A Mean Iterative Approach for Multiple Polynomial Zeros and Convergence
Keywords:
Local convergence,, Polynomial zeros, Multiple zeros, Initial conditionsAbstract
In this study, we propose a mean iterative approach of third-order for solving a polyno-
mial equation that has multiple type zeros. We used three prominent third-order algorithms
for this build: Chebyshev, Halley, and Super-Hallev. For this CHS Combined Mean Method,
we developed two forms of local convergence theorems to determine the convergence of a
polynomial that has multiple zeros. We used the gauge function to determine the conver-
gence of our technique. We employed two distinct forms of initialconditions on a field with
norm to prove the local convergence theorems for the CHS combined mean technique. Our
convergence analysis includes error estimates.
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References
N. Osada, Asymptotic error constants of cubically convergent zero nding meth-
ods, Journal of Computational and Applied Mathematics 196 (2006), 347 { 357,
doi:10.1016/j.cam.2005.09.016.
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 modi cation of Newtons method for multiple roots, Ap-
plied Mathematics and Computation 211 (2009), 474 { 479, doi:10.1016/j.amc.2009.01.087.
H. Ren and I.K. Argyros, Convergence radius of the modi ed Newton method for multi-
ple zeros under Holder continuous derivative, Applied Mathematics and Computation 217
(2006), 612 { 621, doi:10.1016/j.amc.2010.05.098.
P. Chebychev, Computation roots of an equation. In: Complete Works of
P.L.Chebishev USSR Academy of Sciences, Moscow (in Russian) 5 (1964), 7 { 25,
doi:10.1098/rstl.1694.0029.
J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company:
Chelsea, VT, USA (1982).
N. Obreshkov, On the numerical solution of equations (in Bulgarian), Annuaire Univ. So a
Fac. Sci. Phys. Math. 56 (1963), 73 { 83.
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 { 148, doi:10.1098/rstl.1694.0029.
J.M. Gutirrez and M.A. Hernandez, An acceleration of Newtons method: Super-Halley
method, Applied Mathematics and Computation 117 (2001), 223 { 239, doi:10.1016/S0096-
(99)00175-7.
P.D. Proinov, New general convergence theory for iterative processes and its applica-
tions to NewtonKantorovich type theorems. Journal of Complexity 26 (2016), 3 { 42,
doi:10.1016/j.jco.2009.05.001.
P.D. Proinov, General convergence theorems for iterative processes and applications
to the Weierstrass root- nding method, Journal of Complexity 33 (2016), 118 { 144,
doi:10.1016/j.jco.2015.10.001.
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.
S.I. Ivanov, Uni ed Convergence Analysis of ChebyshevHalley Methods for Multiple Poly-
nomial Zeros. Mathematics 10 (2022), 1599, doi:10.3390/math10010135.
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