Certain improvements of Newton’s method with fourth-order convergence

Chun, Changbum; Neta, Beny

doi:10.1016/j.amc.2009.06.007

Cited by 14 publications

(11 citation statements)

References 28 publications

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“…We did not insert numerical results obtained by iterative methods of lower computational efficiency since these methods are not competitive and their lower rank is predictable. Another optimal two-point methods of the fourth order with the efficiency index 4 1/3 were considered in [16][17][18][19][20][21][22], but the cited papers do not exhaust all sources.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Derivative free two-point methods with and without memory for solving nonlinear equations

Petković

Ilić

Džunić

2010

Applied Mathematics and Computation

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Section: Numerical Resultsmentioning

confidence: 99%

“…Since 1 À b k f 0 (a) = O(e kÀ1 ) (see (20)), from a relation similar to (21) we obtain e kþ1 $ e 4 k e 2 kÀ1 :…”

Section: Methods (Ii)mentioning

confidence: 95%

Derivative free two-point methods with and without memory for solving nonlinear equations

Petković

Ilić

Džunić

2010

Applied Mathematics and Computation

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“…We present some numerical test results for various fourth-order iterative schemes in Table 4. We compare our method with King's method with β = 3 [13] (KM), Kou's method [14] (KouM) and fourth order Chun's method [5](CM).…”

Section: − −mentioning

confidence: 99%

“…For numerical illustrations in this section we used the fixed stopping criterion ǫ < 10 −25 . We used the test functions given in [5,15,22]. We present some numerical test results for various fourth-order iterative schemes in Table 4.…”

Section: − −mentioning

confidence: 99%

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A fourth order method for finding a simple root of univariate function

Sbibih

Serghini

Tijini

et al. 2015

bspm

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In this paper, we describe an iterative method for approximating a simple zero z of a real defined function. This method is a essentially based on the idea to extend Newton's method to be the inverse quadratic interpolation. We prove that for a sufficiently smooth function f in a neighborhood of z the order of the convergence is quartic. Using Mathematica with its high precision compatibility, we present some numerical examples to confirm the theoretical results and to compare our method with the others given in the literature.

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Semianalytical minimum‐time solution for the optimal control of a vehicle subject to limited acceleration

Bertolazzi

Frego

2017

Optim Control Appl Methods

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Summary A semianalytic solution of the minimum‐time optimal control problem of a carlike vehicle is herein presented. The vehicle is subject to the effect of laminar (linear) and aerodynamic (quadratic) drag, taking into account the asymmetric bounded longitudinal accelerations. This yields a nonlinear differential equation of the Riccati kind. The associated analytic solution exists in closed form, but it is numerically ill conditioned in some situations; hence, it is reformulated using asymptotic expansions that keep the numerical computations accurate and robust in all cases. Therefore, the proposed algorithm is designed to be fast and robust in sight to be the fundamental module of a more extended optimal control problem that considers a given clothoid curve as the trajectory and the presence of a constraint on the lateral acceleration of the vehicle. The numerical stability of the computation for limit values of the parameters is essential, as shown in the numerical tests.

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Certain improvements of Newton’s method with fourth-order convergence

Cited by 14 publications

References 28 publications

Derivative free two-point methods with and without memory for solving nonlinear equations

Derivative free two-point methods with and without memory for solving nonlinear equations

A fourth order method for finding a simple root of univariate function

Semianalytical minimum‐time solution for the optimal control of a vehicle subject to limited acceleration

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