Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications

The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones.

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Extended Convergence of a Two-Step-Secant-Type Method Under a Restricted Convergence Domain

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2101.155a ◽

2020 ◽

Vol 45 (01) ◽

pp. 155-164

Author(s):

IOANNIS K. ARGYROS ◽

GEORGE SANTHOSH

Keyword(s):

Computational Effort ◽

Convergence Domain ◽

Type Method ◽

Precise Information ◽

Numerical Examples ◽

Special Cases ◽

Lipschitz Constants ◽

Local Convergence Analysis ◽

Lipschitz Conditions ◽

Theoretical Results

We present a local as well as a semi-local convergence analysis of a two-step secant-type method for solving nonlinear equations involving Banach space valued operators. By using weakened Lipschitz and center Lipschitz conditions in combination with a more precise domain containing the iterates, we obtain tighter Lipschitz constants than in earlier studies. This technique lead to an extended convergence domain, more precise information on the location of the solution and tighter error bounds on the distances involved. These advantages are obtained under the same computational eﬀort, since the new constants are special cases of the old ones used in earlier studies. The new technique can be used on other iterative methods. The numerical examples further illustrate the theoretical results.

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Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

Mathematics ◽

10.3390/math7050463 ◽

2019 ◽

Vol 7 (5) ◽

pp. 463 ◽

Cited By ~ 2

Author(s):

Ioannis K. Argyros ◽

Ángel Alberto Magreñán ◽

Lara Orcos ◽

Íñigo Sarría

Keyword(s):

Banach Space ◽

Newton’S Method ◽

Newton's Method ◽

Convergence Region ◽

Precise Information ◽

Solution Of Equations ◽

Lipschitz Constants ◽

Uniqueness Of The Solution ◽

Generalized Newton ◽

Theoretical Results

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

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New Improvement of the Domain of Parameters for Newton’s Method

Mathematics ◽

10.3390/math8010103 ◽

2020 ◽

Vol 8 (1) ◽

pp. 103 ◽

Cited By ~ 1

Author(s):

Cristina Amorós ◽

Ioannis K. Argyros ◽

Daniel González ◽

Ángel Alberto Magreñán ◽

Samundra Regmi ◽

...

Keyword(s):

Newton’S Method ◽

Newton's Method ◽

Convergence Domain ◽

Operator Equations ◽

Numerical Examples ◽

New Information ◽

Analogous Manner ◽

The One ◽

Domain Of Parameters ◽

Local Convergence Analysis

There is a need to extend the convergence domain of iterative methods for computing a locally unique solution of Banach space valued operator equations. This is because the domain is small in general, limiting the applicability of the methods. The new idea involves the construction of a tighter set than the ones used before also containing the iterates leading to at least as tight Lipschitz parameters and consequently a finer local as well as a semi-local convergence analysis. We used Newton’s method to demonstrate our technique. However, our technique can be used to extend the applicability of other methods too in an analogous manner. In particular, the new information related to the location of the solution improves the one in previous studies. This work also includes numerical examples that validate the proven results.

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Extending the convergence domain of Newton’s method for twice Fréchet differentiable operators

Analysis and Applications ◽

10.1142/s0219530515500013 ◽

2016 ◽

Vol 14 (02) ◽

pp. 303-319

Author(s):

Ioannis K. Argyros ◽

Á. Alberto Magreñán

Keyword(s):

Convergence Analysis ◽

Newton’S Method ◽

Newton's Method ◽

Computational Cost ◽

Semilocal Convergence ◽

Convergence Criteria ◽

Fréchet Differentiable ◽

Local Convergence Analysis ◽

Appl Math ◽

Semilocal Convergence Theorem

We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79 (1997) 131–145]. Numerical examples where the old convergence criteria cannot apply to solve nonlinear equations but the new convergence criteria are satisfied are also presented at the concluding section of this paper.

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On the quadratic convergence of Newton’s method under center-Lipschitz but not necessarily Lipschitz hypotheses

Mathematica Slovaca ◽

10.2478/s12175-013-0123-4 ◽

2013 ◽

Vol 63 (3) ◽

Cited By ~ 5

Author(s):

Ioannis Argyros ◽

Saïd Hilout

Keyword(s):

Banach Space ◽

Newton’S Method ◽

Newton's Method ◽

Quadratic Convergence ◽

Lipschitz Constant ◽

Semilocal Convergence ◽

Convergence Conditions ◽

Numerical Examples ◽

Convergence Results

AbstractWe provide new local and semilocal convergence results for Newton’s method in a Banach space. The sufficient convergence conditions do not include the Lipschitz constant usually associated with Newton’s method. Numerical examples demonstrating the expansion of Newton’s method are also provided in this study.

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Extending the Usage of Newton’s Method with Applications to the Solution of Bratu’s Equation

Mathematics ◽

10.3390/math6120274 ◽

2018 ◽

Vol 6 (12) ◽

pp. 274

Author(s):

Ioannis Argyros ◽

Daniel González

Keyword(s):

Convergence Analysis ◽

Newton’S Method ◽

Newton's Method ◽

New Developments ◽

Numerical Examples ◽

Special Cases ◽

Restricted Domains ◽

Lipschitz Constants

We use Newton’s method to solve previously unsolved problems, expanding the applicability of the method. To achieve this, we used the idea of restricted domains which allows for tighter Lipschitz constants than previously seen, this in turn led to a tighter convergence analysis. The new developments were obtained using special cases of functions which had been used in earlier works. Numerical examples are used to illustrate the superiority of the new results.

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A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽

10.3390/math9161855 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1855 ◽

Cited By ~ 1

Author(s):

Petko D. Proinov ◽

Maria T. Vasileva

Keyword(s):

Iterative Methods ◽

Local Convergence ◽

Semilocal Convergence ◽

High Order ◽

Convergence Theorems ◽

New Family ◽

Special Cases ◽

Iteration Functions ◽

Iteration Function ◽

Local Convergence Analysis

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

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ON THE SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SECTIONS ON RIEMANNIAN MANIFOLDS

Asian-European Journal of Mathematics ◽

10.1142/s1793557114500077 ◽

2014 ◽

Vol 07 (01) ◽

pp. 1450007

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Convergence Analysis ◽

Lipschitz Condition ◽

Riemannian Manifolds ◽

Newton’S Method ◽

Newton's Method ◽

Geometric Model ◽

Semilocal Convergence ◽

Convergence Result ◽

Numer Anal ◽

The One

We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] in combination with the weaker center 2-piece L1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal.22 (2002) 359–390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math.8 (2008) 197–226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant α-theory, IMA J. Numer. Anal.23 (2003) 395–419].

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