Optimal fourth-order schemes in (p,q)-calculus: Structural asymmetry and dynamical behavior

We develop and analyze a fourth-order optimal multipoint iterative family constructed intrinsically within the (p,q)-calculus framework. Although the schemes preserve optimal efficiency in the sense of Kung–Traub, the two-scale structure of the (p,q)-derivative fundamentally alters the mechanism of error propagation. A rigorous local convergence analysis is established via a (p,q)-Taylor expansion, yielding an explicit error equation and sharp conditions under which fourth-order convergence is attained. From a dynamical viewpoint, we prove that classical normalization techniques based on affine or Möbius conjugacy fail in the (p,q)-setting, so that the iteration must be investigated through its intrinsic rational map. Extensive basin-of-attraction experiments are carried out for polynomial models of increasing degree and for a nonlinear transcendental equation motivated by a scaled diode model. Across the normalized polynomial benchmarks, a clear stability hierarchy emerges among the proposed weight functions, with M1 forming the most robust subclass; nonsymmetric tests further show that the precise ordering may depend on the geometry of the nonlinear problem. To explain this phenomenon, we provide an analytic singularity analysis of the associated weight functions and identify structural features that promote global stability. The results reveal that in the (p,q)-framework, dynamical robustness depends not only on formal convergence order but also on the analytic structure of the weight function and its interaction with the two-scale derivative. These findings offer practical guidance for the design of stable higher-order methods in generalized calculus settings.