A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm
In this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish th...
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| Format: | Article |
| Language: | English |
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241701 |
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| author | Miguel Vivas-Cortez Usama Asif Muhammad Zakria Javed Muhammad Uzair Awan Yahya Almalki Omar Mutab Alsalami |
| author_facet | Miguel Vivas-Cortez Usama Asif Muhammad Zakria Javed Muhammad Uzair Awan Yahya Almalki Omar Mutab Alsalami |
| author_sort | Miguel Vivas-Cortez |
| collection | DOAJ |
| description | In this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish this task, we built an identity and demonstrated new inequalities. With the help of a new auxiliary result and some well-known ones, like Hölder's, the power mean, improved Hölder, improved power mean, convexity, and bounded features of the function, we obtained new bounds for Euler-Maclaurin's inequality. From an applicable perspective, we developed several intriguing applications of our results, which illustrated the relationship between the means of real numbers and the error bounds of quadrature schemes. We also included a graphical breakdown of our outcomes to demonstrate their validity. Additionally, we constructed a new iterative scheme for non-linear equations that is cubically convergent. Afterwards, we provided a comparative study between the proposed algorithm and standard methods. We also discussed the proposed algorithm's impact on the basins of attraction. |
| format | Article |
| id | doaj-art-ab302e7cf5f4438ba9ec412a19baf6b0 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-ab302e7cf5f4438ba9ec412a19baf6b02025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912358853590910.3934/math.20241701A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithmMiguel Vivas-Cortez0Usama Asif1Muhammad Zakria Javed2Muhammad Uzair Awan3Yahya Almalki4Omar Mutab Alsalami5Escuela de Ciencias Físicasy Matemáticas, Facultad de Ciencias Exactasy Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, EcuadorDepartment of Mathematics, Government College University, Faisalabad, PakistanDepartment of Mathematics, Government College University, Faisalabad, PakistanDepartment of Mathematics, Government College University, Faisalabad, PakistanDepartment of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi ArabiaDepartment of Electrical Engineering, College of Engineering, Taif University, P. O. Box 11099, Taif 21944, Saudi ArabiaIn this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish this task, we built an identity and demonstrated new inequalities. With the help of a new auxiliary result and some well-known ones, like Hölder's, the power mean, improved Hölder, improved power mean, convexity, and bounded features of the function, we obtained new bounds for Euler-Maclaurin's inequality. From an applicable perspective, we developed several intriguing applications of our results, which illustrated the relationship between the means of real numbers and the error bounds of quadrature schemes. We also included a graphical breakdown of our outcomes to demonstrate their validity. Additionally, we constructed a new iterative scheme for non-linear equations that is cubically convergent. Afterwards, we provided a comparative study between the proposed algorithm and standard methods. We also discussed the proposed algorithm's impact on the basins of attraction.https://www.aimspress.com/article/doi/10.3934/math.20241701convex functionssimspon's ruleeuler-maclaurin's inequalityhölder's inequalityiterative scheme |
| spellingShingle | Miguel Vivas-Cortez Usama Asif Muhammad Zakria Javed Muhammad Uzair Awan Yahya Almalki Omar Mutab Alsalami A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm AIMS Mathematics convex functions simspon's rule euler-maclaurin's inequality hölder's inequality iterative scheme |
| title | A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm |
| title_full | A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm |
| title_fullStr | A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm |
| title_full_unstemmed | A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm |
| title_short | A new approach to error inequalities: From Euler-Maclaurin bounds to cubically convergent algorithm |
| title_sort | new approach to error inequalities from euler maclaurin bounds to cubically convergent algorithm |
| topic | convex functions simspon's rule euler-maclaurin's inequality hölder's inequality iterative scheme |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241701 |
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