This paper proposes a new Newton-like method which defines new iterates using a linear system with the same coefficient matrix in each iterate, while the correction is performed on the right-hand-side vector of the Newton system. In this way a method is obtained which is less costly than the Newton method and faster than the fixed Newton method. Local convergence is proved for nonsingular syste...

We derive Newton-type solution algorithms for a Bernoulli-type freeboundary problem at the continuous level. The Newton schemes are obtained by applying Hadamard shape derivatives to a suitable weak formulation of the freeboundary problem. At each Newton iteration, an updated free boundary position is obtained by solving a boundary-value problem at the current approximate domain. Since the boun...

In this paper we introduce a local convergence theory for Least Change Secant Update methods. This theory includes most known methods of this class, as well as some new interesting quasi-Newton methods. Further, we prove that this class of LCSU updates may be used to generate iterative linear methods to solve the Newton linear equation in the Inexact-Newton context. Convergence at a ¡j-superlin...

We consider a large-scale convex minimization problem with nonnegativity constraints that arises in astronomical imaging. We develop a cost functional which incorporates the statistics of the noise in the image data and Tikhonov regularization to induce stability. We introduce an efficient hybrid gradient projection-reduced Newton (active set) method. By “reduced Newton” we mean taking Newton s...

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the KušnirenkoBernštein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic r...

This paper analyzes local convergence rates of primal-dual interior point methods for general nonlinearly constrained optimization problems. For this purpose, we first discuss modified Newton methods and modified quasi-Newton methods for solving a nonlinear system of equations, and show local and Qquadratic/Q-superlinear convergence of these methods. These methods are characterized by a perturb...

We consider implicit integration methods for the numerical solution of sti initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonsti or mildly sti with respect to the stepsize. In these nonsti subintervals, we do not need the (expensive) Newton ...

We study certain plane graphs, called Newton graphs, representing a special class of dynamical systems which are closely related to Newton’s iteration method for finding zeros of (rational) functions defined on the complex plane. These Newton graphs are defined in terms of nonvanishing angles between edges at the same vertex. We derive necessary and sufftcient conditions-of purely combinatorial...

We develop certified tracking procedures for Newton homotopies, which are homotopies for which only the constant terms are changed. For these homotopies, our certified procedures include using a constant predictor with Newton corrections, an Euler predictor with no corrections, and an Euler predictor with Newton corrections. In each case, the predictor is guaranteed to produce a point in the qu...

In this paper, we show that the Newton polytope of an observation Y from a two-dimensional hidden Markov model (2D HMM) lies in a three-dimensional subspace of its ambient eight-dimensional space, whose vertices correspond to the most likely explanations (“hidden” states) for Y given the model. For each Newton polytope, there exists a set of “essential” vertices, which form a skeleton for the p...