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), 2022. Published by Science Publishing Group |
Numerical Integration, Trapezoid Formula, Parabola Formula, Quadrature Formula, Error Formula
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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
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
@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} }
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 -