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Vol. 11, No. 3, August 2026 (Article in Progress)

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Accuracy Comparison of Multivariate Newton-Raphson, Newton-Kantorovich, and Levenberg–Marquardt Methods for Solving Nonlinear Systems Using Numerical Simulation

https://doi.org/10.22219/kinetik.v11i3.2603
Syaharuddin Syaharuddin
Universitas Muhammadiyah Mataram
Hendi Hidayah
Universitas Muhammadiyah Mataram
Vera Mandailina
Universitas Muhammadiyah Mataram
Saba Mehmood
University of Management and Technology
Wasim Raza
Universidade Federal Do Rio de Janeiro

Corresponding Author(s) : Syaharuddin Syaharuddin

syaharuddin.ntb@gmail.com

Kinetik: Game Technology, Information System, Computer Network, Computing, Electronics, and Control, Vol. 11, No. 3, August 2026 (Article in Progress)

Abstract

Multivariable nonlinear equation systems often appear in engineering, physics, economics, and artificial intelligence modeling, but often do not have closed analytical solutions. Therefore, accurate, efficient, and stable numerical methods are needed. This study aims to comparatively evaluate three iterative methods, namely Multivariate Newton-Raphson, Newton-Kantorovich, and Levenberg–Marquardt, in solving identical high-complexity multivariable nonlinear systems. Simulations were performed using MATLAB with an error tolerance of 0.001 and a maximum iteration limit of 100. The test system consisted of a combination of trigonometric, exponential, and polynomial functions, resulting in nonlinear interactions that were challenging for each method. The simulation results show that Levenberg–Marquardt excelled with only 6 iterations and a final error of 3.246 × 10⁻¹⁰, indicating high stability and efficiency, followed by Multivariate Newton-Raphson with 13 iterations and an error of 4.606 × 10⁻⁹, while Newton-Kantorovich requires 27 iterations with an error of 5.770 × 10⁻⁷, reflecting slower semi-local corrections.Three-dimensional visualization shows the intersection point of the surface as a solution, providing an intuitive understanding of the iteration trajectory characteristics of each method. The novelty of this research lies in the integrated numerical simulation framework that allows direct quantitative comparison of the three methods on identical systems with the same initial conditions, tolerance, and iteration limits. These findings provide important empirical references for selecting efficient and stable iterative methods for multivariable nonlinear systems, as well as practical guidance for numerical applications in engineering, physics, and scientific computing.

Keywords

Numerical Methods Multivariate Newton Rapshon Newton Kontrovich Levenberg-Marquardt Nonlinear Equation Systems
Syaharuddin, S., Hidayah, H., Mandailina, V., Mehmood, S., & Raza, W. (2026). Accuracy Comparison of Multivariate Newton-Raphson, Newton-Kantorovich, and Levenberg–Marquardt Methods for Solving Nonlinear Systems Using Numerical Simulation. Kinetik: Game Technology, Information System, Computer Network, Computing, Electronics, and Control, 11(3). https://doi.org/10.22219/kinetik.v11i3.2603
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