The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

Kchouk, Bilel; Dussault, Jean-Pièrre

doi:10.1007/s10957-012-0159-6

Cited by 6 publications

(7 citation statements)

References 13 publications

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“…From the analysis, the researchers conclude that that Shamanskii method has shown superior performance compared to Newton method in terms of efficiency whenever work is measured in terms of function evaluations [9]. Also, if the value of t is sufficiently chosen, then, as the dimension increases, the performance of the Shamanskii method improves and thus reduces the limit of complexity of factoring the approximate Jacobian for two pseudo-Newton iterations [14]. Please refer to [15] for the proof of the convergence theorem described below.…”

Section: Shamanskii Methodsmentioning

confidence: 98%

“…Motivated by this, a method due originally to Shamanskii [11] was developed and analyzed by [7,13,14,16,24]. Starting with an initial approximation x 0 , this method uses the multiple pseudo-Newton approach as described below:…”

Section: Shamanskii Methodsmentioning

confidence: 99%

“…(7) can be referred to [9,10]. Motivated by the excellent convergence of Newton method and low cost of Jacobian evaluation of chord method, a method due originally to Shamanskii [11,12] that lies between Newton method and chord method was proposed and has been analyzed in Kelly [9,[13][14][15]. Other variants of Newton methods with different Jacobian approximation schemes include [9,14,[16][17][18].…”

Section: Fx ðþ¼0mentioning

confidence: 99%

“…Motivated by the excellent convergence of Newton method and low cost of Jacobian evaluation of chord method, a method due originally to Shamanskii [11,12] that lies between Newton method and chord method was proposed and has been analyzed in Kelly [9,[13][14][15]. Other variants of Newton methods with different Jacobian approximation schemes include [9,14,[16][17][18]. However, most of these methods require the computation and storage of the full or approximate Jacobian, which become very difficult and time-consuming as the dimension of systems increases [10,19].…”

Section: Fx ðþ¼0mentioning

confidence: 99%

See 3 more Smart Citations

A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations

Sulaiman¹,

Mamat²,

Omesa³

2020

Nonlinear Systems -Theoretical Aspects and Recent Applications

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Newton-type methods with diagonal update to the Jacobian matrix are regarded as one most efficient and low memory scheme for system of nonlinear equations. One of the main advantages of these methods is solving nonlinear system of equations having singular Fréchet derivative at the root. In this chapter, we present a Jacobian approximation to the Shamanskii method, to obtain a convergent and accelerated scheme for systems of nonlinear equations. Precisely, we will focus on the efficiency of our proposed method and compare the performance with other existing methods. Numerical examples illustrate the efficiency and the theoretical analysis of the proposed methods.

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Section: Shamanskii Methodsmentioning

confidence: 98%

Section: Shamanskii Methodsmentioning

confidence: 99%

Section: Fx ðþ¼0mentioning

confidence: 99%

Section: Fx ðþ¼0mentioning

confidence: 99%

See 2 more Smart Citations

A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations

Sulaiman¹,

Mamat²,

Omesa³

2020

Nonlinear Systems -Theoretical Aspects and Recent Applications

View full text Add to dashboard Cite

show abstract

“…Most of the existing methods are of order between two and six, then a natural question is whether we can construct an iterative method which achieves higher order convergence. Recently, it is worth mentioning that a number of novel methods are introduced for solving systems of nonlinear equations (see [15][16][17][18][19][20][21][22][23][24]). …”

Section: Introductionmentioning

confidence: 99%

A simple and efficient method with high order convergence for solving systems of nonlinear equations

Xiao

Yin

2015

Computers & Mathematics with Applications

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Available online xxxx Keywords: Systems of nonlinear equations Modified Newton method Order of convergence Higher order methods Computational efficiency a b s t r a c tWe propose an m + 1-step modified Newton method of convergence order m + 2 to solve systems of nonlinear equations which are third Fréchet differentiable in a convex set containing the zero. Computational efficiency in the general form for a positive integer m is discussed, which shows that the efficiency increases with m when applied to large systems. Moreover, a comparison between the efficiency of this technique and some existing efficient methods is made, which implies that the present method is more efficient particularly for solving large systems of equations. Theoretical results about order of convergence and computational efficiency are largely verified in numerical examples.

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On per-iteration complexity of high order Chebyshev methods for sparse functions with banded Hessians

Kchouk

Dussault

2013

Numer Algor

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The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

Cited by 6 publications

References 13 publications

A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations

A Shamanskii-Like Accelerated Scheme for Nonlinear Systems of Equations

A simple and efficient method with high order convergence for solving systems of nonlinear equations

On per-iteration complexity of high order Chebyshev methods for sparse functions with banded Hessians

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