On large-scale unconstrained optimization problems and higher order methods

Third order methods will in most cases 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 a second order method. Newton’s method is the most commonly used second order method and Halley’s method is the most well-known third order method. Newton’s method is more used in practical applications than any third order method. We will show that for a large class of problems the ratio of the number of arithmetic operations of Halley’s method and Newton’s method is constant per iteration. It is shown that