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A neural network-based approach for approximating arbitrary roots of polynomials
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Advisor(s)
Abstract(s)
Finding arbitrary roots of polynomials is a fundamental problem in various areas of
science and engineering. A myriad of methods was suggested to address this problem, such as the
sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The
sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute
approximations to all the roots of a given polynomial, which can produce inaccurate results due to
the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require
good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by
their capacity to find complex mappings between the dependent and independent variables. In view
of this, this paper aims to determine, based on comparative results, whether ANNs can be used to
compute approximations to the real and complex roots of a given polynomial, as an alternative to
simultaneous iterative algorithms like the D–K method. Although the results are very encouraging
and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able
to surpass the accuracy of the D–K method. The results indicated, however, that the use of the
approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial
to the accuracy of this method
Description
Keywords
Polynomial roots Artificial neural networks Particle swarm optimization Durand– Kerner method Performance analysis . Faculdade de Ciências Exatas e da Engenharia
Citation
Freitas, D., Lopes Guerreiro, L., & Morgado-Dias, F. (2021). A neural network-based approach for approximating arbitrary roots of polynomials. Mathematics, 9(4), 317. https://doi.org/10.3390/math9040317
Publisher
MDPI