A new iterative multi-step method for solving nonlinear equation
This study introduces an advanced iterative technique designed to solve nonlinear equations with simple roots efficiently. The newly developed algorithm achieves an impressive convergence order of sixteen, utilizing only five functional evaluations per iteration. By incorporating appropriate finite...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-12-01
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| Series: | MethodsX |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2215016125002407 |
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| author | Muhammad Usman Javed Iqbal Alamgir Khan Ikram Ullah Hasib Khan Jehad Alzabut Hisham Mohammad Alkhawar |
| author_facet | Muhammad Usman Javed Iqbal Alamgir Khan Ikram Ullah Hasib Khan Jehad Alzabut Hisham Mohammad Alkhawar |
| author_sort | Muhammad Usman |
| collection | DOAJ |
| description | This study introduces an advanced iterative technique designed to solve nonlinear equations with simple roots efficiently. The newly developed algorithm achieves an impressive convergence order of sixteen, utilizing only five functional evaluations per iteration. By incorporating appropriate finite difference approximations, the method avoids the need for second derivatives, enhancing its computational efficiency and broad applicability. A detailed theoretical analysis of convergence is provided, highlighting the method's superior performance. Furthermore, numerical experiments are carried out to assess its reliability and effectiveness against established methods. The findings reveal that the proposed technique surpasses several renowned iterative methods in both accuracy and computational efficiency. • High-order convergence: The method demonstrates a sixteenth-order convergence rate, requiring just five function evaluations per iteration, which enhances computational efficiency. • Derivative-free approach: The algorithm employs finite difference approximations, eliminating the need for second derivative calculations and increasing its applicability to complex problems. • Numerical validation: Comprehensive numerical tests and comparative studies validate the exceptional accuracy and efficiency of the proposed iterative approach. |
| format | Article |
| id | doaj-art-85c903d9ab024f6e9f41a6b80487b9f0 |
| institution | Matheson Library |
| issn | 2215-0161 |
| language | English |
| publishDate | 2025-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | MethodsX |
| spelling | doaj-art-85c903d9ab024f6e9f41a6b80487b9f02025-06-28T05:30:11ZengElsevierMethodsX2215-01612025-12-0115103394A new iterative multi-step method for solving nonlinear equationMuhammad Usman0Javed Iqbal1Alamgir Khan2Ikram Ullah3Hasib Khan4Jehad Alzabut5Hisham Mohammad Alkhawar6Department of Mathematics, Government Postgraduate College Mardan, 23200 Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Abdul Wali Khan University Mardan, KPK, PakistanSchool of Mathematics and Statistics, Central South University, Changsha 410083, Hunan PR China; Corresponding authors.School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan PR China; Corresponding authors.Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia; Department of Mathematics, Shaheed Benazir Bhutto Uniersity, Sheringal, Dir Upper, 18000 Khyber Pakhtunkhwa, PakistanDepartment of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia; Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, SIMATS, Chennai, India; Department of Industrial Engineering, OSTIM Technical University, 06374 Ankara, TürkiyePreparatory Year Program, Computer Department, Prince Sultan University 11586 Riyadh, Saudi ArabiaThis study introduces an advanced iterative technique designed to solve nonlinear equations with simple roots efficiently. The newly developed algorithm achieves an impressive convergence order of sixteen, utilizing only five functional evaluations per iteration. By incorporating appropriate finite difference approximations, the method avoids the need for second derivatives, enhancing its computational efficiency and broad applicability. A detailed theoretical analysis of convergence is provided, highlighting the method's superior performance. Furthermore, numerical experiments are carried out to assess its reliability and effectiveness against established methods. The findings reveal that the proposed technique surpasses several renowned iterative methods in both accuracy and computational efficiency. • High-order convergence: The method demonstrates a sixteenth-order convergence rate, requiring just five function evaluations per iteration, which enhances computational efficiency. • Derivative-free approach: The algorithm employs finite difference approximations, eliminating the need for second derivative calculations and increasing its applicability to complex problems. • Numerical validation: Comprehensive numerical tests and comparative studies validate the exceptional accuracy and efficiency of the proposed iterative approach.http://www.sciencedirect.com/science/article/pii/S2215016125002407A New Multi-Step Method for Non Linear Equation |
| spellingShingle | Muhammad Usman Javed Iqbal Alamgir Khan Ikram Ullah Hasib Khan Jehad Alzabut Hisham Mohammad Alkhawar A new iterative multi-step method for solving nonlinear equation MethodsX A New Multi-Step Method for Non Linear Equation |
| title | A new iterative multi-step method for solving nonlinear equation |
| title_full | A new iterative multi-step method for solving nonlinear equation |
| title_fullStr | A new iterative multi-step method for solving nonlinear equation |
| title_full_unstemmed | A new iterative multi-step method for solving nonlinear equation |
| title_short | A new iterative multi-step method for solving nonlinear equation |
| title_sort | new iterative multi step method for solving nonlinear equation |
| topic | A New Multi-Step Method for Non Linear Equation |
| url | http://www.sciencedirect.com/science/article/pii/S2215016125002407 |
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