An efficient scaled DFP algorithm based on singular value analysis for solving systems of monotone nonlinear equations

In this study, we introduce a one-parameter scaled memoryless Davidon–Fletcher–Powell (DFP) method. It uses singular value analysis of the search direction matrix and is designed for solving nonlinear systems. Building on the classical DFP method, this variant employs a memoryless approximation to the inverse Jacobian matrix, and incorporating a scaled version of the three-term memoryless DFP formula. The proposed method is well-suited for large-scale nonlinear systems and demonstrates particular efficacy in addressing monotone equations, where the Jacobian matrix is positive definite. The scaling parameters are optimally determined by minimizing the difference between the smallest and largest singular values. The method also combines a projection technique with the memoryless DFP framework, which ensures global convergence. Numerical experiments on several test problems highlight the method’s efficiency and accuracy, confirming its potential for solving nonlinear systems.