A diagonal Hestenes-Stiefel and Polak–Ribière–Polyak algorithm for solving nonlinear system of equations

We address the solution of large-scale nonlinear systems of equations and propose a Jacobian-free diagonal Hestenes-Stiefel and Polak–Ribière–Polyak hybrid algorithm (hybridplus) that uses only residual information and avoids matrix storage. The search direction of the algorithm is obtained by combining a diagonal Jacobian approximation with a positive hybrid of Hestenes-Stiefel and Polak–Ribière–Polyak parameters. We globalized the algorithm using a nonmonotone line search technique that utilizes two-sided trial steps and achieved the global convergence results under standard bounded level-set and Lipschitz continuity assumptions on the Jacobian. Furthermore, when the Jacobian positive definite in the neighborhood of a solution, we prove a uniform lower bound on accepted step sizes and derive an R-linear convergence rate. Extensive experiments on ten benchmark families across dimensions up to 10 6 and ten diverse initializations (500 runs per algorithm) demonstrated that hybridplus consistently outperforms similar algorithms in the literature.