values. This spline may be identical to classical cubic spline for “good” data. Damping of oscillations is achieved at the cost of reducing smoothness till C1. To restore C2 smoothness fifth degree terms are added to third degree polynomials in second step. These terms are chosen to provide continuity of the spline second derivative. Fifth degree polynomial spline is observed to produce lesser oscillations, then classical cubic spline applied to data with break points. These splines have nearly the same accuracy for smooth interpolated functions and sufficiently large knot numbers.

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 6)
DOI 10.11648/j.pamj.20150406.18
Page(s) 269-274
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Polynomial Spline, Classical Cubic Spline, Interpolation, Undesirable Oscillations

References
[1] S. A. A. Karim, Positivity preserving interpolation by using GC1 Rational cubic spline, Applied Mathematical Sciences, V. 8, N. 42, 2014, 2053-2065.
[2] F. Bao, J. Pan, Q. Sun, Q. Duan, A C2 rational interpolation for visualization of shaped data, J. of Information & Computational Sciences, 12:2, 2015, 815-824.
[3] M. Abbas, A.A. Majid, M.N.H. Awang, J.M. Ali, Constrained shape preserving rational b-cubic spline interpolation, World Applied Sciences J., V. 20, N. 6, 2012, 790-800.
[4] X. Han., Shape-preserving piecewise rational interpolant with quartic numerator and quadratic denominator, Applied Mathematics and Computation, V. 251, 2015, 258-274.
[5] Q. Sun, F. Bao, Q. Duan, Shape-preserving Weighted Rational Cubic Interpolation, J. of Computational Information Systems. 8:18, 2012, 7721-7728.
[6] M.Shrivastava, J. Joseph, C2 rational cubic spline involving tensor parameters. Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, N. 3, 2000, 305-314.
[7] R. Delbourgo, J. A. Gregory, C2 rational quadratic spline interpolation to monotonic data. IMA J. Numer. Anal. Vol. 5, 1983, 141-152.
[8] H. Akima, A new method of interpolation and smooth curve fitting based on local procedures, J Assoc. Comput. Machinery. Vol. 17, 1970, 589-602.
[9] N. S. Sapidis, P. D. Kaklis, An algorithm for constructing convexity and monotonicity preserving splines in tensions // Comput. Aided Geometric Design. Vol. 5, 1988, 127-137.
[10] A. A. Harten, A High resolution scheme for the computation of weak solutions of hyperbolic conservation laws J. Comput. Phys. Vol. 49, N. 3, 1983, 357-393.
[11] C.-W. Shu, S. Osher, Efficient implementation of Essentially non-oscillatory shock capturing schemes, J. of Computational Physics, V. 77, 1988, 439-471.
[12] X.-D. Liu, S. Osher, T. Cheng, Weighted Essentially non-oscillatory schemes, J. of Computational Physics, V. 115, 1994, 200-212.
[13] V. I. Pinchukov, Monotonic global cubic spline, J. of Comput. & Mathem. Phys., Vol. 41, N. 2, 2001, 200-206. (Russian).
[14] V. I. Pinchukov, Shape preserving third and fifth degrees polynomial splines. American J. of Applied Mathematics. Vol. 2, No. 5, 2014, pp. 162-169.
[15] V. I. Pinchukov, ENO and WENO Algorithms of Spline Interpolation, Vychislitelnye Technologii, Vol. 14, N. 4, 2009, 100-107. (Russian).
Cite This Article
  • APA Style

    Vladimir Ivanovich Pinchukov. (2015). Weighted Fifth Degree Polynomial Spline. Pure and Applied Mathematics Journal, 4(6), 269-274. https://doi.org/10.11648/j.pamj.20150406.18

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    ACS Style

    Vladimir Ivanovich Pinchukov. Weighted Fifth Degree Polynomial Spline. Pure Appl. Math. J. 2015, 4(6), 269-274. doi: 10.11648/j.pamj.20150406.18

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    AMA Style

    Vladimir Ivanovich Pinchukov. Weighted Fifth Degree Polynomial Spline. Pure Appl Math J. 2015;4(6):269-274. doi: 10.11648/j.pamj.20150406.18

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  • @article{10.11648/j.pamj.20150406.18,
      author = {Vladimir Ivanovich Pinchukov},
      title = {Weighted Fifth Degree Polynomial Spline},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {6},
      pages = {269-274},
      doi = {10.11648/j.pamj.20150406.18},
      url = {https://doi.org/10.11648/j.pamj.20150406.18},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150406.18},
      abstract = {Global fifth degree polynomial spline is developed. Ideas applied in the field of high order WENO (Weighted Essentially non Oscillating) methods for numerical solving compressible flow equations are used to construct interpolation which has accuracy closed to accuracy of classical cubic spline for smooth interpolated functions, and which reduces undesirable oscillations often appearing in the case of data with break points. Fifth degree polynomial spline is constructed in two steps. Third degree spline is developed in first step by usage of additional stencils above three point central stencil, dealt in classical cubic splines. The Procedure of weights calculation provides choice of preferable stencils. Compensating terms are introduced to left side of governing equations for calculation of spline derivative knot values. This spline may be identical to classical cubic spline for “good” data. Damping of oscillations is achieved at the cost of reducing smoothness till C1. To restore C2 smoothness fifth degree terms are added to third degree polynomials in second step. These terms are chosen to provide continuity of the spline second derivative. Fifth degree polynomial spline is observed to produce lesser oscillations, then classical cubic spline applied to data with break points. These splines have nearly the same accuracy for smooth interpolated functions and sufficiently large knot numbers.},
     year = {2015}
    }
    

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    T1  - Weighted Fifth Degree Polynomial Spline
    AU  - Vladimir Ivanovich Pinchukov
    Y1  - 2015/12/22
    PY  - 2015
    N1  - https://doi.org/10.11648/j.pamj.20150406.18
    DO  - 10.11648/j.pamj.20150406.18
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    EP  - 274
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20150406.18
    AB  - Global fifth degree polynomial spline is developed. Ideas applied in the field of high order WENO (Weighted Essentially non Oscillating) methods for numerical solving compressible flow equations are used to construct interpolation which has accuracy closed to accuracy of classical cubic spline for smooth interpolated functions, and which reduces undesirable oscillations often appearing in the case of data with break points. Fifth degree polynomial spline is constructed in two steps. Third degree spline is developed in first step by usage of additional stencils above three point central stencil, dealt in classical cubic splines. The Procedure of weights calculation provides choice of preferable stencils. Compensating terms are introduced to left side of governing equations for calculation of spline derivative knot values. This spline may be identical to classical cubic spline for “good” data. Damping of oscillations is achieved at the cost of reducing smoothness till C1. To restore C2 smoothness fifth degree terms are added to third degree polynomials in second step. These terms are chosen to provide continuity of the spline second derivative. Fifth degree polynomial spline is observed to produce lesser oscillations, then classical cubic spline applied to data with break points. These splines have nearly the same accuracy for smooth interpolated functions and sufficiently large knot numbers.
    VL  - 4
    IS  - 6
    ER  - 

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Author Information
  • Institute of Computational Technologies, Siberian Division of Russian Academy of Sc., Novosibirsk, Russia

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