Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

Shams, Mudassir; Rafiq, Naila; Ahmad, Babar; Mir, Nazir Ahmad

doi:10.1155/2021/6643514

Cited by 7 publications

(6 citation statements)

References 24 publications

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“…which confirms the optimal eight order for the Modified Method 8a (MM a 8 ) (21). Similarly, we can prove the optimal eighth order convegence for the modified method MM b 8 (22). This completes the proof of the theorem.…”

Section: Modified Families Of Four-parametric Four-point Without-memo...supporting

confidence: 82%

“…Here, we introduce new derivative-free with-memory methods which are extensions of the newly suggested modified eighth order derivative-free families of without-memory methods MM a 8 (21) and MM b 8 (22). It is evident from error Equation ( 23) that the convergence order of the methods MM a 8 (21), and MM b 8 (22) can be increased from 8 to 16 if we take γ = − 1…”

Section: Four-parametric Four-point With-memory Methodsmentioning

confidence: 99%

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Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

et al. 2023

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In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation.

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Section: Modified Families Of Four-parametric Four-point Without-memo...supporting

confidence: 82%

Section: Four-parametric Four-point With-memory Methodsmentioning

confidence: 99%

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

et al. 2023

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“…, and the two-step inverse Weierstrass method [51] with fourth-order convergence, locally (IWDKI) defined as:…”

Section: Construction Of Inversementioning

confidence: 99%

Efficient Inverse Fractional Neural Network-Based Simultaneous Schemes for Nonlinear Engineering Applications

Shams,

Carpentieri

2023

Fractal Fract

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Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time.

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“…The lower bound of the convergence is obtained until the first non-zero element of row is found zero (see [19]). The Mathematica program is given for each of the considered method as:…”

Section: Using Cas-mathematica For Finding Lower Bound Of Convergence...mentioning

confidence: 99%

Computer Geometries for Finding All Real Zeros of Polynomial Equations Simultaneously

Rafiq¹,

Akram²,

Shams³

et al. 2021

Computers, Materials &Amp; Continua

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In this research article, we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation. We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab. Lower bound of convergence of simultaneous schemes is calculated using Mathematica. Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented. Some real life engineering applications along with some higher degree polynomials are considered as numerical test problems to show performance and efficiency of the derivative free family of numerical methods with comparison of an existing method of same order in literature. Local computational order of convergence, CPU time, graph of computational order of convergence and residual error graphs elaborate efficiency, robustness and authentication of the suggested family of numerical methods in its domain.

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Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

Cited by 7 publications

References 24 publications

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

Efficient Inverse Fractional Neural Network-Based Simultaneous Schemes for Nonlinear Engineering Applications

Computer Geometries for Finding All Real Zeros of Polynomial Equations Simultaneously

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