Reflected Box Method: A Fast Quadratic‐Convergent Root Solver

A simple and derivative‐free root‐finding method based on geometric reflection and box construction is introduced. The proposed reflected box method achieves quadratic convergence like Newton's method, but operates without requiring derivatives. Its arithmetic simplicity makes the method naturally amenable to fixed‐point and embedded implementation, though this paper focuses on floating‐point evaluation to establish baseline performance. Numerical experiments confirm competitive performance against Brent's and Steffensen's methods, with fewer iterations, simpler arithmetic and comparable residual accuracy.