Curvature-Based Stability in Second-Order Root-Finding Methods

We introduce a second-order, single-point root-finding method based on a quadratic surrogate that guarantees rootward updates under explicit curvature conditions. The method employs single-point quadratic and inverse-quadratic approximations to construct a stable update that remains well-posed beyond the local neighborhood in which Newton’s method is directionally reliable. We prove existence and stability of the update on curvature-defined regions where monotonicity and concavity are fixed, and show that the resulting iteration enlarges the region of provable convergence relative to Newton’s method without requiring external globalization. The analysis isolates the geometric mechanism responsible for misdirection in Newton-type schemes and demonstrates how quadratic surrogate modeling restores directional correctness while retaining second-order local convergence.