Modified iterative double step length model for solving nonlinear systems with application to motion control
Abubakar Sani Halilu1,2, Mohamad Afendee Mohamed1, Mohammed A Saleh3
1Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, 22200, Besut, Terengganu, Malaysia.
Scientific Reports
|November 19, 2025
Summary
This study introduces a robust method for solving nonlinear equations, combining double-step length iteration with Picard-Mann techniques. This approach enhances convergence and efficiency, proving effective in motion control applications.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Nonlinear Systems
Background:
- Solving systems of nonlinear equations is crucial in many scientific and engineering fields.
- Divergence is a common challenge in iterative methods for nonlinear solvers.
- Existing methods may lack robustness or efficiency in handling complex nonlinear systems.
Purpose of the Study:
- To develop a more effective and robust iterative method for solving systems of nonlinear equations.
- To combine the double-step length iterative method with the Picard-Mann hybrid technique for improved performance.
- To demonstrate the global convergence and practical applicability of the proposed method.
Main Methods:
- A novel double-step-length procedure incorporating two corrections per iteration.
- Integration of the Picard-Mann hybrid technique with the double-step-length method.
- Approximation of the Jacobian matrix using a positive scalar to avoid derivative calculations.
- An inexact line search technique to determine two-step lengths.
Main Results:
- The proposed method exhibits enhanced robustness, reducing the likelihood of divergence.
- Global convergence is achieved under mild conditions.
- Numerical tests confirm superior efficiency compared to existing double-direction and length methods.
- Successful implementation in motion control applications.
Conclusions:
- The combined double-step length and Picard-Mann method offers a powerful approach to solving nonlinear equations.
- The method is efficient, robust, and globally convergent.
- Its practical efficacy is demonstrated through successful application in motion control.
Keywords:
Acceleration parameterDouble direction approachDouble step sizeJacobian matrixMotion trackingOptimization problemsPicard–Mann processMore Related Videos
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