The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
Ioannis K. Argyros - One of the best experts on this subject based on the ideXlab platform.
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A New Semi-local Convergence Analysis of the Secant Method
International Journal of Applied and Computational Mathematics, 2017Co-Authors: Ioannis K. Argyros, Ekaterina NathansonAbstract:We provide a new semi-local convergence analysis for the Secant Method in a Banach space setting. Argyros and other authors have analyzed the Method using a Lipschitz condition and a simple center Lipschitz condition. However, the Secant Method has two starting vectors \(u_0\), \(u_{-1}\), and it makes sense to analyze it using a mixed center-Lipschitz condition based on both vectors. The direct analysis of the majorizing sequence employed in this paper can be used to obtain weaker convergence conditions than those used in earlier studies. A numerical example is given to further justify the theoretical results.
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new improved convergence analysis for the Secant Method
Mathematics and Computers in Simulation, 2016Co-Authors: Alberto A Magrenan, Ioannis K. ArgyrosAbstract:We present a new convergence analysis, for the Secant Method in order to approximate a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center–Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence analysis leads to more precise error bounds and to a better information on the location of the solution than the corresponding ones in earlier studies. Numerical examples validating the theoretical results are also provided in this study.
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expanding the applicability of the Secant Method under weaker conditions
Applied Mathematics and Computation, 2015Co-Authors: Ioannis K. Argyros, Alberto A MagrenanAbstract:Abstract We present a new semilocal convergence analysis for Secant Method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies such as (Amat and Busquier, 2003; Amat et al., in press; Argyros and Hilout, 2012; Argyros et al., 2014; Argyros and Magrenan, 2014, 2015; Dennis, 1971; Ezquerro et al., 2000; Ortega and Rheinboldt, 1970; Potra and Ptak, 1984; Schmidt, 1978). Numerical examples are also presented to illustrate the theoretical results obtained in this study.
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new semilocal and local convergence analysis for the Secant Method
Applied Mathematics and Computation, 2015Co-Authors: Alberto A Magrenan, Ioannis K. ArgyrosAbstract:We present a new convergence analysis, for the Secant Method in order to approximate a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence analysis leads to more precise error bounds and to a better information on the location of the solution than the corresponding ones in earlier studies such as 2,6,9,11,14,15,17,20,22-26]. Numerical examples validating the theoretical results are also provided in this study.
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expanding the applicability of Secant Method with applications
Bulletin of The Korean Mathematical Society, 2015Co-Authors: Alberto A Magrenan, Ioannis K. ArgyrosAbstract:Abstract. We present new sufficient convergence criteria for the con-vergence of the Secant-Method to a locally unique solution of a nonlinearequation in a Banach space. Our idea uses Lipschitz and center–Lipschitzinstead of just Lipschitz conditions in the convergence analysis. The newconvergence criteria can always be weaker than the corresponding onesin earlier studies. Numerical examples are also provided in this study tosolve equations in cases not possible before. 1. IntroductionIn this study we are concerned with the problem of approximating a locallyunique solution x ⋆ of equation(1.1) F(x) = 0,where F is a Fr´echet–differentiable operator defined on a convex subset D of aBanach space X with values in a Banach space Y.A vast number of problems from applied science including engineering canbe solved by means of finding the solutions equations in a form like (1.1) us-ing mathematical modelling [7,11,16,19]. For example, dynamic systems aremathematically modeled by difference or differential equations, and their so-lutions usually represent the states of the systems. Except in special cases,the solutions of these equations cannot be found in closed form. This is themain reason why the most commonly used solution Methods are iterative. It-eration Methods are also applied for solving optimization problems. In suchcases, the iteration sequences converge to an optimal solution of the problemat hand. Since all of these Methods have the same recursive structure, they canbe introduced and discussed in a general framework. The convergence analy-sis of iterative Methods is usually divided into two categories: semilocal andlocal convergence analysis. In the semilocal convergence analysis one derivesconvergence criteria from the information around an initial point whereas in
Qingbiao Wu - One of the best experts on this subject based on the ideXlab platform.
