The column-updating method for solving nonlinear equations in Hilbert space

Gomes-Ruggiero, Márcia A.; Martínez, José Mário

doi:10.1051/m2an/1992260203091

Cited by 14 publications

(5 citation statements)

References 22 publications

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“…As in the case of Newton's method, the secant-type methods have also been extended and analyzed for solving nonlinear problems in Hilbert spaces; see e.g. [77], [156]. Newton's method has been extended for solving nonsmooth systems of nonlinear equations [38], [98], [129], [155], and has been combined with interior point methods [78].…”

Section: Vector Problemsmentioning

confidence: 99%

“…for B kþ1 10: end for Solving a secant equation that deals with n Â n matrices is the most attractive feature of Algorithms 3 and 4, in sharp contrast with the standard extension of quasi-Newton methods for general Hilbert spaces, (see e.g. [77], [156]), that in this context would involve n 2 Â n 2 linear operators to approximate the derivative of F . Clearly, dealing with n Â n matrices for solving the related linear systems significantly reduces the computational cost associated with the linear algebra of the algorithm.…”

Section: Matrix Problemsmentioning

confidence: 99%

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Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Monsalve¹,

Raydan²

2011

Port. Math.

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Nonlinear matrix equations arise in di¤erent scientific topics, such as applied statistics, control theory, and financial mathematics, among others. As in many other scientific areas, Newton's method has played an important role when solving these matrix problems. Under standard assumptions, the specialized Newton methods that have been developed for specific problems exhibit local and q-quadratic convergence and require a suitable initial guess. They also require, as usual, a significant amount of computational work per iteration, that in this case involve several matrix factorizations per iterations. As expected, whenever a Newton method can be developed, a secant method can also be developed. Indeed, more recently, secant methods for solving specific nonlinear matrix problems have been developed opening a new line of research. As in previous scenarios, these specialized secant methods exhibit local and q-superlinear convergence, also require a suitable initial guess, and avoid the use of derivatives in the formulation of the schemes.In this review we start by recalling the presence of Newton's method and the secant methods, and also their classical relationship, in di¤erent and sometimes unexpected scenarios for vector problems. Then we present and describe the state of the art in the use of Newton's method and also the secant method in the space of matrices. A second objective is to present a unified approach for describing the features of these classical schemes, that in the space of matrices represent an interesting research area with special features to be explored.

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Section: Vector Problemsmentioning

confidence: 99%

Section: Matrix Problemsmentioning

confidence: 99%

Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Monsalve¹,

Raydan²

2011

Port. Math.

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“…In Nightingale, Newton's method and seven quasi-Newton methods are implemented. Our numerical experience in the last few years indicate that the best quasi-Newton methods included in the package are Broyden's good method [I] and the method based on the Column-Updating formula ( [38], [45], [24]). Of course, Newton's method must also be kept within a package for solving nonlinear systems, since it tends to be the most efficient method when the cost of solving linear systems is not very high.…”

Section: Introductionmentioning

confidence: 99%

Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy^∗

Friedlander

Gomes-Ruggiero

Kozakevich

et al. 1997

Optimization Methods and Software

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1997) Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy , Optimization Methods and Software, 8:1, 25-51, A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists of combining efficient local methods with an algorithm that reduces monotonicallv the squared norm oflthe system in a proper way. he-local methods used are ~e w t o n ' s method and two quasi-Newton algorithms. Global iterations are based on recently introduced box-constrained minimization algorithms. Numerical experiments are presented.

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“…CUM: Column-Updating method. Implementação de Gomes-Ruggiero e Martínez [27] com Bk = P 7 (J(xk)) quando k =O (mod m) . Pesquisas futuras são necessárias, tanto do ponto de vista teórico quanto prático.…”

Section: Experimentos Numericosunclassified

“…Entre eles, podemos citar os métodos introduzidos por Tewarson [54], Tewarson e Zhang [55], Dennis e Marwil [15) e Martínez [38). Gomes-Ruggiero e Martínez [27] mostraram que o "Column-Updating method" (CUM) introduzido em Martínez [39) é muito efetivo quando comparado com a implementação do primeiro método de Broyden com memória limitada (Gomes-Ruggiero, Martínez e Moretti [28), Griewank [30), Matthies e Strang [46]) na resolução de sistemas não lineares de grande porte.…”

Section: Introduçaounclassified

Novos resultados sobre formulas secantes e aplicações

Zambaldi¹

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IntroduçãoOs Sistemas de Equações não Lineares aparecem em muitos problemas da vida real. São os casos, por exemplo, de problemas de Mecânica dos fluidos, fluxo de cargas em redes de transmissão de energia elétrica e simulações numéricas de reservatórios. Um trabalho recente de Moré [4 7] relaciona uma coleção de problemas práticos, oriundos de diversas áreas e aplicações, cuja formulação conduz a um sistema de equações não lineares. O ProblemaO problema típico em que estamos interessados é o seguinte: Dada uma função, F : IRn ~ IRn, F = (! 1 , ..• , fnf, buscamos obter a solução de F(x) =O. (1.1) Denotaremos a matriz de derivadas parciais de F, a matriz Jacobiana, ou simplesmente o Jacobiano, por J(x). Portanto, ( \7 f1(x)T )aberto e c.p: {}X V---+ JRnXn uma função contínua. Para um ponto arbitráiio Xo E n ' Eo E v' Bo = c.p(xo' Eo)' consideramos a seqüência gerada por (2.3) onde 9

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The column-updating method for solving nonlinear equations in Hilbert space

Cited by 14 publications

References 22 publications

Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy^∗

Novos resultados sobre formulas secantes e aplicações

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The column-updating method for solving nonlinear equations in Hilbert space

Cited by 14 publications

References 22 publications

Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Newton’s method and secant methods: A longstanding relationship from vectors to matrices

Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy∗

Novos resultados sobre formulas secantes e aplicações

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Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy^∗