Computing multiple zeros by using a parameter in Newton–Secant method

Феррара, Массимилиано; Sharifi, Somayeh; Salimi, Mehdi

doi:10.1007/s40324-016-0074-0

Cited by 10 publications

(8 citation statements)

References 25 publications

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“…Whereas if used to solve multiple zeros of nonlinear equations the convergence to be linear. Therefore, in [15] modified Newton-Secant method with the addition of parameter 𝜃. The purpose of this modification, namely to maintain the order of convergence Newton-Secant method to remain cubic, if used to find the roots of nonlinear equations [15].…”

Section: Theorem 2 (Bolzano's Theorem)mentioning

confidence: 99%

“…Therefore, in [15] modified Newton-Secant method with the addition of parameter 𝜃. The purpose of this modification, namely to maintain the order of convergence Newton-Secant method to remain cubic, if used to find the roots of nonlinear equations [15]. The following theorem related to the parameter 𝜃 used to modify the Newton-Secant method :…”

Section: Theorem 2 (Bolzano's Theorem)mentioning

confidence: 99%

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On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function

Juhari

2021

CAUCHY

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This study discusses the analysis of the modification of Newton-Secant method and solving nonlinear equations having a multiplicity of by using a modified Newton-Secant method. A nonlinear equation that has a multiplicity is an equation that has more than one root. The first step is to analyze the modification of the Newton-Secant method, namely to construct a mathematical model of the Newton-Secant method using the concept of the Newton method and the concept of the Secant method. The second step is to construct a modified mathematical model of the Newton-Secant method by adding the parameter . After obtaining the modified formula for the Newton-Secant method, then applying the method to solve a nonlinear equations that have a multiplicity . In this case, it is applied to the nonlinear equation which has a multiplicity of . The solution is done by selecting two different initial values, namely and . Furthermore, to determine the effectivity of this method, the researcher compared the result with the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified. The obtained results from the analysis of modification of Newton-Secant method is an iteration formula of the modified Newton-Secant method. And for the result of using a modified Newton-Secant method with two different initial values, the root of is obtained approximately, namely with less than iterations. whereas when using the Newton-Raphson method, the Secant method, and the Newton-Secant method, the root is also approximated, namely with more than iterations. Based on the problem to find the root of the nonlinear equation it can be concluded that the modified Newton-Secant method is more effective than the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified

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Section: Theorem 2 (Bolzano's Theorem)mentioning

confidence: 99%

Section: Theorem 2 (Bolzano's Theorem)mentioning

confidence: 99%

On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function

Juhari

2021

CAUCHY

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“…The above expression interprets real and ideal gas behavior with variables a 1 and a 2 , respectively. For calculating the gas volume V, we can rewrite the above expression (27) in the following way:…”

Section: Casesmentioning

confidence: 99%

“…In order to graphically compare by means of attraction basins, we investigate the dynamics of the new methods PM1 8 , PM2 8 , PM3 8 and PM4 8 and compare them with available methods from the literature, namely SM 8 , KT 8 and KM 8 . For more details and many other examples of the study of the dynamic behavior for iterative methods, one can consult [26][27][28][29].…”

Section: Graphical Comparison By Means Of Attraction Basinsmentioning

confidence: 99%

Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

et al. 2019

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In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

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“…[2], [4][5][6][7]. Using inverse interpolation, Kung and Traub [3] constructed two general optimal classes without memory.…”

Section: Introductionmentioning

confidence: 99%

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

et al. 2016

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Abstract:In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari's method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to 8 1 4 1:682. We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

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Computing multiple zeros by using a parameter in Newton–Secant method

Cited by 10 publications

References 25 publications

On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function

On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function

Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

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