Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, the hypotheses of the present study are based on the first Fréchet-derivative only, thereby the application of methods is expanded. New analysis also provides the radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of derivatives of higher order. Moreover, the order of convergence for the methods is verified by using computational order of convergence or approximate computational order of convergence without using higher order derivatives. Numerical examples are provided to verify the theoretical results and to show the good convergence behavior.

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Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽

10.3390/math6110260 ◽

2018 ◽

Vol 6 (11) ◽

pp. 260 ◽

Cited By ~ 3

Author(s):

Janak Sharma ◽

Ioannis Argyros ◽

Sunil Kumar

Keyword(s):

Iterative Method ◽

Iterative Methods ◽

Higher Order ◽

High Order ◽

Convergence Order ◽

Eighth Order ◽

First Derivative ◽

Ball Convergence ◽

Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

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Convergence Ball and Complex Geometry of an Iteration Function of Higher Order

Mathematics ◽

10.3390/math7010028 ◽

2018 ◽

Vol 7 (1) ◽

pp. 28 ◽

Cited By ~ 2

Author(s):

Deepak Kumar ◽

Ioannis Argyros ◽

Janak Sharma

Keyword(s):

Complex Geometry ◽

Basins Of Attraction ◽

Higher Order ◽

Order Method ◽

Eighth Order ◽

Science And Engineering ◽

Convergence Ball ◽

Convergence Results ◽

Iteration Function ◽

Derivatives Of

Higher-order derivatives are used to determine the convergence order of iterative methods. However, such derivatives are not present in the formulas. Therefore, the assumptions on the higher-order derivatives of the function restrict the applicability of methods. Our convergence analysis of an eighth-order method uses only the derivative of order one. The convergence results so obtained are applied to some real problems, which arise in science and engineering. Finally, stability of the method is checked through complex geometry shown by drawing basins of attraction of the solutions.

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Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces

Mathematics ◽

10.3390/math9192510 ◽

2021 ◽

Vol 9 (19) ◽

pp. 2510

Author(s):

Deepak Kumar ◽

Sunil Kumar ◽

Janak Raj Sharma ◽

Lorentz Jantschi

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Complex Geometry ◽

Basins Of Attraction ◽

New Approach ◽

Convergence Error ◽

Uniqueness Of The Solution ◽

Dynamical Nature ◽

Fifth Order ◽

Local Convergence Analysis

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric.

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Why using symbolic computation for proving the convergence of iterative methods?

Creative Mathematics and Informatics ◽

10.37193/cmi.2013.02.14 ◽

2013 ◽

Vol 22 (2) ◽

pp. 127-134

Author(s):

GHEORGHE ARDELEAN ◽

◽

LASZLO BALOG ◽

Keyword(s):

Nonlinear Equations ◽

Symbolic Computation ◽

Newton’S Method ◽

Newton's Method ◽

Order Of Convergence ◽

Higher Order ◽

Order Convergence ◽

Convergence Error ◽

Higher Order Convergence ◽

Appl Math

In [YoonMe Ham et al., Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477–486], some higher-order modifications of Newton’s method for solving nonlinear equations are presented. In [Liang Fang et al., Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math., 228 (2009) 296–303], the authors point out some flaws in the results of YoonMe Ham et al. and present some modified variants of the method. In this paper we point out that the paper of Liang Fang et al. itself contains some flaw results and we correct them by using symbolic computation in Mathematica. Moreover, we show that the main result in Theorem 3 of Liang Fang et al. is wrong. The order of convergence of the method is’nt 3m+2, but is 2m+4. We give the general expression of convergence error too.

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Constructing a High-Order Globally Convergent Iterative Method for Calculating the Matrix Sign Function

Mathematical Problems in Engineering ◽

10.1155/2018/8973867 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9

Author(s):

Haifa Bin Jebreen

Keyword(s):

Iterative Method ◽

Order Of Convergence ◽

Eighth Order ◽

Matrix Sign Function ◽

Sign Function ◽

Globally Convergent ◽

The Matrix ◽

Matrix Iteration ◽

Particulate Matrix ◽

Theoretical Results

This work is concerned with the construction of a new matrix iteration in the form of an iterative method which is globally convergent for finding the sign of a square matrix having no eigenvalues on the axis of imaginary. Toward this goal, a new method is built via an application of a new four-step nonlinear equation solver on a particulate matrix equation. It is discussed that the proposed scheme has global convergence with eighth order of convergence. To illustrate the effectiveness of the theoretical results, several computational experiments are worked out.

