On the Monoid Generated by Newton Algorithm in Solving a Particular Cubic Polynomial

M.M.  Zubairu

M.M. Zubairu Department of Mathematics, Bayero University Kano, P.M.B. 3011, Kano, Nigeria

Keywords: Rank properties, Order decreasing and Monotone transformations, Abundant semigroup

Abstract

We initiate the study of the algebraic interpretations of the Newton algorithm via transformation semigroup. We use MATLAB to solve the equation x³ + 4x² - 10 = 0 with an error tolerance of ε = 10⁻⁴. In each iteration, we obtain a transformation on a set of seven elements. With the aid of GAP 4.0, we construct a monoid that interprets the algebraic phenomena of all the iterations. The study reveals that the monoid obtained is left-adequate with 64 elements.

References

Howie, J. M. Fundamentals of semigroup theory. London Mathematical Society, New series 12. The Clarendon Press, Oxford University Press, 1995.

Higgins, P. M. Techniques of semigroup theory. Oxford university Press, 1992.

Ali, B., Umar, A. and Zubairu, M. M. Regularity and Green's relations for the semigroups of partial and full contractions of a finite chain. Scientific African, 21, (2023) p.e01890.

Fernandes, V. H., Gomes, G. M. S. and Jesus, M. M. Congruences on monoids of orderpreserving or order-reversing transformations on a finite chain. Glasg. Math. J. 47 (2005) 413-424.

Garba, G. U. On the nilpotent rank of certain semigroups of transformations, Glasgow Math. J. 36 (1), (1994), 1-9.

Gomes, G. M. S. and Howie, J. M. On the ranks of certain finite semigroups of transformations, Math. Proc. Cambridge Phil. Soc., 101 (1987), 395-403.

Laradji, A. and Umar, A. On certain finite semigroups of order-decreasing transformations I, Semigroup Forum, 69 (2004), 1847-200.

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278 (2004), 3427-359.

Umar, A. and Zubairu, M. M. On certain semigroups of contraction mappings of a finite chain. Algebra Discrete Math. 32 (2021), No. 2, 299-320.

Fountain, J. B. Abundant Semigroups. Proc. Lond. Math. Soc. 44 (1982), 103-129.

Ali, B., Jada, M. A. and Zubairu, M. M. On Rank of Semigroup of Order-Preserving Order-Decreasing Partial Contraction Mappings on a Finite Chain. International Journal of Mathematical Sciences and Optimization: Theory and Applications 10(4), (2024), 1-11

Zubairu, M. M., Umar, A. and Al-Kharousi, F. S. The decreasing and monotone injective partial monoid on a finite chain. Algebra and Discrete Mathematics 40 (2), (2025), 281-7314. DOI:10.12958/adm2388.

Zubairu, M. M., Umar, A. and Aliyu, J. A. On certain semigroups of order decreasing full contraction mappings of a finite chain. Recent Developments in Algebra and Analysis: (Trend in Mathematics), Springer Int. Pub., 1, (2024), 35-45.

Fountain, J. B. Adequate Semigroups. Proc. Edinb. Math. Soc. 22 (1979), 113-125.

Howie, J. M. and Marques Ribeiro, M. I. Rank properties in finite semigroups, Comm. Algebra., 27: 11, (1999), 5333-5347.

Howie, J. M. and Marques Ribeiro, M. I. Rank Properties in Finite Semigroups II: The Small Rank and the Large Rank. Southeast Asian Bulletin of Mathematics, 24, (2000), 2317-237.