Shape Preserving Third and Fifth Degrees Polynomial Splines

Vladimir Ivanovich Pinchukov

doi:doi:10.11648/j.ajam.20140205.13

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Shape Preserving Third and Fifth Degrees Polynomial Splines

Vladimir Ivanovich Pinchukov

Published in American Journal of Applied Mathematics (Volume 2, Issue 5)

Received: 22 August 2014 Accepted: 12 September 2014 Published: 30 September 2014

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Abstract

This paper is devoted to the development of positivity and monotonisity preserving linear spline techniques, namely, techniques which are based on ideas applied in the field of high order TVD (Total Variation Diminishing) methods for numerical solving compressible flow equations. Third and fifth degrees polynomial splines are constructed. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for “good” data. These splines get shape preserviation at the cost of reducing smoothness till C^1. To restore C^2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative. These C^2fifth degree polynomial splines are observed to preserve monotonisity or positivity for all considered data with these properties.

Published in	American Journal of Applied Mathematics (Volume 2, Issue 5)
DOI	10.11648/j.ajam.20140205.13
Page(s)	162-169
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), 2014. Published by Science Publishing Group

Keywords

Classical Cubic Spline, Interpolation, Shape Preserving, Positivity, Monotonisity, Polynomial Functions

References

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[2]	S. Butt, and K. W. Brodlie, Preserving positivity using piecewise cubic interpolation, Computers and Graphics, 17(1), 1993, 55-64.
[3]	D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice–Hall, Englewood Cliffs, NJ, 1989, 27.
[4]	M. Abbas, J. M. Ali, A. A. Majid, Positivity Preserving Interpolation of Positive Data by Cubic Trigonometric Spline. MATEMATIKA, Vol. 27, N. 1, 2011, 41–50.
[5]	Xi-An. Han, Yi M. Chen, and X. Huang, The cubic trigonometric Bezier curve with two shape parameters. Applied Math. Letters. 2009. 22: 226–231.
[6]	L. L. Schumaker, Constructive aspects of discrete polynomial spline functions. Approximation Theory. G.G. Lorentz (ed.). Akademic Press. New York. 1973, 469-476.
[7]	S. de Boor, Splines as linear combinations of B-splines: A survey. Approximation Theory II. G.G. Lorentz, C.K. Chui, and L.L. Schumaker (Eds.) Academic Press. New York. 1976, 1-47.
[8]	Q. Sun, F. Bao, Q. Duan, Shape-preserving Weighted Rational Cubic Interpolation, J. of Computational Information Systems. 8:18, 2012, 7721-7728.
[9]	M. Shrivastava, J. Joseph, C2 rational cubic spline involving tensor parameters. Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, N. 3, 2000, 305-314.
[10]	R. Delbourgo, J. A. Gregory, C2 rational quadratic spline interpolation to monotonic data. IMA J. Numer. Anal. Vol. 5, 1983, 141-152.
[11]	H. Akima, A new method of interpolation and smooth curve fitting based on local procedures, J Assoc. Comput. Machinery. Vol. 17, 1970, 589-602.
[12]	N. S. Sapidis, P. D. Kaklis, An algorithm for constructing convexity and monotonicity pre- serving splines in tensions // Comput. Aided Geometric Design. Vol. 5, 1988, 127-137.
[13]	V. I. Pinchukov, Monotonic global cubic spline, J. of Comput. & Mathem. Phys., Vol. 41, N. 2, 2001, 200-206. (Russian)
[14]	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.

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Vladimir Ivanovich Pinchukov. (2014). Shape Preserving Third and Fifth Degrees Polynomial Splines. American Journal of Applied Mathematics, 2(5), 162-169. https://doi.org/10.11648/j.ajam.20140205.13

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ACS Style

Vladimir Ivanovich Pinchukov. Shape Preserving Third and Fifth Degrees Polynomial Splines. Am. J. Appl. Math. 2014, 2(5), 162-169. doi: 10.11648/j.ajam.20140205.13

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AMA Style

Vladimir Ivanovich Pinchukov. Shape Preserving Third and Fifth Degrees Polynomial Splines. Am J Appl Math. 2014;2(5):162-169. doi: 10.11648/j.ajam.20140205.13

