The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula

Yuxin Zhou; Jun Zhang; Yufeng Diao

doi:doi:10.11648/j.acm.20221105.11

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The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula

Yuxin Zhou, Jun Zhang, Yufeng Diao

Published in Applied and Computational Mathematics (Volume 11, Issue 5)

Received: 7 September 2022 Accepted: 5 October 2022 Published: 11 October 2022

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Abstract

In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentiable and fourth-order differentiable respectively, these conditions limit the wide application of the formula. For this reason, recent relevant documents have studied the error estimation of trapezoidal formula and parabolic formula under the condition that the integrand has a continuous first derivative in the integral interval except for the most limited points, but sometimes the integral integrand of practical problems can be derived almost everywhere, and the breakpoints between its derivatives are countable. In this paper, the unified integral formula format and its complex quadrature formula of two classical quadrature formulas are constructed firstly, and then appropriately relaxed the limiting conditions of the integrand function, under the condition that the integral interval is almost everywhere differentiable and the non-differentiable points are the first kind of discontinuities. Finally, the error estimation of the quadrature formula is studied. The research results weaken the restrictions of the integrand, thus expand the conditions for the use of the complex trapezoidal quadrature formula and the complex parabolic quadrature formula, and modify and improve the existing literature results.

Published in	Applied and Computational Mathematics (Volume 11, Issue 5)
DOI	10.11648/j.acm.20221105.11
Page(s)	116-122
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), 2024. Published by Science Publishing Group

Keywords

Numerical Integration, Trapezoid Formula, Parabola Formula, Quadrature Formula, Error Formula

References

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[4]	Pearce C E M, Pečarić J. Inequalities for differentiable mappings with application to special Means and quadrature formulae [J]. Appl. Math. Lett., 2000, 13 (2): 51-55.
[5]	Kirmaci U S. Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula [J]. Appl. Math. Comp., 2004, 147: 137-146.
[6]	LIU Zheng, DING Gui-Yan. Some approximate calculations of definite integral on the error estimates. Journal of Anshan University of Science and Technology, 2003, 26 (4): 313-317.
[7]	LAI Zhizhu, ZHANG Yunyan. Solving numerical integration based on genetic algorithms. Computer Engineering and Applications, 2014, 50 (2): 54-57.
[8]	LAI Zhizhu. Solving numerical integration based on shuffled frog leaping algorithm. Journal of Liupanshui Normal University, 2013, 25 (2): 77-80.
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[10]	ZHENG Xiaoyang. Several New Numerical Solutions of Integral Equation and Differential Equation Based on Wavelet [D]. Chongqing University, 2011.
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[15]	WEN Weibin. Study of Numerical Manifold Method and Time Integration Method based on B-spline Interpolation [D]. Chongqing University, 2014.
[16]	CHEN Yating, JI Degang, JIA Li, ZHANG Yajing. Numerical Integration Formula by Second Derivative [J]. Journal of Hebei University (Natural Science Edition), 2014, 34 (04): 16-19.

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APA Style

Yuxin Zhou, Jun Zhang, Yufeng Diao. (2022). The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula. Applied and Computational Mathematics, 11(5), 116-122. https://doi.org/10.11648/j.acm.20221105.11

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ACS Style

Yuxin Zhou; Jun Zhang; Yufeng Diao. The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula. Appl. Comput. Math. 2022, 11(5), 116-122. doi: 10.11648/j.acm.20221105.11

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AMA Style

Yuxin Zhou, Jun Zhang, Yufeng Diao. The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula. Appl Comput Math. 2022;11(5):116-122. doi: 10.11648/j.acm.20221105.11

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@article{10.11648/j.acm.20221105.11,
  author = {Yuxin Zhou and Jun Zhang and Yufeng Diao},
  title = {The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {5},
  pages = {116-122},
  doi = {10.11648/j.acm.20221105.11},
  url = {https://doi.org/10.11648/j.acm.20221105.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221105.11},
  abstract = {In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentiable and fourth-order differentiable respectively, these conditions limit the wide application of the formula. For this reason, recent relevant documents have studied the error estimation of trapezoidal formula and parabolic formula under the condition that the integrand has a continuous first derivative in the integral interval except for the most limited points, but sometimes the integral integrand of practical problems can be derived almost everywhere, and the breakpoints between its derivatives are countable. In this paper, the unified integral formula format and its complex quadrature formula of two classical quadrature formulas are constructed firstly, and then appropriately relaxed the limiting conditions of the integrand function, under the condition that the integral interval is almost everywhere differentiable and the non-differentiable points are the first kind of discontinuities. Finally, the error estimation of the quadrature formula is studied. The research results weaken the restrictions of the integrand, thus expand the conditions for the use of the complex trapezoidal quadrature formula and the complex parabolic quadrature formula, and modify and improve the existing literature results.},
 year = {2022}
}

