Original Halley Method and its Improvement with Automatic Differentiation

Original Halley method, also referred to as the tangent hyperbolas method, is the classical third order method which can be used to solve the optimization problems. However, computing the derivative terms in solving the Halley equation becomes a key problem to reduce its computational cost. By using efficiently the techniques of automatic differentiation and preconditioned conjugate graduate method, original Halley method is established and improved in this paper. Theoretical and numerical results show that the improved version is more efficient than the original method.