Approximating solutions of equations using Newton's method with a modified Newton's method iterate as a starting point

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [6]-[8]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations in a Banach space setting. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [6]-[8]. Numerical examples are also provided

Solving equations using Newton's method under weak conditions on Banach spaces with a convergence structure

We provide new semilocal results for Newton's method on Banach spaces with a convergence structure. Using more precise majorizing sequence we show that, under weaker convergence conditions than before, we can obtain finer error bounds on the distances involved and a more precise information on the location of the solution

Newton's method and regularly smooth operators

A semilocal convergence analysis for Newton's method in a Banach space setting is provided in this study. Using a combination of regularly smooth and center regularly smooth conditions on the operator involved, we obtain more precise majorizing sequences than in [7]. It then follows that under the same computational cost and the same or weaker hypotheses than in [7] the following benefits are obtained: larger convergence domain; finer estimates on the distances involved, and an at least as precise information on the location of the solution of the corresponding equation. Numerical examples are given to further validate the results obtained in this study

Algorithms for computing normally hyperbolic invariant manifolds

An effcient algorithm is developed for the numerical computation of normally hyperbolic invariant manifolds, based on the graph transform and Newton's method. It fits in the perturbation theory of discrete dynamical systems and therefore allows application to the setting of continuation. A convergence proof is included. The scope of application is not restricted to hyperbolic attractors, but extends to normally hyperbolic manifolds of saddle type. It also computes stable and unstable manifolds. The method is robust and needs only little specification of the dynamics, which makes it applicable to e.g. Poincaré maps. Its performance is illustrated on examples in 2D and 3D, where a numerical discussion is included.

A modified secant method for unconstrained minimization

A gradient-secant algorithm for unconstrained optimization problems is presented. The algorithm uses Armijo gradient method iterations until it reaches a region where the Newton method is more efficient, and then switches over to a secant form of operation. It is concluded that an efficient method for unconstrained minimization has been developed, and that any convergent minimization method can be substituted for the Armijo gradient method

A short survey on Kantorovich-like theorems for Newton's method

We survey influential quantitative results on the convergence of the Newton iterator towards simple roots of continuously differentiable maps defined over Banach spaces. We present a general statement of Kantorovich's theorem, with a concise proof from scratch, dedicated to wide audience. From it, we quickly recover known results, and gather historical notes together with pointers to recent articles

A Newton Collocation Method for Solving Dynamic Bargaining Games

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.

A Recursive Algorithm for Mixture of Densities Estimation

In the framework of the so-called extended linear sigma model (eLSM), we include a pseudoscalar glueball with a mass of 2.6 GeV (as predicted by Lattice-QCD simulations) and we compute the two- and three-body decays into scalar and pseudoscalar mesons. This study is relevant for the future PANDA experiment at the FAIR facility. As a second step, we extend the eLSM by including the charm quark according to the global U(4)R × U(4)L chiral symmetry. We compute the masses, weak decay constants and strong decay widths of open charmed mesons. The precise description of the decays of open charmed states is important for the CBM experiment at FAIR

The theory and some applications of Pták's method of non-discrete mathematical induction

Bibliography: p. 79-80.The aim of this thesis is three-fold: (1) to develop the theory of small functions; (2) to synthesize Pták's work presented in his papers [10], [11], ..., [16] into a coherent body of knowledge; (3) to elaborate on Pták's work (i) by providing small function generalizations of Banach's Fixed Point Theorem and Edelstein's Extended Contraction Principle; (ii) by connecting the Induction Theorem to Baire's Category Theorem and Cantor's Intersection Theorem. Throughout the exposition the editorial "we" is to be understood in the sense of Halmos [ 18]; "we" means "the author and the reader"