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TY  - JOUR
T1  - The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula
AU  - Yuxin Zhou
AU  - Jun Zhang
AU  - Yufeng Diao
Y1  - 2022/10/11
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221105.11
DO  - 10.11648/j.acm.20221105.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 116
EP  - 122
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221105.11
AB  - In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentiable and fourth-order differentiable respectively, these conditions limit the wide application of the formula. For this reason, recent relevant documents have studied the error estimation of trapezoidal formula and parabolic formula under the condition that the integrand has a continuous first derivative in the integral interval except for the most limited points, but sometimes the integral integrand of practical problems can be derived almost everywhere, and the breakpoints between its derivatives are countable. In this paper, the unified integral formula format and its complex quadrature formula of two classical quadrature formulas are constructed firstly, and then appropriately relaxed the limiting conditions of the integrand function, under the condition that the integral interval is almost everywhere differentiable and the non-differentiable points are the first kind of discontinuities. Finally, the error estimation of the quadrature formula is studied. The research results weaken the restrictions of the integrand, thus expand the conditions for the use of the complex trapezoidal quadrature formula and the complex parabolic quadrature formula, and modify and improve the existing literature results.
VL  - 11
IS  - 5
ER  -

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Author Information

Yuxin Zhou

College of Computer Science and Technology, Inner Mongolia Minzu University, Tongliao, China
Jun Zhang

College of Computer Science and Technology, Inner Mongolia Minzu University, Tongliao, China
Yufeng Diao

College of Computer Science and Technology, Inner Mongolia Minzu University, Tongliao, China

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Plain Text BibTeX RIS

APA Style

Yuxin Zhou, Jun Zhang, Yufeng Diao. (2022). The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula. Applied and Computational Mathematics, 11(5), 116-122. https://doi.org/10.11648/j.acm.20221105.11

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ACS Style

Yuxin Zhou; Jun Zhang; Yufeng Diao. The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula. Appl. Comput. Math. 2022, 11(5), 116-122. doi: 10.11648/j.acm.20221105.11

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AMA Style

Yuxin Zhou, Jun Zhang, Yufeng Diao. The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula. Appl Comput Math. 2022;11(5):116-122. doi: 10.11648/j.acm.20221105.11

Copy | Download

@article{10.11648/j.acm.20221105.11,
  author = {Yuxin Zhou and Jun Zhang and Yufeng Diao},
  title = {The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {5},
  pages = {116-122},
  doi = {10.11648/j.acm.20221105.11},
  url = {https://doi.org/10.11648/j.acm.20221105.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221105.11},
  abstract = {In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentiable and fourth-order differentiable respectively, these conditions limit the wide application of the formula. For this reason, recent relevant documents have studied the error estimation of trapezoidal formula and parabolic formula under the condition that the integrand has a continuous first derivative in the integral interval except for the most limited points, but sometimes the integral integrand of practical problems can be derived almost everywhere, and the breakpoints between its derivatives are countable. In this paper, the unified integral formula format and its complex quadrature formula of two classical quadrature formulas are constructed firstly, and then appropriately relaxed the limiting conditions of the integrand function, under the condition that the integral interval is almost everywhere differentiable and the non-differentiable points are the first kind of discontinuities. Finally, the error estimation of the quadrature formula is studied. The research results weaken the restrictions of the integrand, thus expand the conditions for the use of the complex trapezoidal quadrature formula and the complex parabolic quadrature formula, and modify and improve the existing literature results.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - The Unified Format of Trapezoid and Parabola Quadrature Formula and Its Complex Formula
AU  - Yuxin Zhou
AU  - Jun Zhang
AU  - Yufeng Diao
Y1  - 2022/10/11
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221105.11
DO  - 10.11648/j.acm.20221105.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 116
EP  - 122
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221105.11
AB  - In numerical integration, classical trapezoidal formula and parabolic formula play an important role in the theory and application of numerical integration, but trapezoidal formula and parabolic formula are relatively independent quadrature formulas, and the reasoning of error formula requires that the integrand function be second-order differentiable and fourth-order differentiable respectively, these conditions limit the wide application of the formula. For this reason, recent relevant documents have studied the error estimation of trapezoidal formula and parabolic formula under the condition that the integrand has a continuous first derivative in the integral interval except for the most limited points, but sometimes the integral integrand of practical problems can be derived almost everywhere, and the breakpoints between its derivatives are countable. In this paper, the unified integral formula format and its complex quadrature formula of two classical quadrature formulas are constructed firstly, and then appropriately relaxed the limiting conditions of the integrand function, under the condition that the integral interval is almost everywhere differentiable and the non-differentiable points are the first kind of discontinuities. Finally, the error estimation of the quadrature formula is studied. The research results weaken the restrictions of the integrand, thus expand the conditions for the use of the complex trapezoidal quadrature formula and the complex parabolic quadrature formula, and modify and improve the existing literature results.
VL  - 11
IS  - 5
ER  -

Copy | Download