Algorithms of predictor-corrector type with convergence and stability analysis for solving nonlinear systems

Dalal Khalid Almutairi; Ioannis K. Argyros; Krzysztof Gdawiec; Sania Qureshi; Amanullah Soomro; Khalid H. Jamali; Marwan Alquran; Asifa Tassaddiq; Dalal Khalid Almutairi; Ioannis K. Argyros; Krzysztof Gdawiec; Sania Qureshi; Amanullah Soomro; Khalid H. Jamali; Marwan Alquran; Asifa Tassaddiq

doi:10.3934/math.20241538

AIMS Mathematics

2024, Volume 9, Issue 11: 32014-32044. doi: 10.3934/math.20241538

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Research article

Algorithms of predictor-corrector type with convergence and stability analysis for solving nonlinear systems

1.
Department of Mathematics, College of Science, Al-Zulfi Majmaah University, Al-Majmaah, 11952, Saudi Arabia
2.
Department of Computing and Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
3.
Institute of Computer Science, University of Silesia, Bedzinska 39, 41-200, Sosnowiec, Poland
4.
Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon
5.
Department of Mathematics, Near East University, 99138, Mersin, Turkey
6.
Department of Basic Sciences and Related Studies, Mehran University of Engineering & Technology, Jamshoro – 76062, Pakistan
7.
Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box (3030), Irbid 22110, Jordan, Jordan
8.
Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al-Majmaah, 11952, Saudi Arabia

Received: 15 August 2024 Revised: 30 September 2024 Accepted: 25 October 2024 Published: 11 November 2024
MSC : 65H04, 65H05, 26C10, 30C15

Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature.
- local convergence,
- zeros,
- efficiency index,
- Newton's algorithm,
- polynomiography,
- system of non-linear equations
Citation: Dalal Khalid Almutairi, Ioannis K. Argyros, Krzysztof Gdawiec, Sania Qureshi, Amanullah Soomro, Khalid H. Jamali, Marwan Alquran, Asifa Tassaddiq. Algorithms of predictor-corrector type with convergence and stability analysis for solving nonlinear systems[J]. AIMS Mathematics, 2024, 9(11): 32014-32044. doi: 10.3934/math.20241538

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Abstract

Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature.

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