On the Bernoulli–Euler–Lagrange–Aitken numerical method for roots of polynomials
https://doi.org/10.29235/1561-8323-2023-67-359-365
Abstract
The article presents a development of the Euler–Lagrange method for calculation of all roots of an arbitrary polynomial P(z) with complex coefficients based on the calculation of the limits of ratios of determinants (as in the Bernoulli–Aitken–Nikiporets methods) built by means of the Taylor and Laurent series coefficients for the function P′(z) / P(z).
About the Authors
A. V. Lebedev
Belarusian State University
Belarus
Belarus
Andrei V. Lebedev, D. Sc. (Physics and Mathematics), Professor, Head of the Department
220050
4, Nezavisimosti Ave.
Minsk
Yu. V. Trubnikov
Vitebsk State University named after P. M. Masherov
Belarus
Belarus
Yurii V. Trubnikov, D. Sc. (Physics and Mathematics), Professor
210038
33, Moskovskiy Ave.
Vitebsk
M. M. Chernyavsky
Vitebsk State University named after P. M. Masherov
Belarus
Belarus
Mikhail M. Chernyavsky, Lecturer
210038
33, Moskovskiy Ave.
Vitebsk
References
1. Bernoulli D. Observationes de serbus recurrentibus. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, 1732 (1728), no. 3, pp. 85–100.
2. McNamee J. M., Pan V. Y. Numerical methods for roots of polynomials, part II. Boston, Amsterdam, Oxford, 2013. 741 p.
3. Euler L. Introduction to the analysis of infinite (in two volumes). Vol. 1. Moscow, 1961. 315 p. (in Russian).
4. Lagrange J. L. Sur la Méthode d’Approximation tirée des séries récurrentes (1798). Traité de la résolution des équations numériques de tous les degrés. Paris, 1826, vol. 6, pp. 130–137.
5. Aitken A. C. On Bernulli’s Numerical Solution of Algebraic Equations. Proceedings of the Royal Society of Edinburgh, 1927, vol. 46, pp. 289–305. doi: 10.1017/s0370164600022070
6. Shmoylov V. I., Savchenko D. I. Some applications of the summation algorithm of continued fractions. Vestnik Voronezhskogo gosudarstvennoro universiteta. Seriya Fizika. Matematika = Prоceedings of Voronezh State University. Series: Physics. Mathematics, 2013, no. 2, pp. 258–276 (in Russian).
7. Shmoylov V. I., Kirichenko G. A. Solution of Algebraic Equations by Continuous Fractions of Nikiportsa. Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika. Mekhanika. Informatika = Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, no. 4-1, pp. 428–439 (in Russian). doi: 10.18500/1816-9791-2014-14-4-428-439
8. Shmoylov V. I. Continued Fractions and the r / φ-algorithm. Taganrog, 2012. 608 p. (in Russian).
9. Trubnikov Yu. V., Chernyavsky M. M. Divergent Power Series and Formulas of the Approached Analytical Solution of Algebraic Equations. Vesnіk Vіtsebskaga dzyarzhaўnaga ўnіversіteta = Bulletin of Vitebsk State University, 2018, no. 4 (101), pp. 5–17 (in Russian).
10. Chernyavsky M. M., Trubnikov Yu. V. Modification of Aitken’s formulas and algorithms for analytical finding of multiple roots of polynomials. Vesnіk Vіtsebskaga dzyarzhaўnaga ўnіversіteta = Bulletin of Vitebsk State University, 2021, no. 1 (110), pp. 13–25 (in Russian).
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