# Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

## Authors

• Rajat Subhra Das Department of Mathematics, Dr. L.K.V.D. College, Tajpur, Samastipur, Bihar, India
• Abhimanyu Kumar University Department of Mathematics, L. N. Mithila University, Darbhanga, Bihar, India

## Keywords:

Local convergence, Halley method, Super-Halley method, Initial conditions, Multiple zeros, Normed field

## Abstract

In this article, we have constructed an iterative method of third order for solving polynomial equations with multiple polynomial zeros. We have combined two well known third order methods one is Halley and another is Super-Halley for this construction purpose. This constructed method is basically the mean of the methods Halley and Super-Halley, so we name the method as H-S Combined Mean Method. We have proposed some local convergence theorems of this H-S Combined Mean Method to establish the computation of a polynomial with known multiple zeros. For the establishment of this local convergence theorem, the key role is performed by a function (Real valued) termed as the function of initial conditions. Function of initial conditions $$I$$ is a mapping from the set $$D$$ into the set $$X$$, where $$D$$ (subset of $$X$$) is the domain of the H-S Combined Mean iterative scheme. Here the initial conditions uses the information only at the initial point and are given in the form $$I(w_0)$$ which belongs to $$J$$, where $$J$$ is an in interval on the positive real line which also contains zero and $$w_0$$ is the starting point. We have used the notion of gauge function which also plays very important role in establishing the convergence theorem. Here we have used two types of initial conditions over an arbitrary normed field and established local convergence theorems of the constructed H-S Combined Mean Method. The error estimations are also found in our convergence analysis. For simple zero, the method as well as the results hold good.

## References

B. Neta, New third order nonlinear solvers for multiple roots, Applied Mathematics and Computation 202 (2008), 162 – 170, DOI: 10.1016/j.amc.2008.01.031.

C. Chun and B. Neta, A third-order modification of Newton’s method for multiple roots, Applied Mathematics and Computation 211 (2009), 474 – 479, DOI: 10.1016/j.amc.2009.01.087.

E. Halley, A New Exact and Easy Method for Finding the Roots of Equations Generally and without any Previous Reduction, Philosophical Transactions of the Royal Society of London 18 (1964), 136 – 147, DOI: 10.1098/rstl.1694.0029.

H. Ren and I.K. Argyros, Convergence radius of the modified Newton method for multiple zeros under Holder continuous derivative, Applied Mathematics and Computation 217 (2006), 612 – 621, DOI: 10.1016/j.amc.2010.05.098.

J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company: Chelsea, VT, USA (1982).

J.M. Gutiérrez and M.A. Hernandez, An acceleration of Newton’s method: Super-Halley method, Applied Mathematics and Computation 117 (2001), 223 – 239, DOI: 10.1016/S0096-3003(99)00175-7.

N. Obreshkov, On the numerical solution of equations (in Bulgarian), Annuaire Univ. Sofia Fac. Sci. Phys. Math. 56 (1963), 73 – 83.

N. Osada, Asymptotic error constants of cubically convergent zero finding methods, Journal of Computational and Applied Mathematics 196 (2006), 347 – 357.

P.D. Proinov, General convergence theorems for iterative processes and applications to the Weierstrass root-finding method, Journal of Complexity 33 (2016), 118 – 144, DOI: 10.1016/j.jco.2015.10.001.

P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. Journal of Complexity 26 (2016), 3 – 42, DOI: 10.1016/j.jco.2009.05.001.

S.I. Ivanov, General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super-Halley Method for Multiple Polynomial Zeros, Mathematics 8 (2020), 1599, DOI: 10.3390/math8091599.

16-04-2024
CITATION

## How to Cite

Das, R. S., & Kumar, A. (2024). Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros. Communications in Mathematics and Applications, 14(5), 1679–1692. https://doi.org/10.26713/cma.v14i5.2329

Research Article