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The Convergence Ball and Error Analysis of the Relaxed Secant Method
Advances in Mathematical Physics, 2020Co-Authors: Qingbiao Wu, Minhong ChenAbstract:A relaxed Secant Method is proposed. Radius estimate of the convergence ball of the relaxed Secant Method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.
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on the convergence ball and error analysis of the modified Secant Method
Advances in Mathematical Physics, 2018Co-Authors: Qingbiao Wu, Minhong ChenAbstract:We aim to study the convergence properties of a modification of Secant iteration Methods. We present a new local convergence theorem for the modified Secant Method, where the derivative of the nonlinear operator satisfies Lipchitz condition. We introduce the convergence ball and error estimate of the modified Secant Method, respectively. For that, we use a technique based on Fibonacci series. At last, some numerical examples are given.
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the convergence ball and error analysis of the two step Secant Method
Applied Mathematics-a Journal of Chinese Universities Series B, 2017Co-Authors: Qingbiao Wu, Minhong Chen, Yasir KhanAbstract:Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order, we obtain an estimate of the radius of the convergence ball for the two-step Secant Method. Moreover, we also provide an error estimate that matches the convergence order of the two-step Secant Method. At last, we give an application of the proposed theorem.
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Convergence analysis for the Secant Method based on new recurrence relations
Applied Mathematics-a Journal of Chinese Universities Series B, 2008Co-Authors: Weihong Bi, Qingbiao WuAbstract:A new convergence theorem for the Secant Method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.
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convergence ball of a modified Secant Method with convergence order 1 839
Applied Mathematics and Computation, 2007Co-Authors: Qingbiao WuAbstract:Abstract A local convergence theorem on a modified Secant Method with convergence order 1.839… for solving nonlinear equations is established under the hypotheses that the second-order and third-order derivative of the function involved are bounded. An estimate of the radius of the convergence ball of this Method is obtained. Moreover, an error estimate is provided which matches the convergence order of the Method.
M J Rubio - One of the best experts on this subject based on the ideXlab platform.
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improving the applicability of the Secant Method to solve nonlinear systems of equations
Applied Mathematics and Computation, 2014Co-Authors: Sergio Amat, M A Hernandezveron, M J RubioAbstract:A modification of the Secant Method for the approximation of nonlinear system of equations is considered to improve the applicability of the Secant Method. This modification changes the resolution of a linear system in each step, necessary to apply the Secant Method, for several matrix multiplications. In this way, the numerical stability of the Secant Method can be improved. In addition, in this paper, we prove that the modification considered keeps two important properties of the Secant Method, such as: does not use derivatives in its algorithm and has R-order of convergence at least 1 + 5 / 2 .
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semilocal convergence of the Secant Method under mild convergence conditions of differentiability
Computers & Mathematics With Applications, 2002Co-Authors: M A Hernandez, M J RubioAbstract:Abstract In this work, we obtain a semilocal convergence result for the Secant Method in Banach spaces under mild convergence conditions. We consider a condition for divided differences which generalizes those usual ones, i.e., Lipschitz continuous and Holder continuous conditions. Also, we obtain a result for uniqueness of solutions.
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the Secant Method for nondifferentiable operators
Applied Mathematics Letters, 2002Co-Authors: M A Hernandez, M J RubioAbstract:In this paper, we use the Secant Method to find a solution of a nonlinear operator equation in Banach spaces. A semilocal convergence result is obtained. For that, we consider a condition for divided differences which generalizes the usual ones, i.e., Lipschitz continuous or Holder continuous conditions. Besides, we apply our results to approximate the solution of a nonlinear equation.
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the Secant Method and divided differences holder continuous
Applied Mathematics and Computation, 2001Co-Authors: M A Hernandez, M J RubioAbstract:We apply the Secant Method to solve non-linear operator equations in Banach spaces. A semilocal convergence result is obtained, where the first-order divided difference of the non-linear operator is Holder continuous. For that, we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution and give an explicit expression for the a priori error bounds. Moreover, we apply our results to the numerical solution of a non-linear boundary value problem of second-order.
