In this paper, we propose and analyze a family of multi-step iterative methods for solving nonlinear equations, inspired by the decomposition technique introduced by Noor and Noor (Appl. Math. Comput. 183, 322–327, 2006). The newly developed methods derived from this family can be regarded as \(\left( \varvec{p}\varvec{,}\varvec{q}\right) -\) extensions of existing approaches for solving nonlinear equations. In particular, we consider the \(\left( \varvec{p}\varvec{,}\varvec{q}\right) -\) generalization of Chun’s two-step iterative scheme, enhanced via the Adomian decomposition method. The convergence order of the method is analytically established within the framework of \(\left( \varvec{p},\varvec{q}\right) -\) calculus. In addition to a rigorous theoretical analysis, a geometric interpretation is provided to offer deeper insight into the behavior of the method. We also evaluate the method’s performance for various values of the quantum parameters \(\varvec{p}\) and \(\varvec{q}\) . Numerical experiments are conducted for various values of the quantum parameters \(\varvec{p}\) and \(\varvec{q}\) , demonstrating that the generalized method achieves comparable or improved accuracy compared to the classical third-order Chun method, while offering enhanced flexibility and derivative-free implementation.
