On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Chacha, Chacha Stephen

doi:10.1007/s42967-021-00152-3

Cited by 2 publications

(2 citation statements)

References 27 publications

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“…If 𝐶 𝑓 (𝑦 0 ) is arbitrarily large, then the equation is ill-conditioned. More details on condition numbers can be found in the articles by Chacha and Naqvi (2018) and Chacha (2022).…”

Section: Condition Number Of Nonlinear Equationsmentioning

confidence: 99%

An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

Nguni,

Chacha,

Mtunya

2023

Tanz. J. Sci.

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Recent researches have shown a great interest in developing novel numerical methods for solving nonlinear equations of the form arising from real world phenomena. However, very little attention has been given to the study of condition numbers which is an important aspect in measuring the sensitivity of the problem in response to slight perturbations in the input data. In this article, we present an efficient free derivative iterative scheme constructed by refining Newton-Raphson method standard form in which the derivative term is approximated by using finite difference scheme; hence making it derivative free. We also conducted an in-depth analysis of the condition numbers to explore sensitivity and efficiency comparisons between the proposed algorithm and existing methods for the given problems. Our investigation focused on iteration numbers, residuals, and convergence under mild error tolerances. Based on five numerical case studies, results revealed that the proposed Algorithm 2 outperforms the existing Algorithm 1 in terms of accuracy of the approximate solution. The results for the condition numbers indicated that all problems considered were ill-conditioned, highlighting the significance of studying condition numbers in the context of solving nonlinear equations.

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Section: Condition Number Of Nonlinear Equationsmentioning

confidence: 99%

An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

Nguni,

Chacha,

Mtunya

2023

Tanz. J. Sci.

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“…We now consider a matrix used in a model for the population of the bilby for the quasistationary behaviour of quasi-birth-death processes. The bilby is an endangered Australian marsupial ( [25,26]). Define the 5 × 5 matrix B = βA T 2 , where β = 0.5, d = [0, 0.5, 0.55, 0.8, 1] is the vector of probability that the population moves down a level given phase j and g = 0.2.…”

Section: Examplementioning

confidence: 99%

On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$

Chacha¹

2022

J Egypt Math Soc

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This work incorporates an efficient inversion free iterative scheme into Newton’s method to solve Newton’s step regardless of the singularity of the Fr$${\acute{\text {e}}}$$ e ´ chet derivative. The proposed iterative scheme is constructed by extending the idea of the foundational form of the conjugate gradient method. Moreover, the resulting scheme is refined and employed to obtain a symmetric solution of the nonlinear matrix equation $$X-A^{*}e^{X}A=I.$$ X - A ∗ e X A = I . Furthermore, explicit expressions for the perturbation and residual bound estimates of the approximate positive definite solution are derived. Finally, five numerical case studies provided confirm both the preciseness of theoretical results and the effectiveness of the propounded iterative method.

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On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Cited by 2 publications

References 27 publications

An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

On solution and perturbation estimates for the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$

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