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a new type of recurrence relations for the Secant Method
International Journal of Computer Mathematics, 1999Co-Authors: M A Hernandez, M J RubioAbstract:We apply the Secant Method to solve nonlinear operator equations in Banach spaces. We establish a Newton-Kantorovich convergence theorem using a new system of recurrence relations and give an explicit expression for the a priori error bounds. Moreover, we apply our results to the numerical resolution of a nonlinear boundary value problem of second order and improve the error bounds obtained by other authors.
Toshiyuki Koyama - One of the best experts on this subject based on the ideXlab platform.
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Calculation of Stress-Strain Curve of Two-Phase Microstructure on the Basis of the Extended Secant Method
Materials Science Forum, 2020Co-Authors: Toshiyuki Koyama, Hidehiro OnoderaAbstract:The Secant Method proposed by Weng [1] is a practical calculation Method to evaluate the stress-strain curve of the two-phase materials, but the shape of the inclusion phase has been often assumed to be a sphere or an ellipsoid in the calculation. In this study, we modified the Secant Method by utilizing the phase-field micro-elasticity theory [2,3] so as to be able to calculate the SS-curve of the materials consisting arbitrary morphology of microstructure, and applied this Method to the conventional microstructures in steels, i.e., the ferrite-bainite two-phase microstructure.
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calculation of stress strain curve of the two phase mixture including ellipsoidal inclusion on the basis of the modified Secant Method
Tetsu To Hagane-journal of The Iron and Steel Institute of Japan, 2011Co-Authors: Toshiyuki KoyamaAbstract:The stress–strain curve of the composite material has been calculated on the basis of the Secant Method proposed by Weng. Although the effect of inclusion shape on the stress–strain curve has been evaluated by Zhao and Weng, they have assumed the inclusion phase to be a rigid elastic medium which dose not deform plastically. In this study, we modified their Secant Method so as to be able to calculate the stress–strain curve of the composite material which contains the plastically deformed inclusion phase, and evaluated the effect of the inclusion shape on the deformation behavior of the stress–strain curve. The results obtained are as follows: When both phases are soft, i.e. the matrix and inclusion phases are both deformable, the stress–strain curve is determined only from the volume fraction of the inclusion phase. In the case that the inclusion phase is hard, i.e. the matrix phase is deformable but the inclusion phase is less deformable, the stress–strain curve depends not only on the volume fraction of the inclusion phase but also on the shape of the inclusion phase. The modified Secant Method proposed in this study will be useful for understanding the effect of inclusion shape on the mechanical property of the composite materials, comprehensively.
Said Hilout - One of the best experts on this subject based on the ideXlab platform.
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Weaker convergence conditions for the Secant Method
Applications of Mathematics, 2014Co-Authors: Ioannis K. Argyros, Said HiloutAbstract:We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the Secant Method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.
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Semilocal convergence conditions for the Secant Method, using recurrent functions
2011Co-Authors: Ioannis K. Argyros, Said HiloutAbstract:Using our new concept of recurrent functions, we present new sufficient convergence conditions for the Secant Method to a locally unique solution of a nonlinear equation in a Banach space. We combine Lipschitz and center-Lipschitz conditions on the divided difference operator to obtain the semilocal convergence analysis of the Secant Method. Our error bounds are tighter than earlier ones. Moreover, under our convergence hypotheses, we can expand the applicability of the Secant Method in cases not covered before [8], [9], [12]-[14], [16], [19]-[21]. Application and examples are also provided in this study.
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extended sufficient semilocal convergence for the Secant Method
Computers & Mathematics With Applications, 2011Co-Authors: Ioannis K. Argyros, Said HiloutAbstract:We establish new sufficient convergence conditions for the Secant Method to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center-Lipschitz conditions on the divided difference operator, we obtain a new semilocal convergence analysis of the Secant Method. Moreover, our sufficient convergence conditions expand the applicability of the Secant Method in cases not covered before (Dennis, 1971 [9], Hernandez et al., 2005 [8], Laasonen, 1969 [15], Ortega and Rheinboldt, 1970 [11], Potra, 1982 [5], Potra, 1985 [7], Schmidt, 1978 [18], Yamamoto, 1987 [12], Wolfe, 1978 [19]). Numerical examples are also provided in this study.