Leap-frogging Newton’s method is constructed by combining the classical Newton’s method with the pseudo-secant method. Under the assumptions that a given function f : R → R has a simple real zero α and is sufficiently smooth in a small neighborhood of α, we investigate the convergence behavior of Leap-frogging Newton’s method near α. The order of convergence is shown to be cubic and the asymptotic error constant is proven to be 1 4 · f ″ ( α ) f ′ ( α ) 2 . The numerical method based on this proposed theory have shown satisfactory results via programming in Mathematica with its high-precision computability.