15 results on '"Local convergence"'
Search Results
2. An alternative analysis for the local convergence of iterative methods for multiple roots including when the multiplicity is unknown.
- Author
-
Alarcón, Diego, Hueso, Jose L., and Martínez, Eulalia
- Subjects
- *
MULTIPLICITY (Mathematics) , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *NUMERICAL analysis , *RADIUS (Geometry) , *NONLINEAR equations , *ALGEBRAIC equations - Abstract
In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the (m + i) th derivative of the function f (x) with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
3. A quasi-Newton modified LP-Newton method.
- Author
-
Martínez, María de los Ángeles and Fernández, Damián
- Subjects
- *
NEWTON-Raphson method , *NONLINEAR equations , *QUASI-Newton methods - Abstract
We consider a method to solve constrained system of nonlinear equations based on a modification of the Linear-Programming-Newton method and replacing the first-order information with a quasi-Newton secant update, providing a computationally simple method. The proposed strategy combines good properties of two methods: the least change secant update for unconstrained system of nonlinear equations with isolated solutions and the Linear-Programming-Newton for constrained nonlinear system of equations with possible nonisolated solutions. We analyse the local convergence of the proposed method under a standard error bound condition proving its linear convergence for nonisolated solutions. Numerical experiments were done in order to show the claimed convergence rate. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
4. A central path interior point method for nonlinear programming and its local convergence.
- Author
-
Qiu, Songqiang and Chen, Zhongwen
- Subjects
- *
NONLINEAR programming , *STOCHASTIC convergence , *ALGORITHMS , *NUMERICAL analysis , *MATHEMATICAL models - Abstract
In this paper, we present an interior point method for nonlinear programming that avoids the use of penalty function or filter. We use an adaptively perturbed primal dual interior point framework to computer trial steps and a central path technique is used to keep the iterate bounded away from 0 and not to deviate too much from the central path. A trust-funnel-like strategy is adopted to drive convergence. We also use second-order correction (SOC) steps to achieve fast local convergence by avoiding Maratos effect. Furthermore, the presented algorithm can avoid the blocking effect. It also does not suffer the blocking of productive steps that other trust-funnel-like algorithm may suffer. We show that, under second-order sufficient conditions and strict complementarity, the full Newton step (combined with an SOC step) will be accepted by the algorithm near the solution, and hence the algorithm is superlinearly local convergent. Numerical experiments results, which are encouraging, are reported. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
5. Local convergence of Newton’s method in the classical calculus of variations.
- Author
-
Gockenbach, Mark and Liu, Chang
- Subjects
- *
STOCHASTIC convergence , *NEWTON-Raphson method , *CALCULUS of variations , *QUADRATIC fields , *APPROXIMATION theory - Abstract
Sufficient conditions for a weak local minimizer in the classical calculus of variations can be expressed without reference to conjugate points. The local quadratic convergence of Newton’s method follows from these sufficient conditions. Newton’s method is applied in the minimization form; that is, the step is generated by minimizing the local quadratic approximation. This allows the extension to a globally convergent line search based algorithm (which will be presented in a future paper). [ABSTRACT FROM PUBLISHER]
- Published
- 2015
- Full Text
- View/download PDF
6. Local convergence for deformed Chebyshev-type method in Banach space under weak conditions.
- Author
-
Argyros, Ioannis K. and George, Santhosh
- Subjects
- *
FRECHET spaces , *CHEBYSHEV approximation , *ITERATIVE methods (Mathematics) , *BANACH spaces , *STOCHASTIC convergence - Abstract
We present a local convergence analysis for deformed Chebyshev methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Chebyshev and other high-order methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second or third Fréchet derivative. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
7. On the Local Convergence of a Penalty-Function-Free SQP Method.
- Author
-
Shen, Chungen, Shao, Wenqiong, and Xue, Wenjuan
- Subjects
- *
STOCHASTIC convergence , *MATHEMATICAL functions , *MATHEMATICAL optimization , *MATHEMATICAL analysis , *NUMBER theory - Abstract
In this article, we propose a nonmonotone linesearch sequential quadratic programming method for general constrained optimization problems without a penalty function or a filter. The algorithm proposed here is a development of the algorithm in Xue et al. [17]. Compared with the former, the novelty of the method we propose is that the new algorithm will achieve the local convergence under weaker assumptions. In order to avoid the Maratos effect, we use the second-order correction in this method, which need not be computed at each iteration. In other words, after a certain number of iterations, there is no need to compute the second-order correction step any more. The global convergence and the locally superlinear convergence of our method are proved under some suitable conditions. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
8. Convergence analysis of a method for variational inclusions.
- Author
-
Rashid, M.H., Wang, J.H., and Li, C.
- Subjects
- *
STOCHASTIC convergence , *SET-valued maps , *PROBLEM solving , *NUMERICAL analysis , *MATHEMATICS theorems , *LIPSCHITZ spaces , *VARIATIONAL approach (Mathematics) - Abstract
Consider the following variational inclusion problem: where f is differentiable in a neighbourhood of a solution and g is differentiable at , and F is a set-valued mapping, and the method introduced in Jean-Alexis and Pietrus [C. Jean-Alexis and A. Pietrus, On the convergence of some methods for variational inclusions, Rev. R. Acad. Cien. serie A. Mat. 102(2) (2008), pp. 355–361] for solving this problem: where ∇f(x) denotes the Fréchet derivative of f at x and [x, y; g] the first-order divided difference of g on the points x and y. Local converge analysis are provided for the method under the weaker conditions than Jean-Alexis and Pietrus (2008). Moreover, if ∇f and the first-order divided difference of g are p-Hölder continuous at a solution, then we show that this method converges superlinearly. In particular, our results extend the corresponding ones Jean-Alexis and Pietrus (2008), and fix a gap in the proof in (Jean-Alexis and Pietrus (2008), Theorem 1). [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
9. Local convergence of filter methods for equality constrained non-linear programming.
- Author
-
Karas, Elizabeth W., Gonzaga, Clóvis C., and Ribeiro, Ademir A.
