A Method to Accelerate the Convergence of the Secant Algorithm

Nijmeijer, M. J. P.

doi:10.1155/2014/321592

Cited by 5 publications

(5 citation statements)

References 17 publications

(26 reference statements)

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“…Both tables use the function f that is also used in Table 1. It is shown in appendix C of [17] that C 2 = −2 and C n = (−1) n+1 for n > 2 for this function. This gives A …”

Section: Appendix a Recursion With Disturbancementioning

confidence: 89%

“…Lemma 3 in [17] states that instead of the condition f ∈ C n+2 (I), the two conditions f ∈ C n+1 (I) and f (n+1) being Lipschitz continuous on I will already guarantee the boundedness of A n in the case of the method of improved approximants. Hence we can weaken the condition f ∈ C n+2 (I) in Lemma 1 to these two conditions in the case where the parallel algorithm is implemented with improved approximants.…”

Section: Basic Convergencementioning

confidence: 99%

“…An algorithm with a higher order of convergence requires fewer iterations if the required accuracy of the approximation is high enough. The fastest derivative-free methods with one function evaluation per iteration achieve orders of convergence asymptotically close to two [5,17,24,27].…”

Section: Introductionmentioning

confidence: 97%

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A parallel root-finding algorithm

Nijmeijer¹

2015

LMS J. Comput. Math.

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We present a parallel algorithm to calculate a numerical approximation of a single, isolated root α of a function f : R → R which is sufficiently regular at and around α. The algorithm is derivative free and performs one function evaluation on each processor per iteration. It requires at least three processors and can be scaled up to any number of these. The order with which the generated sequence of approximants converges to α is equal to (n + √ n 2 + 4)/2 for n + 1 processors with n 2. This assumes that particular combinations of the derivatives of f do not vanish at α.

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“…Both tables use the function f that is also used in Table 1. It is shown in appendix C of [17] that C 2 = −2 and C n = (−1) n+1 for n > 2 for this function. This gives A …”

Section: Appendix a Recursion With Disturbancementioning

confidence: 89%

Section: Basic Convergencementioning

confidence: 99%

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A parallel root-finding algorithm

Nijmeijer¹

2015

LMS J. Comput. Math.

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“…Yet another generalization of the secant method for finding simple real roots of f (x) was recently given by Nijmeijer [10]. This method too requires no derivative information, requires one evaluation of f (x) per iteration, and has the same order of convergence as our method.…”

mentioning

confidence: 92%

Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Sidi

2021

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The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

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“…Due to the well performance of the secant method, secant method and secant-like methods have been widely studied by many authors [3][4][5][6][7][8][9][10][11]. The authors [12] proposed a new method for solving the nonlinear equation.…”

Section: Introductionmentioning

confidence: 99%

On the Convergence Ball and Error Analysis of the Modified Secant Method

Lin

Chen

et al. 2018

Advances in Mathematical Physics

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We aim to study the convergence properties of a modification of secant iteration methods. We present a new local convergence theorem for the modified secant method, where the derivative of the nonlinear operator satisfies Lipchitz condition. We introduce the convergence ball and error estimate of the modified secant method, respectively. For that, we use a technique based on Fibonacci series. At last, some numerical examples are given.

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A Method to Accelerate the Convergence of the Secant Algorithm

Cited by 5 publications

References 17 publications

A parallel root-finding algorithm

A parallel root-finding algorithm

Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

On the Convergence Ball and Error Analysis of the Modified Secant Method

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