Computable Implementation of ``Fundamental Theorem of Algebra"

Sjogren, Jon A.; Li, X.; Zhao, Minghui; Lu, Chin-Pi

doi:10.12732/ijpam.v86i1.9

Cited by 1 publication

(1 citation statement)

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“…Niel Hernor Abel [1] proved Abel's Impossibility theorem "There is no solution in radical to general polynomial with arbitrary coefficient of degree five or higher" in 1824. The fact established in the 17th century that every generic polynomial equation of positive degree has a solution, possibly non-real, was completely demonstrated at the beginning of the 19th century as the "Fundamental theorem of algebra" [2].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%