Newton-Raphson method, which dates back to 1669–1670, is widely used to solve systems of equations and unconstrained optimization problems. Newton-Raphson consists in linearizing the system of equations and provides quadratic local convergence order. Quite soon after Newton and Raphson introduced their iterative process, Halley in 1694 proposed a higherorder method providing cubic asymptotic convergence order. Chebyshev in 1838 proposed another high-order variant. In High-order Newton-penalty Algorithms [2], by interpreting Newton’s iteration as a linear extrapolation, formulæ were proposed to compute higher-order extrapolations generalizing Newton-Raphson’s and Chebyshev’s methods.

In this paper, we provide details using an automatic differentiation (AD) tool to implement those high-order extrapolations. We present a complexity analysis allowing to predict the efficiency of those high-order strategies.