Two-step diagonal Newton method for large-scale systems of nonlinear equations

Waziri, Mohammed Yusuf; Leong, Wah June; Hassan, Musa Abu; Monsi, Mansor

doi:10.1109/icmsao.2011.5775576

Cited by 6 publications

(17 citation statements)

References 10 publications

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“…Waziri et al [7] have set the step length ( = 1, for all ); this approach is mostly used in many Newton-like methods.…”

Section: The Improved Secant Diagonal Updatingmentioning

confidence: 99%

“…Here, we continue in the spirit of diagonal updating, using a new line search strategy to obtain a good step length ( ) in every iteration, anticipating to produce a more accurate approximation of the Jacobian inverse matrix and then employing restating strategy whenever the updating matrix is undefined. To this end, +1 would be obtained almost similar to the diagonal updating scheme presented in [7] in which = − ( ) instead of = − ( ). Now, let the deviation between +1 and denoted as Φ = +1 − be minimized under some norms; the optimal solution is given as…”

Section: The Improved Secant Diagonal Updatingmentioning

confidence: 99%

See 1 more Smart Citation

An Enhanced Matrix-Free Secant Method via Predictor-Corrector Modified Line Search Strategies for Solving Systems of Nonlinear Equations

Waziri

Majid

2013

International Journal of Mathematics and Mathematical Sciences

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Diagonal updating scheme is among the cheapest Newton-like methods for solving system of nonlinear equations. Nevertheless, the method has some shortcomings. In this paper, we proposed an improved matrix-free secant updating scheme via line search strategies, by using the steps of backtracking in the Armijo-type line search as a step length predictor and Wolfe-Like condition as corrector. Our approach aims at improving the overall performance of diagonal secant updating scheme. Under mild assumptions, the global convergence results have been presented. Numerical experiments verify that the proposed approach is very promising.

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“…Waziri et al [7] have set the step length ( = 1, for all ); this approach is mostly used in many Newton-like methods.…”

Section: The Improved Secant Diagonal Updatingmentioning

confidence: 99%

Section: The Improved Secant Diagonal Updatingmentioning

confidence: 99%

An Enhanced Matrix-Free Secant Method via Predictor-Corrector Modified Line Search Strategies for Solving Systems of Nonlinear Equations

Waziri

Majid

2013

International Journal of Mathematics and Mathematical Sciences

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“…One of the famous method for solving is the Newton method, which generates a sequence { x k } using the formula

x_{k + 1} = x_{k} - J {(x_{k})}^{- 1} F (x_{k}), \forall k = 0, 1, 2, … .

The Newton method has a very good convergence property but have some shortcomings such as Jacobian computation per iteration, which is costly. Other methods for solving are quasi Newton methods, spectral gradient methods, conjugate gradient methods, etc . For simplicity, we denote F k = F ( x k ) and J k = J ( x k ).…”

Section: Introductionmentioning

confidence: 99%

“…Other methods for solving (1) are quasi Newton methods, spectral gradient methods, conjugate gradient methods, etc. [2][3][4][5][6][7][8][9] For simplicity, we denote F k = F(x k ) and J k = J(x k ). Methods for solving symmetric nonlinear problem (1) has been proposed by many authors.…”

mentioning

confidence: 99%

An inexact conjugate gradient method for symmetric nonlinear equations

Abubakar

Awwal

2019

Comp and Math Methods

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In this article, we present a conjugate gradient method for large‐scale nonlinear equations with symmetric Jacobian. The method is a modification of the descent Dai‐Liao conjugate gradient method for unconstrained optimization problem proposed by Kafaki and Gambari. Under some assumptions, which include symmetric property of the Jacobian, we establish the global convergence of the proposed method and, finally, some numerical results to show its efficiency.

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“…It is vital to mention that, [5] have reported that, the undesirable performance behaviors of Newton's chord methodespecially when solving high dimensional systems of nonlinear equations is associated with the insufficient Jacobian information in each iteration. The validation associated to our procedure is to enhance the convergence properties as well as improving numerical stability.…”

Section: Derivation Processmentioning

confidence: 99%