Ikhwanul Muslimin, Syaharuddin, Vera Mandailina, Saba Mehmood, Wasim Raza

PERFORMANCE EVALUATION OF NEWTON–KONTOROVICH AND ADAPTIVE NEWTON LINE SEARCH ON MULTIVARIATE NONLINEAR SYSTEMS

Introduction

Performance evaluation of newton–kontorovich and adaptive newton line search on multivariate nonlinear systems. Compare Newton–Kontorovich and Adaptive Newton Line Search for multivariate nonlinear systems. Discover which method offers superior convergence, accuracy, and efficiency for engineering and scientific applications.

Abstract

Solving multivariate nonlinear systems is essential in engineering, physics, and applied sciences. This study compares the performance of two numerical methods—Newton–Kontorovich and Interactive Newton–Raphson with Line Search—on trigonometric and exponential nonlinear systems. The methods are evaluated based on convergence rate, accuracy, and iteration efficiency through numerical simulations using MATLAB. The Newton–Kontorovich method, typically used for integral or differential equations, is compared with the adaptive line search strategy that enhances global convergence. Results show that the Interactive Newton–Raphson method achieves a smaller final error (5.95×10⁻²) with stable convergence, while Newton–Kontorovich converges in fewer iterations but with larger error (3.126). These findings highlight the superiority of adaptive strategies for complex nonlinear systems. Practical implications include improved numerical reliability for applications in structural engineering, optimization, and scientific modeling.

Review

This study tackles the pertinent problem of solving multivariate nonlinear systems, a foundational challenge across various scientific and engineering disciplines. The authors compare two numerical methods, Newton–Kontorovich (NK) and Adaptive Newton with Line Search (ANLS), evaluating their performance on trigonometric and exponential systems using MATLAB simulations. The primary metrics for comparison are convergence rate, accuracy, and iteration efficiency. This comparative analysis is valuable given the widespread application of such systems and the continuous need for robust and efficient solution techniques. The abstract effectively highlights the key findings: the ANLS method achieves a significantly smaller final error (5.95×10⁻²) with stable convergence, while the Newton–Kontorovich method, though converging in fewer iterations, yields a substantially larger error (3.126). This disparity strongly supports the authors' conclusion regarding the superiority of adaptive strategies for complex nonlinear systems, particularly in achieving higher accuracy. The stated practical implications, including improved numerical reliability for applications in structural engineering, optimization, and scientific modeling, underscore the potential impact of these findings. While the abstract presents clear results, several aspects would benefit from further elaboration in a full paper. Firstly, clarification on the term "Interactive Newton–Raphson with Line Search" from the abstract versus "Adaptive Newton Line Search" from the title is needed for consistency. More importantly, given that Newton–Kontorovich is typically applied to operator equations (integral or differential), a detailed explanation of its specific formulation and implementation for general *multivariate algebraic nonlinear systems* would be crucial. Additionally, the definition of "final error" and why the error for the Newton–Kontorovich method is so high (3.126) despite "converging" in fewer iterations requires deeper analysis; does this imply convergence to a different, less accurate solution, or premature termination? A more thorough discussion of the conditions under which each method performs optimally, or fails, would significantly enhance the paper's contribution.

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