Download or read book Newton s Method as a Dynamical System Global Convergence and Predictability written by R. G. Holt and published by . This book was released on 1985 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: Newton's method as an iterative scheme to compute both unstable and stable fixed points of a discrete dynamical system is considered. It is shown for Newton iterations that the basins of attraction are intertwined in a complicated manner. This complex structure appears to be fractal, and its dimension is estimated. Consequences of predictability for the final state are given in terms of imprecision in the initial data. Keywords include: Newton's method, Predictability, Basin boundaries, Fractal, Nonlinear dynamic.
Download or read book Newton s Method and Dynamical Systems written by H.-O. Peitgen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Newton Methods for Nonlinear Problems written by Peter Deuflhard and published by Springer Science & Business Media. This book was released on 2011-09-18 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1995 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Newton Type Methods for Optimization and Variational Problems written by Alexey F. Izmailov and published by Springer. This book was released on 2014-07-08 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents comprehensive state-of-the-art theoretical analysis of the fundamental Newtonian and Newtonian-related approaches to solving optimization and variational problems. A central focus is the relationship between the basic Newton scheme for a given problem and algorithms that also enjoy fast local convergence. The authors develop general perturbed Newtonian frameworks that preserve fast convergence and consider specific algorithms as particular cases within those frameworks, i.e., as perturbations of the associated basic Newton iterations. This approach yields a set of tools for the unified treatment of various algorithms, including some not of the Newton type per se. Among the new subjects addressed is the class of degenerate problems. In particular, the phenomenon of attraction of Newton iterates to critical Lagrange multipliers and its consequences as well as stabilized Newton methods for variational problems and stabilized sequential quadratic programming for optimization. This volume will be useful to researchers and graduate students in the fields of optimization and variational analysis.
Download or read book Newton s Method and the Effect of Damping on the Basins of Attraction written by Christopher Richard McClure and published by . This book was released on 2002 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: The intricately interwoven basins of attraction stemming from Newton's Method applied to a simple complex polynomial are a common sight in fractal, dynamical systems, and numerical analysis literature. In this work, the author investigates how this workhorse of root-finding algorithms works for complex polynomials, in addition to a variety of other settings, from the simple, one-dimensional real function with a simple root, to the infinite-dimension Banach space. The rapid, quadratic convergence of Newton's method to a simple root is well known, but this performance is not guaranteed for all roots and for all starting points. Damping is one modification to the Newton algorithm that can be used to overcome difficulties in global convergence. We explore computationally how this damping affects the fractal geometry of the Newton basins of attraction for a simple function.
Download or read book A Study of the Global Convergence Properties of Newton s Method written by Donald Allen Celarier and published by . This book was released on 1978 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: A construct is developed which is useful in the investigation of the global convergence properties of Newton's method. This construct is used to study the application of Newton's method to polynomials. A proof that Newton's method converges almost globally for polynomials with only real zeroes is extended to a larger class of polynomials. A conjecture is advanced concerning conditions which are necessary and sufficient for almost global convergence for both real and complex polynomials. Another application of the construct involves the use of Bairstow's method on cubic polynomials having one real zero. The question of whether a certain fixed set is stable is resolved.
Download or read book Reviews in Global Analysis 1980 86 as Printed in Mathematical Reviews written by and published by . This book was released on 1988 with total page 848 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Global Convergence for Newton Methods in Mathematical Programming written by James W. Daniel and published by . This book was released on 1972 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Global Convergence of Damped Newton s Method for Nonsmooth Equations Via the Path Search written by Cornell University. Department of Computer Science and published by . This book was released on 1990 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational examples from nonlinear programming are given.
Download or read book Newton s Method as a Dynamical System written by Johannes Rückert and published by . This book was released on 2006 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study transcendental and rational mappings that arise as Newton maps of entire functions. Our first result is that "in between" any two accesses to infinity of an immediate basin, a Newton map exhibits either another immediate basin, a virtual immediate basin or infinitely many preimages of some point. This result is joint work with Dierk Schleicher and allows to locate virtual immediate basins. An important corollary is a proof of the folklore result that for Newton maps of polynomials, every complementary component of an immediate basin contains another immediate basin. Our second main result, which is joint work with Xavier Buff, shows an interesting connection between virtual immediate basins of the Newton map N_f and asymptotic values of f, answering a 2003 question of Douady: in many cases, 0 is an asymptotic value of f if N_f has a virtual immediate basin. Conversely, if f has an asymptotic value of logarithmic type at 0, then N_f has a virtual immediate basin. We show by way of counterexamples that this is not true for other types of asymptotic values. Our third main result gives a combinatorial classification of a class of Newton maps of polynomials. Let N be the Newton map of a polynomial such that all critical points of N land on a fixed point after finitely many iterations. In this case, we construct a graph that characterizes N uniquely up to Möbius conjugation. Conversely, we show that every graph with an associated map that satisfies several natural conditions is realized by a unique Newton map. In an appendix, we introduce a class of bounded type transcendental entire functions with the property that its set of escaping points is organized in the form of unbounded rays. This fourth main result is joint work with Günter Rottenfußer, Lasse Rempe and Dierk Schleicher, and is part of an answer to a long-standing conjecture of Fatou and Eremenko.
Download or read book A GLOBAL CONVERGENCE TECHNIQUE FOR NEWTON S METHOD WITH PERIODIC HESSIAN EVALUATION written by Francesco LAMPARIELLO and published by . This book was released on 1999 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Structural Dynamic Systems Computational Techniques and Optimization written by Cornelius T. Leondes and published by Routledge. This book was released on 2021-09-01 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: The finite element, an approximation method for solving differential equations of mathematical physics, is a highly effective technique in the analysis and design, or synthesis, of structural dynamic systems. Starting from the system differential equations and its boundary conditions, what is referred to as a weak form of the problem (elaborated in the text) is developed in a variational sense. This variational statement is used to define elemental properties that may be written as matrices and vectors as well as to identify primary and secondary boundaries and all possible boundary conditions. Specific equilibrium problems are also solved. This book clearly reveals the effectiveness and great significance of the finite element method available and the essential role it will play in the future as further development occurs.
Download or read book A Total quasi newton Method with Global Convergence for General Nonlinear Optimisation Problems written by and published by . This book was released on 2008 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Identification of Nonlinear Systems Using Neural Networks and Polynomial Models written by Andrzej Janczak and published by Springer Science & Business Media. This book was released on 2004-11-18 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph systematically presents the existing identification methods of nonlinear systems using the block-oriented approach It surveys various known approaches to the identification of Wiener and Hammerstein systems which are applicable to both neural network and polynomial models. The book gives a comparative study of their gradient approximation accuracy, computational complexity, and convergence rates and furthermore presents some new and original methods concerning the model parameter adjusting with gradient-based techniques. "Identification of Nonlinear Systems Using Neural Networks and Polynomal Models" is useful for researchers, engineers and graduate students in nonlinear systems and neural network theory.
Download or read book Newton s Method and Dynamical Systems written by Jianping Yang and published by . This book was released on 1992 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Predictability of Complex Dynamical Systems written by James B. Kadtke and published by Springer. This book was released on 1996-10-04 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: This practical guide to data analysis and algorithms addresses researchers and practitioners interested in modeling, prediction, and forecasting natural systems based on nonlinear dynamics. It explores many interesting topics in complex dynamical systems and shows how probabilistic techniques, such as signal and time-series analysis, financial and stochastic modeling, and political modeling can be used to predict them.
