Abstract
We present a new method to calculate analytically the roots of the general complex polynomial of degree three. This method is based on the approach of appropriate changes of variable involving an arbitrary parameter. The advantage of this method is to calculate the roots of the cubic polynomial as closed formula using the standard convention of the square and cubic roots. In contrast, the reference methods for this problem, as Cardan–Tartaglia and Lagrange, give the roots of the cubic polynomial as expressions with case distinctions which require a convention of roots depending on coefficients.
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References
van der Waerden, B.: A History of Algebra. Springer, Berlin (1985)
Lane, S.M., Birkhoff, G.: Algebra. American Mathematical Society, Providence (1999)
Hager, W.W., Pederson, R.N.: Perturbation in eigenvalues. Linear Algebra Appl. 42, 39–55 (1982). https://doi.org/10.1016/0024-3795(82)90137-9
Ipsen, I.C.F., Nadler, B.: Refined perturbation bounds for eigenvalues of Hermitian and non-Hermitian matrices. SIAM J. Matrix Anal. Appl. 31, 40–53 (2009)
Kirillov, O.N., Mailybaev, A.A., Seyranian, A.P.: Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation. J. Phys. A: Math. Gen. 38(24), 5531 (2005)
Anderson, J.: A secular equation for the eigenvalues of a diagonal matrix perturbation. Linear Algebra Appl. 246, 49–70 (1996). https://doi.org/10.1016/0024-3795(94)00314-9
Adelakun, A.O.: Resonance oscillation and transition to chaos in \(\phi ^8\)-duffing-van der pol oscillator. Int. J. Appl. Comput. Math. 7, 1–16 (2021)
Tsang, T., Park, H.-Y.: Sound velocity anisotropy in cubic crystals. Phys. Lett. A 99(8), 377–380 (1983). https://doi.org/10.1016/0375-9601(83)90297-9
Baydoun, I., Savin, É., Cottereau, R., Clouteau, D., Guilleminot, J.: Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media. Wave Motion 51(8), 1325–1348 (2014)
Mensch, T., Rasolofosaon, P.: Elastic-wave velocities in anisotropic media of arbitrary symmetry-generalization of Thomsen’s parameters. Geophys. J. Int. 128, 43–64 (1997)
Bourbaki, N.: Éléments d’histoire des Mathématiques. Springer, Berlin (2006)
Lagrange, J.L.: Réflexions sur la résolution algébrique des équations, In: Nouveaux Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin, Oeurves complètes, tome 3, 1770
Zhao, T., Wang, D., Hong, H.: Solution formulas for cubic equations without or with constraints. J. Symb. Comput. 46, 904–918 (2011)
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The author would like to thank, in particularly, Dalia Ibrahim for her support during the difficult moment to work this paper. Also, the author wish to thank Dr. Thomas Sturm for his useful comments.
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Baydoun, I. Analytical Formula for the Roots of the General Complex Cubic Polynomial. Int. J. Appl. Comput. Math 10, 55 (2024). https://doi.org/10.1007/s40819-024-01680-1
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DOI: https://doi.org/10.1007/s40819-024-01680-1