On the complexity of extending the convergence region for Traub’s method

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.

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“…By the Banach lemma on invertible operators [5,[19][20][21], expression (17), F (x) −1 ∈ (Y, X), and…”

Section: Local Convergencementioning

confidence: 99%

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

2020

Self Cite

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“…There exist continuous and nondecreasing functions ω : (16) where functions f j are given previously and ρ is defined in (10). Furthermore, the limit point x * is the only solution of equation F(x) = 0 in the set U 1 given in (Γ 5 ).…”

Section: Convergencementioning

confidence: 99%

“…so w 0 (t) = w(t) = t. Then, obtain the roots by solving (6), (7), (9) (knowing now w 0 , w, w 1 , ψ 1 , ψ 2 ) using Mathematica. Then, we determine the radii using (10). The same is done in the rest of the examples.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The results obtained here can be used for operators G satisfying autonomous differential equations [3][4][5][6][7][8][9][10] of the form…”

mentioning

confidence: 99%

“…3. The local results obtained here can be used for projection schemes such as the Arnoldi's algorithm, the generalized minimum residual algorithm (GMRES), the generalized conjugate algorithm (GCA) for combined Newton/finite projection schemes and in connection to the mesh independence principle can be used to develop the cheapest and most efficient mesh refinement strategies [3][4][5][6][7][8][9][10]17] .…”

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confidence: 99%

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Extended Local Convergence for High Order Schemes Under ω-Continuity Conditions

Argyros

et al. 2020

Contemporary Mathematics

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There is a plethora of schemes of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them challenging and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives that do not even appear on these schemes. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these schemes and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other schemes along the same lines.

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An algorithm with feasible inexact projections for solving constrained generalized equations

Regmi,

Argyros,

George

2024

Math Methods in App Sciences

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The goal of this article is to design a more flexible algorithm than the ones used previously for solving constrained generalized equations. It turns out that the new algorithm even if specialized provides a finer error analysis with advantages: larger radius of convergence; tighter upper error bounds on the distances; and a more precise information on the isolation of the solution. Moreover, the same advantages exist even if the generalized equation reduces to a nonlinear equation. These advantages are obtained under the same computational cost, since the new parameters and majorant functions are special cases of the ones used in earlier studies. Applications complement the theoretical results.

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On the complexity of extending the convergence region for Traub’s method

Cited by 34 publications

References 12 publications

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Extended Local Convergence for High Order Schemes Under ω-Continuity Conditions

An algorithm with feasible inexact projections for solving constrained generalized equations

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