A discrete Newton algorithm for minimizing a function of many variables

Many algorithms have been proposed for finding a local minimum of a function f ( x ) f rom R" into R ~ using only function and gradient values. These include discrete Newton methods, quasi -Newton methods, and the conjugate gradient algorithm. For a discussion of these methods, see, for example [1]. There are classes of problems for which each of these methods is the most efficient. In this paper, we describe a variant of the discrete Newton algorithm which requires fewer operat ions than the standard method when n > 39. If a precondit ioned form of the matrix of second derivatives has many multiple or clustered eigenvalues, this can be exploited, and even greater efficiency can be achieved. Storage is of O(n) rather than the O(n 2) of the standard discrete Newton method, and this method requires less storage if n > 12. The algorithm uses a precondit ioned conjugate gradient method to generate and solve the linear sys tem which determines the Newton step. In Section 2 we present the algorithm and discuss its convergence rate and properties. Section 3 provides numerical examples. A Fortran code is provided in [2].