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. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Polynomial Spline, Classical Cubic Spline, Interpolation, Undesirable Oscillations
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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
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
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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Download@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}
}
TY - JOUR 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 SP - 269 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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