scholarly journals On extensions of the Newton-Raphson iterative scheme to arbitrary orders

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Gilbert Labelle
Keyword(s):  
Differential Operators ◽  
Iterative Scheme ◽  
Successive Approximations ◽  
Series Expansions ◽  
Special Series ◽  
International Audience ◽  
Number Of Leaves ◽  
Newton Raphson

International audience The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation $f(t)=0$ has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves. Le schéma itératif classique à convergence quadratique de Newton-Raphson pour engendrer des approximations successives d'une racine d'une équation $f(t)=0$ a été étendu de plusieurs façons par divers auteurs, allant de la convergence cubique à des convergences d'ordres arbitraires. Nous introduisons deux telles extensions en utilisant des opérateurs différentiels appropriés ainsi que des arguments combinatoires. Nous terminons avec quelques applications incluant des développements en séries exprimant des fonctions de la racine et l'énumération de classes de structures arborescentes selon leur nombre de feuilles.

On metastable approximations in co-operative assemblies

1956 ◽  
Vol 235 (1201) ◽  
pp. 247-259 ◽  
Keyword(s):  
Exact Solution ◽  
Specific Heat ◽  
Curie Point ◽  
Lattice Structure ◽  
Three Dimensional ◽  
Entropy Change ◽  
Successive Approximations ◽  
Series Expansions ◽  
Temperature Expansion ◽  
Heat Curve

The exact solution of the three-dimensional Ising model of a ferromagnetic presents diffi­culties of a very fundamental nature. It therefore seems that the most reliable information on the behaviour of the model is provided by exact series expansions of the partition func­tion at low and high temperatures. However, the usual low -temperature expansion fails to converge in the neighbourhood of the critical point. By rearranging the terms of the series on the basis of physical considerations, it is possible to obtain a systematic set of successive approximations, each approximation taking exact account of clusters of a given size or less (metastable approximations). By extrapolation accurate estimates can be derived of the Curie point and critical values of the energy and entropy. It is found that there is a marked difference in behaviour between two- and three-dimensional lattices, a far larger proportion of the entropy change taking place in the temperature region below the Curie point in the latter case. The corre­sponding specific heat curves are therefore much closer to those observed experimentally. Finally, a brief discussion is given of the dependence of the specific heat curve on lattice structure.

Policy Improvement and the Newton–Raphson Algorithm for Renewal Reward Processes

1989 ◽  
Vol 3 (3) ◽  
pp. 393-396 ◽  
Author(s):  
J. M. McNamara
Keyword(s):  
Continuous Time ◽  
Successive Approximations ◽  
Average Reward ◽  
Policy Improvement ◽  
Unique Root ◽  
Renewal Reward Processes ◽  
Reward Process ◽  
Newton Raphson ◽  
Renewal Reward Process ◽  
Raphson Algorithm

We consider a renewal reward process in continuous time. The supremum average reward, γ* for this process can be characterised as the unique root of a certain function. We show how one can apply the Newton–Raphson algorithm to obtain successive approximations to γ*, and show that the successive approximations so obtained are the same as those obtained by using the policy improvement technique.

scholarly journals On the bialgebra of functional graphs and differential algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Maurice Ginocchio
Keyword(s):  
Dynamical Systems ◽  
Differential Operators ◽  
Discrete Dynamical Systems ◽  
Rooted Trees ◽  
Connected Components ◽  
Differential Algebras ◽  
Noncommutative Polynomials ◽  
International Audience

International audience We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.

scholarly journals Some simple varieties of trees arising in permutation analysis

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Mathilde Bouvel ◽  
Marni Mishna ◽  
Cyril Nicaud
Keyword(s):  
Related Species ◽  
Asymptotic Formulas ◽  
Genome Comparison ◽  
Set Of Permutations ◽  
International Audience ◽  
Interval Trees ◽  
Number Of Leaves ◽  
Permutation Analysis

International audience After extending classical results on simple varieties of trees to trees counted by their number of leaves, we describe a filtration of the set of permutations based on their strong interval trees. For each subclass we provide asymptotic formulas for number of trees (by leaves), average number of nodes of fixed arity, average subtree size sum, and average number of internal nodes. The filtration is motivated by genome comparison of related species. Nous commençons par étendre les résultats classiques sur les variétés simples d'arbres aux arbres comptés selon leur nombre de feuilles, puis nous décrivons une filtration de l'ensemble des permutations qui repose sur leurs arbres des intervalles communs. Pour toute sous-classe, nous donnons des formules asymptotiques pour le nombre d'arbres (comptés selon les feuilles), le nombre moyen de nœuds d'arité fixée, la moyenne de la somme des tailles des sous-arbres, et le nombre moyen de nœuds internes. Cette filtration est motivée par des problématiques de comparaison de génomes.

