Advances in the Efficiency of Computational Methods and Applications

Argyros, Ioannis K.

doi:10.1142/4448

Cited by 34 publications

(62 citation statements)

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“…We also note that the IN model is a particular instance of the IPN model. These facts show the equivalence of these 1 We have recently noticed that Wilkinson [36] has previously considered the iterates…”

Section: The Same Conclusion Holds If the Above Condition Is Replaced Bymentioning

confidence: 57%

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

Cătinaș

2004

Math. Comp.

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Abstract. A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974) which assumes that at each step the linear systems are perturbed and then they are only approximately solved; we have characterized the high convergence orders of these iterates in terms of the perturbations and residuals.In the present paper we show that these three models are in fact equivalent, in the sense that each one may be used to characterize the high convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.

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Section: The Same Conclusion Holds If the Above Condition Is Replaced Bymentioning

confidence: 57%

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

Cătinaș

2004

Math. Comp.

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“…Hypotheses of the form (14), (30) and (31) are as easy to handle as (11) (see also (49) and (50)) and are always present as sufficient convergence conditions in the study of Newton's method (2) [3], [7]. Note that t * can be replaced by 2η/(2 − δ) in condition (26).…”

Section: Theoremmentioning

confidence: 99%

“…A survey of local and semilocal convergence theorems for Newton's method under various conditions on the Fréchet-derivative F (x) of operator F (x) can be found in [3], [4], [7], [9], [11].…”

Section: Introductionmentioning

confidence: 99%

New and Generalized Convergence Conditions for the Newton-Kantorovich Method

Argyros¹

2003

Journal of Applied Analysis

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Abstract. We present new semilocal convergence theorems for Newton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton's method under Lipschitz conditions we show that the famous Newton-Kantorovich hypothesis having gone unchallenged for a long time can be weakened under the same hypotheses/computational cost.

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“…Using (3) we can arrive at the famous Newton-Kantorovich condition (14) which is sufficient for the convergence of Newton's method (2). A survey of local and semilocal convergence results for Newton's method (2) can be found in [1], [2], [5], [6], [8], [10], [11], and the references there.…”

mentioning

confidence: 99%

“…Semilocal convergence analysis for Newton's method (2). It is convenient to define a scalar iteration {t n } (n ≥ 0) for some given η ≥ 0,…”

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confidence: 99%

A new approach for finding weaker conditions for the convergence of Newton's method

Argyros

2005

Appl. Math. (Warsaw)

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Abstract. The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can be obtained this way. Here we show that we can further weaken conditions (18)- (20) A large number of problems in applied mathematics and also in engineering are solved by finding solutions of certain equations. For example, dynamical systems are mathematically modeled by difference or differential equations, and their solutions usually represent states of the systems. For the sake of simplicity, assume that a time-invariant system is driven by the equationẋ = G(x) (for some suitable operator G), where x is the state. Then the equilibrium states are determined by solving equation (1). Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential, and integral equations), 2000 Mathematics Subject Classification: 65H10, 65G99, 47H17, 49M15.

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Advances in the Efficiency of Computational Methods and Applications

Cited by 34 publications

References 0 publications

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

New and Generalized Convergence Conditions for the Newton-Kantorovich Method

A new approach for finding weaker conditions for the convergence of Newton's method

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