<p>Logistic regression and image recovery are among the most compelling scientific applications that demand robust and efficient algorithms. This paper presents an accelerated double-step length method for solving large-scale nonlinear equations with convex constraints. The approach derives a correction parameter using a hybrid iterative procedure that integrates the Picard–Mann method into the search direction. By using the Frobenius norm to quantify the discrepancy between Broyden’s update and its approximation, we derive an acceleration parameter. The proposed algorithm satisfies a sufficient descent condition without requiring a line search. Global convergence is established under weaker assumptions, namely, that the underlying mapping is monotone. Numerical simulations demonstrate the efficiency of the proposed algorithm. Furthermore, the method is successfully applied to logistic regression and the restoration of blurred images, highlighting its relevance to scientific applications.</p>