Rate of convergence of higher order methods

Steihaug, Trond; Suleiman, Sara

doi:10.1016/j.apnum.2011.06.016

Cited by 8 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There exists a class of tensor methods which utilizes information from higher order derivatives to accelerate convergence to a single solution of a SNE. In [125], Steihaug and Suleiman utilize the model:…”

Section: Tensor Methods Utilizing Higher Order Derivativesmentioning

confidence: 99%

“…Furthermore, in order to solve Eq. ( 15), Steihaug and Suleiman in [125] introduce a class of inexact Chebyshev-Halley methods which under some assumptions are locally convergent satisfying Eq. ( 15) and…”

Section: Tensor Methods Utilizing Higher Order Derivativesmentioning

confidence: 99%

“…Eustaquio et al conducted a rigorous quantitative comparative analysis of their method against the Inexact Newton method and the tensor methods utilizing higher order derivatives presented by Steihaug and Suleiman [125]. Eustaquio et al illustrated that the class of tensor-free Chebyshev-Halley methods are capable of efficiently handling problems that have ill-conditioned or singular Jacobian matrices at the solution.…”

Section: End Whilementioning

confidence: 99%

See 2 more Smart Citations

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

View full text Add to dashboard Cite

This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting a thorough description and analysis of the known methods capable of finding one or many solutions to SNEs, and to assist interested readers seeking to identify solution techniques which are well suited for solving the various classes of SNEs which one may encounter in real world applications.To accomplish these objectives, we present a multi-part survey. In part one, we focus on root-finding approaches which can be used to search for solutions to a SNE without transforming it into an optimization problem. In part two, we will introduce the various transformations which have been utilized to transform a SNE into an optimization problem, and we discuss optimization algorithms which can then be used to search for solutions. In part three, we will present a robust quantitative comparative analysis of methods capable of searching for solutions to SNEs.

show abstract

Section: Tensor Methods Utilizing Higher Order Derivativesmentioning

confidence: 99%

Section: Tensor Methods Utilizing Higher Order Derivativesmentioning

confidence: 99%

Section: End Whilementioning

confidence: 99%

See 1 more Smart Citation

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Kotsireas¹,

Pardalos²,

Semenov³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The iterative methods of Chebyshev-Halley family [29], adapted to the nonlinear structural problem, are described by the equation:…”

Section: Super-halley and Chebyshev Methodsmentioning

confidence: 99%

Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

Souza

Castelani

Shirabayashi

et al. 2018

Tend. Mat. Apl. Comput.

View full text Add to dashboard Cite

ABSTRACT. A large part of the numerical procedures for obtaining the equilibrium path or loaddisplacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Pták, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.

show abstract

“…Encontramos na literatura diversos artigos relacionados à convergência do método de Halley, ver, por exemplo, [3,4,26,32,36,37]. A grande maioria das publicações sobre o método de Halley e afins restringe-se a uma abordagem teórica, onde procura-se dar a caracterização mais abrangente possível aos objetos matemáticos relevantes.…”

Section: Convergência Do Método De Halleyunclassified

Métodos de otimização de terceira ordem

Ferreira¹,

Mello²

View full text Add to dashboard Cite

Agradeço à minha família, ao meu noivo Agnaldo e a todos meus amigos, que me apoiaram em todas as minhas decisões e estiveram sempre ao meu lado, me dando muita força para seguir.Agradeço a todos os professores, da educação básica e superior, que contribuíram diretamente na minha formação fornecendo toda a base necessária para a realização desse trabalho. Eu poderia citar os nomes daqueles pelos quais guardo um carinho mais especial, mas prefiro não fazê-lo pois posso ser injusta e esquecer de alguém.Agradeço à UNICAMP e ao IMECC por toda estrutura e apoio oferecidos, principalmente pelos funcionários da secretaria de graduação e de pós-graduação. Foi de fundamental importância para o bom desenvolvimento deste trabalho.E por último, mas não menos importante, agradeço à FAPESP pelo apoio financeiro recebido através do processo 2011/03911-9.xii Resumo Métodos de Otimização de terceira ordem, embora de longa tradição, eram considerados, até passado recente, impraticáveis, devido à taxa com que o esforço computacional cresce em função da dimensão do problema. Avanços no desenvolvimento de estruturas de dados, rotinas que trabalham com estas estruturas e a exploração da esparsidade de grande parte dos problemas encontrados na prática já permitem implementações destes métodos que podem torná-los competitivos com métodos de segunda ordem. O objeto desta dissertação é a apresentação do método de Halley, um método de terceira ordem, sua implementação em MATLAB e a realização de testes computacionais, visando uma comparação empírica de sua eficiência frente ao método de Newton, o método de segunda ordem mais empregado na atualidade.

show abstract

Rate of convergence of higher order methods

Cited by 8 publications

References 23 publications

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

Métodos de otimização de terceira ordem

Contact Info

Product

Resources

About