- Subjects
- *
STOCHASTIC convergence , *ALGORITHMS , *NONLINEAR programming , *LINEAR substitutions , *QUADRATIC programming - Abstract
In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646-669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
10. A new iterative method of asymptotic order 1+√2 for the computation of fixed points.
- Author
-
Argyros, Ioannis K.
- Subjects
- *
ASYMPTOTIC expansions , *TOPOLOGY , *TOPOLOGICAL spaces , *COMPUTATIONAL complexity , *COMPUTATIONAL mathematics , *COMPUTER algorithms - Abstract
We introduce a new iterative method of asymptotic order 1+√2 for approximating fixed points of non-linear equations in a Banach space as well as in a partially topological space. The method requires less computational cost than others in the literature and uses only divided differences of order 1. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
11. On the local convergence of secant-type methods.
- Author
-
Amat *, S., Busquier †, S., and Gutiérrez, J. M.
- Subjects
- *
COMPUTATIONAL mathematics , *STOCHASTIC convergence , *NUMERICAL analysis , *COMPUTERS , *MATHEMATICAL statistics , *HYPOTHESIS - Abstract
In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed. * E-mail: sergio.amat@upct.es † E-mail: sonia.busquier@upct.es [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
12. An Iterative Method for Solving Nonlinear Operator Equations Using Generalized Divided Differences.
- Author
-
Attili, B. S.
- Subjects
- *
NONLINEAR operator equations , *STOCHASTIC convergence , *ITERATIVE methods (Mathematics) , *JACOBIAN matrices , *PARTIAL differential equations , *INTEGRAL equations - Abstract
An iterative method that uses generalized divided differences to solve nonlinear operator equations is proposed. Local and semi local convergence of the proposed method is shown. Numerical examples are also presented to demonstrate the efficiency of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2002
- Full Text
- View/download PDF
13. A quasi-Newton modified LP-Newton method
- Author
-
María de los Ángeles Martínez and Damián Fernández
- Subjects
CONSTRAINED NONLINEAR SYSTEM OF EQUATIONS ,021103 operations research ,Control and Optimization ,Matemáticas ,Applied Mathematics ,0211 other engineering and technologies ,QUASI-NEWTON METHOD ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Local convergence ,Matemática Pura ,symbols.namesake ,LOCAL CONVERGENCE ,Secant method ,NONISOLATED SOLUTIONS ,symbols ,Calculus ,Applied mathematics ,Quasi-Newton method ,0101 mathematics ,Newton's method ,Software ,CIENCIAS NATURALES Y EXACTAS ,Mathematics - Abstract
We consider a method to solve constrained system of nonlinear equations based on a modification of the Linear-Programming-Newton method and replacing the first-order information with a quasi-Newton secant update, providing a computationally simple method. The proposed strategy combines good properties of two methods: the least change secant update for unconstrained system of nonlinear equations with isolated solutions and the Linear-Programming-Newton for constrained nonlinear system of equations with possible nonisolated solutions. We analyse the local convergence of the proposed method under a standard error bound condition proving its linear convergence for nonisolated solutions. Numerical experiments were done in order to show the claimed convergence rate. Fil: Martinez Arraigada, Maria de Los Angeles. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina Fil: Fernández Ferreyra, Damián Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
- Published
- 2017
14. On the dynamics of some newtons type iterative functions
- Author
-
Sonia Busquier, Concepción Bermúdez, P. Leauthier, Sergio Plaza, and Sergio Amat
- Subjects
Nonlinear system ,Conjugacy class ,Computational Theory and Mathematics ,Iterative method ,Applied Mathematics ,Mathematical analysis ,Dynamics (mechanics) ,Applied mathematics ,Type (model theory) ,Complex quadratic polynomial ,Computer Science Applications ,Local convergence ,Mathematics - Abstract
The dynamics of a family of Newton's type iterative methods for second-and third-degree complex polynomials is studied. The conjugacy classes of these methods are presented. Classical properties of rational maps are used.
- Published
- 2009
15. On accelerated iterative methods for the solution of systems of linear equations
- Author
-
Ahmet Yasar Özban and Kırıkkale Üniversitesi
- Subjects
Mathematical optimization ,convergence ,Iterative method ,Applied Mathematics ,nonnegative matrices ,Linear system ,MathematicsofComputing_NUMERICALANALYSIS ,Relaxation (iterative method) ,Stone method ,System of linear equations ,stationary iterative methods ,Computer Science::Numerical Analysis ,Computer Science Applications ,Local convergence ,Computational Theory and Mathematics ,Successive over-relaxation ,Applied mathematics ,Gauss–Seidel method ,irreducible matrices ,Mathematics - Abstract
WOS: 000176035700009 In literature, it has been reported that the convergence of some preconditioned stationary iterative methods using certain type upper triangular matrices as preconditioners are faster than the basic iterative methods. In this paper, a new preconditioned iterative method for the numerical solution of linear systems has been introduced, and the convergence analysis of the proposed method and an existing one have been done. Some numerical examples have also been given, which show the effectiveness of both of the methods.
- Published
- 2002
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.