General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Ivanov, Stoil I.

doi:10.3390/math8091599

Cited by 8 publications

(8 citation statements)

References 18 publications

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“…Proinov [9,10], and later, Ivanov [11] have introduced convergence theorems for the Picard iterative scheme given as:…”

Section: Introductionmentioning

confidence: 99%

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Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Das,

Kumar

2024

Comm. Math. Appl.

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In this article, we have constructed an iterative method of third order for solving polynomial equations with multiple polynomial zeros. We have combined two well known third order methods one is Halley and another is Super-Halley for this construction purpose. This constructed method is basically the mean of the methods Halley and Super-Halley, so we name the method as H-S Combined Mean Method. We have proposed some local convergence theorems of this H-S Combined Mean Method to establish the computation of a polynomial with known multiple zeros. For the establishment of this local convergence theorem, the key role is performed by a function (Real valued) termed as the function of initial conditions. Function of initial conditions I is a mapping from the set D into the set X , where D (subset of X ) is the domain of the H-S Combined Mean iterative scheme. Here the initial conditions uses the information only at the initial point and are given in the form I(w 0 ) which belongs to J, where J is an in interval on the positive real line which also contains zero and w 0 is the starting point. We have used the notion of gauge function which also plays very important role in establishing the convergence theorem. Here we have used two types of initial conditions over an arbitrary normed field and established local convergence theorems of the constructed H-S Combined Mean Method. The error estimations are also found in our convergence analysis. For simple zero, the method as well as the results hold good.

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“…Proinov [9,10], and later, Ivanov [11] have introduced convergence theorems for the Picard iterative scheme given as:…”

Section: Introductionmentioning

confidence: 99%

“…Here, we investigate convergence of the H-S Combined Mean Method for polynomial zeros which are multiple in nature with the help of the same initial conditions as in the Proinov [ [10], [9]] and Ivanov [11].…”

Section: Introductionmentioning

confidence: 99%

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Das,

Kumar

2024

Comm. Math. Appl.

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“…In 2009 Proinov [10] established two forms of local convergence theorems about Newton's technique under two types of initial conditions. Recently, Proinov [11], [12] and later Ivanov [13] have introduced convergence theorems for the Picard iterative scheme given as below wm+1 = T (wm), m = 0, 1, 2, . .…”

Section: Introductionmentioning

confidence: 99%

“…Here, we investigate convergence of the C-S combined mean method for polynomial zeros which are multiple in nature with the help of the same initial conditions as in the Proinov [11], [12], [10] and Ivanov [13].…”

Section: Introductionmentioning

confidence: 99%

Construction and Convergence of the C-S Combined Mean Method for Multiple Polynomial Zeros

Das¹,

Kumar

2023

ARJOM

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In this article, we have combined two well known third order methods one is Chebyshev and another is Super- Halley to form an iterative method of third for solving polynomial equations with multiple polynomial zeros. This constructed method is basically the mean of the methods Chebyshev and Super-Halley, so we name the method as C-S Combined Mean Method. We have proposed some local convergence theorems of this C-S Combined Mean Method to establish the computation of a polynomial with known multiple zeros. For the establishment of this local convergence theorem, the key role is performed by a function(Real valued) termed as the function of initial conditions. Function of initial conditions I is a mapping from the set D into the set M , where D (subset of M ) is the domain of the C-S Combined mean iterative scheme. Here the initial conditions uses the information only at the initial point and are given in the form I(w0) which belongs to J , where J is an in interval on the positive real line which also contains 0 and w0 is the starting point. We have used the notion of gauge function which also plays very important role in establishing the convergence theorem. Here we have used two types of initial conditions over an arbitrary normed field and established local convergence theorems of the constructed C-S Combined mean method. The error estimations are also found in our convergence analysis. For simple zero, the method as well as the results hold good.

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“…Apart from Picard, Mann, and Ishikawa, many iterative schemes with better convergence rates are obtained; see, for example, [3][4][5][6][7][8][9][10][11]. In many cases, these algorithms cannot obtain strong convergence; therefore, it was necessary to investigate new effective algorithms.…”

Section: Introductionmentioning

confidence: 99%

Approximation of the Fixed Point for Unified Three-Step Iterative Algorithm with Convergence Analysis in Busemann Spaces

Almusawa

Hammad

Sharma³

2021

Axioms

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In this manuscript, a new three-step iterative scheme to approximate fixed points in the setting of Busemann spaces is introduced. The proposed algorithms unify and extend most of the existing iterative schemes. Thereafter, by making consequent use of this method, strong and Δ-convergence results of mappings that satisfy the condition (Eμ) in the framework of uniformly convex Busemann space are obtained. Our results generalize several existing results in the same direction.

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General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Cited by 8 publications

References 18 publications

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Construction and Convergence of the C-S Combined Mean Method for Multiple Polynomial Zeros

Approximation of the Fixed Point for Unified Three-Step Iterative Algorithm with Convergence Analysis in Busemann Spaces

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