Nonmonotone quasi-Newton method with diagonal Jacobian approximation for symmetric nonlinear equations

• We propose a quasi-Newton method with a secant-based diagonal approximation of the Jacobian (QNSDAJ) for efficiently solving symmetric and sparse nonlinear equations. • The diagonal Jacobian approximation significantly reduces storage requirements and computational cost compared with full quasi-Newton updates. • Global convergence is established under mild assumptions by combining QN-SDAJ with the nonmonotone linesearch of Li-Fukushima. • The method robustly handles challenges such as non-descent directions, very small stepsizes, and illconditioned or unavailable Jacobians. • Numerical experiments on benchmark problems show that QN-SDAJ achieves competitive or superior performance compared to Newton, the inexact BFGS, the nonlinear conjugate gradient, and the derivative-free spectral residual method DF-SANE. We investigate a quasi-Newton method with a diagonal approximation of the Jacobian based on the secant equation, referred to as QN-SDAJ, for solving sparse, symmetric nonlinear equations arising from unconstrained optimization problems. The search direction is constructed using a Barzilai-Borwein-type scaling derived from the secant equation, and global convergence is established under suitable assumptions when combined with the nonmonotone line search of Li and Fukushima. This globalization strategy addresses the major limitations of classical Armijo-type monotone linesearch rules, such as nondescent directions, unacceptably small steps, or unavailable Jacobians. We conducted several numerical experiments on benchmark problems to demonstrate the computational efficiency and robustness of QN-SDAJ. Our results show that QN-SDAJ is a competitive or superior alternative to methods, including the exact Newton method, a variant of the nonlinear conjugate gradient method, an inexact Broyden-Fletcher-Goldfarb-Shanno method, and a derivative-free spectral residual method, DF-SANE, which is widely regarded as the state-of-the-art.