Consistency and Quantitative Backward Stability Analysis of the Two-Step Jarratt Method for Nonlinear Systems

In this work, we revisit the two-step Jarratt method from the perspective of numerical stability. While high-order iterative schemes are often examined in terms of convergence rate and computational efficiency, their backward stability properties have received comparatively less attention. We begin by establishing the method’s strong consistency. Next, we provide a quantitative backward stability assessment within the standard floating-point arithmetic framework, deriving explicit perturbation bounds that show that the iteration errors remain proportional to machine precision. To support the theoretical findings, we present numerical experiments—including tests under finite-precision perturbations—as well as Python implementations and visualizations of the numerical examples. The results illustrate that the two-step Jarratt method not only achieves a high convergence order but also remains numerically robust for well-conditioned nonlinear systems.