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@article{10.11648/j.ajam.20140205.13,
  author = {Vladimir Ivanovich Pinchukov},
  title = {Shape Preserving Third and Fifth Degrees Polynomial Splines},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {5},
  pages = {162-169},
  doi = {10.11648/j.ajam.20140205.13},
  url = {https://doi.org/10.11648/j.ajam.20140205.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140205.13},
  abstract = {This paper is devoted to the development of positivity and monotonisity preserving linear spline techniques, namely, techniques which are based on ideas applied in the field of high order TVD (Total Variation Diminishing) methods for numerical solving compressible flow equations. Third and fifth degrees polynomial splines are constructed. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for “good” data. These splines get shape preserviation at the cost of reducing smoothness till C^1. To restore C^2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative. These C^2fifth degree polynomial splines are observed to preserve monotonisity or positivity for all considered data with these properties.},
 year = {2014}
}

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TY  - JOUR
T1  - Shape Preserving Third and Fifth Degrees Polynomial Splines
AU  - Vladimir Ivanovich Pinchukov
Y1  - 2014/09/30
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140205.13
DO  - 10.11648/j.ajam.20140205.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 162
EP  - 169
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140205.13
AB  - This paper is devoted to the development of positivity and monotonisity preserving linear spline techniques, namely, techniques which are based on ideas applied in the field of high order TVD (Total Variation Diminishing) methods for numerical solving compressible flow equations. Third and fifth degrees polynomial splines are constructed. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for “good” data. These splines get shape preserviation at the cost of reducing smoothness till C^1. To restore C^2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative. These C^2fifth degree polynomial splines are observed to preserve monotonisity or positivity for all considered data with these properties.
VL  - 2
IS  - 5
ER  -

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Author Information

Vladimir Ivanovich Pinchukov

Siberian division of Russian Academy of Sc., In-te of Computational Technologies, Novosibirsk, 630090, Russia

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APA Style

Vladimir Ivanovich Pinchukov. (2014). Shape Preserving Third and Fifth Degrees Polynomial Splines. American Journal of Applied Mathematics, 2(5), 162-169. https://doi.org/10.11648/j.ajam.20140205.13

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ACS Style

Vladimir Ivanovich Pinchukov. Shape Preserving Third and Fifth Degrees Polynomial Splines. Am. J. Appl. Math. 2014, 2(5), 162-169. doi: 10.11648/j.ajam.20140205.13

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AMA Style

Vladimir Ivanovich Pinchukov. Shape Preserving Third and Fifth Degrees Polynomial Splines. Am J Appl Math. 2014;2(5):162-169. doi: 10.11648/j.ajam.20140205.13

Copy | Download

@article{10.11648/j.ajam.20140205.13,
  author = {Vladimir Ivanovich Pinchukov},
  title = {Shape Preserving Third and Fifth Degrees Polynomial Splines},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {5},
  pages = {162-169},
  doi = {10.11648/j.ajam.20140205.13},
  url = {https://doi.org/10.11648/j.ajam.20140205.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140205.13},
  abstract = {This paper is devoted to the development of positivity and monotonisity preserving linear spline techniques, namely, techniques which are based on ideas applied in the field of high order TVD (Total Variation Diminishing) methods for numerical solving compressible flow equations. Third and fifth degrees polynomial splines are constructed. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for “good” data. These splines get shape preserviation at the cost of reducing smoothness till C^1. To restore C^2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative. These C^2fifth degree polynomial splines are observed to preserve monotonisity or positivity for all considered data with these properties.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Shape Preserving Third and Fifth Degrees Polynomial Splines
AU  - Vladimir Ivanovich Pinchukov
Y1  - 2014/09/30
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140205.13
DO  - 10.11648/j.ajam.20140205.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 162
EP  - 169
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140205.13
AB  - This paper is devoted to the development of positivity and monotonisity preserving linear spline techniques, namely, techniques which are based on ideas applied in the field of high order TVD (Total Variation Diminishing) methods for numerical solving compressible flow equations. Third and fifth degrees polynomial splines are constructed. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for “good” data. These splines get shape preserviation at the cost of reducing smoothness till C^1. To restore C^2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative. These C^2fifth degree polynomial splines are observed to preserve monotonisity or positivity for all considered data with these properties.
VL  - 2
IS  - 5
ER  -

Copy | Download