Geodesic
On the Hadamard and Vandermonde determinants and the Bernoulli–Euler–Lagrange–Aitken method
Matematičeskie zametki, Tome 116 (2024) no. 1, pp. 91-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article develops an Euler–Lagrange method for calculating all the roots of an arbitrary polynomial $P(z)$ with complex coefficients. The method is based on calculating the limits of ratios of determinants (as in the Bernoulli–Aitken methods) constructed from the coefficients of the Taylor and Laurent series expansions of the function $P'(z)/P(z)$.
Keywords: root of polynomial, Taylor series, Vandermonde determinant.
Mots-clés : Laurent series, Hadamard determinant 
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A. V. Lebedev; Yu. V. Trubnikov; M. M. Chernyavsky. On the Hadamard and Vandermonde determinants and the Bernoulli–Euler–Lagrange–Aitken method. Matematičeskie zametki, Tome 116 (2024) no. 1, pp. 91-108. http://geodesic.mathdoc.fr/item/MZM_2024_116_1_a6/

[1] D. Bernulli, “Observationes de serbus recurrentibus”, Comment. acad. sc. Petrop., 3 (1732 (1728)), 85–100

[2] J. M. Mc Namee, V. Y. Pan, Numerical Methods for Roots of Polynomials. Part II, Stud. Comput. Math., 16, Academic Press, Boston, 2013 | MR

[3] L. Eiler, Vvedenie v analiz beskonechnykh, v. 1, GIFML, M., 1961 | MR

[4] J. L. Lagrange, “(1798) Sur la Méthode d'Approximation tirée des séries récurrentes // Traité de la résolution des équations numériques de tous les degrés”, J. L. Lagrange – Paris, 6, 1826, 130–137

[5] A. C. Aitken, “On Bernulli's numerical solution of algebraic equations”, Proc. Royal Soc. Edinburgh, 46 (1927), 289–305 | DOI

[6] Yu. V. Trubnikov, M. M. Chernyavskii, “Raskhodyaschiesya stepennye ryady i formuly priblizhennogo analiticheskogo nakhozhdeniya reshenii algebraicheskikh uravnenii”, Vest. Vit. un-ta, 101:4 (2018), 5–17

[7] Yu. V. Trubnikov, M. M. Chernyavskii, “Modifikatsiya formul Eitkena i algoritmy analiticheskogo nakhozhdeniya kratnykh kornei polinomov”, Vest. Vit. un-ta, 110:1 (2021), 13–25

[8] A. V. Lebedev, Yu. V. Trubnikov, M. M. Chernyavskii, “O metode Bernulli–Eilera–Lagranzha–Eitkena vychisleniya kornei polinomov”, Dokl. NAN Belarusi, 67:5 (2023), 359–365 | MR