Condition Numbers of the Nonlinear Matrix Equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mi>I</mml:mi></mml:math>

Chacha, Chacha Stephen; Naqvi, Syed Muhammad Raza Shah

doi:10.1155/2018/3291867

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…Consider the Eq. (1.2) with the matrix A = B/10, where B is a doubly stochastic matrix used in communication theory and graph theory [3,4]. We observe that A satisfies Condition 2.1.…”

Section: Numerical Experimentsmentioning

confidence: 78%

“…In addition, Liao and Zhang [14] studied preconditioned iteration methods for a class of complex linear algebraic systems (W + iT )u = c for symmetric matrices W and T , one of which is nonsingular. Chacha and Naqvi [4] investigated mixed and condition numbers of the nonlinear matrix equation X p − A * e X A = I (1.4) and proposed a fixed point method for the Eq. (1.4).…”

Section: Introductionmentioning

confidence: 99%

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Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

Kim¹

2019

EAJAM

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“…Consider the Eq. (1.2) with the matrix A = B/10, where B is a doubly stochastic matrix used in communication theory and graph theory [3,4]. We observe that A satisfies Condition 2.1.…”

Section: Numerical Experimentsmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

Kim¹

2019

EAJAM

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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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“…For F(X) = −X n , where n ≥ 2, authors in [22] proved that under mild conditions the iterations converged monotonically to the elementwise minimal nonnegative solutions. Chacha and Naqvi [23] derived the explicit expressions for mixed and componentwise condition numbers for the nonlinear matrix equation X p − A * e X A = I, where p is a positive integer.…”

mentioning

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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Condition Numbers of the Nonlinear Matrix Equation Xp-A⁎eXA=I

Cited by 5 publications

References 11 publications

Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations

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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