A quasi-Newton algorithm for large-scale nonlinear equations.
1School of Information and Statistics, Guangxi University of Finance & Economics, Nanning, Guangxi 530003 P.R. China.
Journal of Inequalities and Applications
|February 21, 2017
Summary
This study introduces a novel algorithm for solving large-scale nonlinear equations, combining conjugate gradient and quasi-Newton methods with a new nonmonotone line search. The approach demonstrates competitive performance and avoids Jacobian matrix computation.
Area of Science:
- Numerical analysis
- Computational mathematics
Background:
- Large-scale nonlinear equations pose significant computational challenges.
- Existing methods often require Jacobian matrices or have convergence limitations.
Purpose of the Study:
- To design an efficient algorithm for large-scale nonlinear equations.
- To improve convergence rates and avoid Jacobian computations.
Main Methods:
- A hybrid approach combining a conjugate gradient (CG) sub-algorithm for initial points with a quasi-Newton main algorithm.
- Introduction of a novel nonmonotone line search technique for step length determination, which bypasses Jacobian matrix calculation.
Main Results:
- The proposed algorithm establishes global convergence under suitable conditions.
- A [Formula: see text]-order convergent rate is achieved for the main algorithm.
- Numerical results indicate the method is competitive for large-scale problems.
Conclusions:
- The developed algorithm offers an efficient and robust solution for large-scale nonlinear equations.
- The nonmonotone line search technique effectively avoids Jacobian computation, simplifying the process.
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