Multistep Quadrature Based Methods for Nonlinear System of Equations with Singular Jacobian

Oghovese, Ogbereyivwe; Muka, K. O.

doi:10.4236/jamp.2019.73049

Cited by 3 publications

(3 citation statements)

References 34 publications

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“…Next we establish the convergence of the families of means-based IM given in (2.12) for different types of mean used in M T . The Taylor's expansion technique also used in literature [15,16,17,13] and some reference therein, was used in proving the convergence of the method. First, the Proposition 2.1 is considered.…”

Section: Methods Formulationmentioning

confidence: 99%

“…In order to improve the CO and EI of (1.2), many authors have used different techniques. A good review on some of these techniques such as the geometric, functional, composition, sampling, Adomain decomposition, rational function and weight function techniques can be found in [1,15,16,17] and reference therein. The Weerakoon and Fernando (W-F) method in [25] is one modified form of the NM (1.2), put forward by replacing the function f (x) in (1.2) with the arithmetic mean of the functions f (x) and f (w).…”

Section: Introductionmentioning

confidence: 99%

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Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

2021

Punjab Univ. j. math.

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In literature, the arithmetic mean of the two functions in the denominator of the second step of order three Weerakoon and Fernando (2000) iterative method have been replaced with other different means. However, these actions have not improve its order of convergence. To improve the order of convergence of these modified methods, a generic family of iterative methods that involve two weight functions and a generic consequential function for replacement of means is proposed. The analysis of convergence carried out on the families of methods, shows that they are of fourth order convergence and requires evaluation of three functions per iteration cycle. Further, the flexibility of the weight functions enables the re-discovery of some existing and construction of new families of iterative methods. Some concrete members of the family of methods are applied to solve some nonlinear equations and real life problems that are modeled into nonlinear equations

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Section: Methods Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

2021

Punjab Univ. j. math.

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“…This domain is adressed by considering classical quadrature formulas see [17,34]. Furthermore, the connection has been re-established by Noor [28] and derived the quadrature-based fifth-order convergent method by using the specially derived Gaussian quadrature formula for solving non-linear equations. By implementing the decomposition technique [9], Ogbereyivwe and Muka [32] have developed quadrature-based families of iterative methods for solving system of nonlinear equations subject to singular jacobian.…”

Section: Introductionmentioning

confidence: 99%

Solution of nonlinear equations using three point Gaussian quadrature formula and decomposition technique

Sana

Noor

2021

Punjab Univ. j. math.

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The problem of solving nonlinear equations (real or complex) is a nontrivial task in many areas of science and engineering. Usually, the analytic methods for such equations are not directly affordable and require an iterative approach for getting an approximate solution. Keeping in view the above facts, we suggest and analyze some new iterative methods for solving nonlinear equation of the form f(u) = 0 by using the decomposition technique coupled with a system of equations and threepoints Gaussian quadrature formula. We also determine the convergence order of our proposed iterative methods. Some test examples are given to endorse and validate the performance of new methods as compared to previously well-known methods.

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Generalized high-order iterative methods for solutions of nonlinear systems and their applications

Thangkhenpau,

Panday,

Panday

et al. 2024

MATH

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<abstract><p>In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.</p></abstract>

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Multistep Quadrature Based Methods for Nonlinear System of Equations with Singular Jacobian

Cited by 3 publications

References 34 publications

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

Solution of nonlinear equations using three point Gaussian quadrature formula and decomposition technique

Generalized high-order iterative methods for solutions of nonlinear systems and their applications

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