We develop a theory of quasi-Newton and least-change update methods for solving systems of nonlinear equations F (x) = 0. In this theory, no diierentiability conditions are necessary. Instead, we assume that F can be approximated, in a weak sense, by an aane function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-diie...

The most important line-search algorithms for solving large-scale unconstrained optimization problems we consider in this paper are the quasi-Newton methods, truncated Newton and conjugate gradient. These methods proved to be efficient, robust and relatively inexpensive in term of computation. In this paper we compare the Dolan-Moré [11] performance profile of line-search algorithms implemented...

We present a new diagonal quasi-Newton update with an improved diagonal Jacobian approximation for solving large-scale systems of nonlinear equations. In this approach, the Jacobian approximation is derived based on the quasi-Cauchy condition. The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as to carry some additional ...

In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.

Computational experience with several limited-memory quasi-Newton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a well-known test library [J. on several synthetic problems allowing control of the clustering of eigenvalues in the Hessian spectrum, and on some large-scale problems in oceanography and meteorology. The result...

An expectation maximization (EM) algorithm is derived to estimate the parameters of a phylogenetic model, a probabilistic model of molecular evolution that considers the phylogeny, or evolutionary tree, by which a set of present-day organisms are related. The EM algorithm is then extended for use with a combined phylogenetic and hidden Markov model. An efficient method is also shown for computi...

Use of data assimilation techniques is becoming increasingly common across many application areas. The inverse Hessian (and its square root) plays an important role in several different aspects of these processes. In geophysical and engineering applications, the Hessian-vector product is typically defined by sequential solution of a tangent linear and adjoint problem; for the inverse Hessian, h...

Newton-type methods and quasi-Newton methods have proven to be very successful in solving dense unconstrained optimization problems. Recently there has been considerable interest in extending these methods to solving large problems when the Hessian matrix has a known a priori sparsity pattern, This paper treats sparse quasi-Newton methods in a uniform fashion and shows the effect of loss of pos...

In this paper, we derive and discuss a new adaptive quasi-Newton eigen-estimation algorithm and compare it with the RLS-type adaptive algorithms and the quasi-Newton algorithm proposed by Mathew et al. through experiments with stationary and nonstationary data.