scholarly journals Iterative strategies associated with the normal flow technique on the nonlinear analysis of structural arches

2014 ◽  
Vol 67 (2) ◽  
pp. 143-150 ◽  
Author(s):  
Dalilah Pires Maximiano ◽  
Andréa Regina Dias da Silva ◽  
Ricardo Azoubel da Mota Silveira
Keyword(s):  
Iterative Scheme ◽  
Displacement Curve ◽  
Equilibrium Path ◽  
Normal Flow ◽  
Nonlinear Load ◽  
Nonlinear Solution ◽  
Highly Nonlinear ◽  
Newton Raphson ◽  
Load Displacement ◽  
Path Following Methods

A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with static nonlinear behavior is based on the Newton-Raphson iterative scheme to which are coupled the path-following methods. In this context, this study uses one technique, referred to as normal flow, in the process of obtaining the approximate nonlinear static response of structural systems. Basically, this technique is an adaptation made with in the Newton-Raphson iterative scheme in an attempt to speed up the nonlinear solution process and/or remove convergence problems. To overcome the critical points and to trace the whole nonlinear equilibrium path, three different strategies are used in association with the normal flow technique: the cylindrical arc-length, the minimum residual displacement norm and the generalized displacement. With this procedure, the performance of these strategies when associated with the normal flow technique is valued. Two arches with highly nonlinear load-displacement curves are used in the study. The results obtained demonstrated that the association of the generalized displacement strategy with the normal flow technique contributes to the improvement of the nonlinear solution methodology.

scholarly journals Complete k-ary trees and generalized meta-Fibonacci sequences

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Chris Deugau ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
International Audience ◽  
Number Of Leaves ◽  
Fibonacci Sequences ◽  
Infinite Sequences

International audience We show that a family of generalized meta-Fibonacci sequences arise when counting the number of leaves at the largest level in certain infinite sequences of k-ary trees and restricted compositions of an integer. For this family of generalized meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations.

scholarly journals Darcy-Weisbach resistance coefficient determination using Newton-Raphson approach for android 4.0

Tecnura ◽  
2019 ◽  
Vol 23 (60) ◽  
pp. 52-58
Author(s):  
Edgar Orlando Ladino Moreno ◽  
César Augusto García Ubaque ◽  
María Camila García Vaca
Keyword(s):  
Friction Coefficient ◽  
Turbulent Flows ◽  
Resistance Coefficient ◽  
Laminar Flows ◽  
Successive Approximations ◽  
Analysis Tool ◽  
Unstable Flow ◽  
Educational Processes ◽  
Newton Raphson ◽  
Initial Seed

Context: The article presents the friction coefficient calculation for pressure pipes, from Newton-Raphson numerical method, under JavaScript source code for Android 4.0 or higher. Methodology: The code development is based on Colebrook-White model for turbulent flows. The app also establishes the friction coefficient for laminar flows from the Poiseuille equation and determines the range for the enter flow in a transition zone (unstable flow). An algorithm of successive approximations was implemented for non-linear equation solution based on the relative roughness and Reynolds number. Results: In the Newton-Raphson method, if the initial value moves away from zero or presents different inflection points, the probability of divergence of the method increases significantly. To overcome this anomaly, a friction coefficient equal to 0.015 was proposed as initial (seed) value; thereby we guarantee the convergence for turbulent flows with an approximation of 0.0000001. The values ​​calculated by the app were calibrated with the Excel analysis tool (Goal Seek). Conclusion: The app is available for free in Google Play Store, and is a tool to support educational processes and the professional field.

Determining the Properties of the Basins of Convergence in the Generalized Hénon–Heiles System

2020 ◽  
Vol 30 (01) ◽  
pp. 2050007 ◽  
Author(s):  
Euaggelos E. Zotos ◽  
Md. Sanam Suraj ◽  
Amit Mittal ◽  
Rajiv Aggarwal
Keyword(s):  
Configuration Space ◽  
Iterative Scheme ◽  
Equilibrium Points ◽  
Perturbation Parameter ◽  
Critical Values ◽  
Convergence Properties ◽  
Newton Raphson

We examine the convergence properties of the generalized Hénon–Heiles system, by using the multivariate version of the Newton–Raphson iterative scheme. In particular, we numerically investigate how the perturbation parameter [Formula: see text] influences several aspects of the method, such as its speed and efficiency. Color-coded diagrams are used for revealing the basins of convergence on the configuration plane. Additionally, we compute the degree of fractality of the convergence basins on the configuration space, as a function of the perturbation parameter, by using different tools, such the uncertainty dimension and the (boundary) basin entropy. Our analysis suggests that the perturbation parameter strongly influences the number of the equilibrium points, as well as the geometry and the structure of the associated basins of convergence. Furthermore, the highest degree of fractality, along with the appearance of nonconverging points, occur near the critical values of the perturbation parameter.

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