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ANALYSIS OF THE GENERAL EQUATIONS OF THE TRANSVERSE VIBRATION OF A PIECEWISE UNIFORM VISCOELASTIC PLATE

Computational nanotechnology ◽

10.33693/2313-223x-2020-7-3-52-56 ◽

2020 ◽

Vol 7 (3) ◽

pp. 52-56

Author(s):

MMATMATISA JALILOV ◽

◽

RUSTAM RAKHIMOV ◽

Keyword(s):

Cross Section ◽

Transverse Vibration ◽

Three Dimensional ◽

Viscoelastic Plate ◽

Constant Thickness ◽

Regional Problem ◽

Rigid Contact ◽

Under Construction ◽

Derivatives Of ◽

Theoretical Results

This article discusses the analysis of the general equations of the transverse vibration of a piecewise homogeneous viscoelastic plate obtained in the “Oscillation of inlayer plates of constant thickness” [1]. In the present work on the basis of a mathematical method, the approached theory of fluctuation of the two-layer plates, based on plate consideration as three dimensional body, on exact statement of a three dimensional mathematical regional problem of fluctuation is stood at the external efforts causing cross-section fluctuations. The general equations of fluctuations of piecewise homogeneous viscoelastic plates of the constant thickness, described in work [1], are difficult on structure and contain derivatives of any order on coordinates x, y and time t and consequently are not suitable for the decision of applied problems and carrying out of engineering calculations. For the decision of applied problems instead of the general equations it is expedient to use confidants who include this or that final order on derivatives. The classical equations of cross-section fluctuation of a plate contain derivatives not above 4th order, and for piecewise homogeneous or two-layer plates the elementary approached equation of fluctuation is the equation of the sixth order. On the basis of the analytical decision of a problem the general and approached decisions of a problem are under construction, are deduced the equation of fluctuation of piecewise homogeneous two-layer plates taking into account rigid contact on border between layers, and also taking into account mechanical and rheological properties of a material of a plate. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely the is intense-deformed status of plates at non-stationary external loadings.

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Experimental and Theoretical Investigations of Complex Formation of Substituted Phenylazo-Derivatives of Methylphloroglucinol

JOURNAL OF ADVANCES IN CHEMISTRY ◽

10.24297/jac.v12i8.2837 ◽

2016 ◽

Vol 12 (8) ◽

pp. 295-300

Author(s):

Olga Kovalchukova ◽

Amangdam A.T. ◽

Strashnova S.B. ◽

Strashnov P.V. ◽

Romashkina E.P. ◽

...

Keyword(s):

Complex Formation ◽

Organic Ligand ◽

Spectrophotometric Titration ◽

Organic Ligands ◽

Oh Groups ◽

Tautomeric Forms ◽

Theoretical Investigations ◽

Derivatives Of ◽

Theoretical Results ◽

Titration Technique

Using spectrophotometric titration technique, the processes of complex formation of some phenylazo-derivatives of methylphloroglucinol (MPG) containing hydroxo-, nitro- and nitroso-substituents were studied. The spectral criteria of neutral and ionized forms of the organic ligands in their different tautomeric forms were determined.It was detected that the complex formation is accompanied by formation of one or two chelate cycles which involve azo- or nitroso-fragments and neighboring OH-groups of the organic ligands. Different types of coordination lead to different changes in the electronic absorption spectra.The DFT-B3LYP modeling of a Ni(II) complex of α-hydroxyphenylazo MPG established the most probable coordination mode of the organic ligand: tridentate chelating dianion, distorted square coordination of Ni-cations including one water molecule. The theoretical results are in a good accordance with the experimental data.

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A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

Journal of Sandwich Structures & Materials ◽

10.1177/1099636217729731 ◽

2017 ◽

Vol 21 (6) ◽

pp. 1820-1842

Author(s):

Wu Zhen ◽

Ma Rui ◽

Chen Wanji

Keyword(s):

Transverse Shear ◽

Equilibrium Equation ◽

Three Dimensional ◽

Higher Order ◽

Triangular Element ◽

Shear Stresses ◽

Sandwich Plates ◽

Transverse Shear Stress ◽

Transverse Shear Stresses ◽

Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

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