Halley and Newton are one step apart

Abstract

One of the central problems of scientific computation is the efficient numerical solution of the system of n equations in n unknowns F ( x ) = 0 where F : R n → R n is sufficiently smooth. While Newton's method is usually used for solving such systems, third order methods will in general use fewer iterations than a second order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third order methods than second order methods. In this note we will consider the case where F = ∇ f , where f is three times continuously differentiable. We will show that for a large class of sparse problems the ratio of the number of arithmetic operations of a third order method and Newton's method is constant per iteration. It is shown that equation image when the structure of the tensor is induced by a general sparse structured Hessian matrix which gives no fill‐ins when we use a direct method to solve a system of linear equations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)