Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 4 (2009), 119 -- 132


Ioannis K. Argyros and Saïd Hilout

Abstract. We introduce a special class of real recurrent polynomials fn (n ≥ 1) of degree n, with unique positive roots sn, which are decreasing as n increases. The first root s1, as well as the last one denoted by s are expressed in closed form, and enclose all sn (n > 1).
This technique is also used to find weaker than before [5] sufficient convergence conditions for some popular iterative processes converging to solutions of equations.

2000 Mathematics Subject Classification: 26C10; 12D10; 30C15; 30C10; 65J15; 47J25.
Keywords: real polynomials; enclosing roots; iterative processes; nonlinear equations.

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  1. I. K. Argyros, On the Newton--Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004), 315-332. MR2072881(2005c:65047). Zbl 1055.65066.

  2. I. K. Argyros, A unifying local--semilocal convergence analysis and applications for two--point Newton--like methods in Banach space, J. Math. Anal. Appl. 298 (2004), 374-397. MR2086964. Zbl 1057.65029.

  3. I. K. Argyros, Convergence and applications of Newton--type iterations, Springer--Verlag Pub., New York, 2008. MR2428779. Zbl 1153.65057.

  4. I. K. Argyros, On a class of Newton--like methods for solving nonlinear equations, J. Comput. Appl. Math. 228 (2009), 115-122. MR2514268. Zbl 1168.65349.

  5. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. MR0664597(83h:46002). Zbl 0484.46003.

  6. J. M. McNamee, Numerical methods for roots of polynomials, part I, 14, Elsevier, 2007. MR2483756. Zbl 1143.65002.

  7. F. A. Potra, On the convergence of a class of Newton--like methods. Iterative solution of nonlinear systems of equations (Oberwolfach, 1982), Lecture Notes in Math., 953 (1982), Springer, Berlin--New York, 125-137. MR0678615(84e:65057). Zbl 0507.65020.

  8. F. A. Potra, On an iterative algorithm of order 1.839dots for solving nonlinear operator equations, Numer. Funct. Anal. Optim. 7 (1984/85), 75-106. MR0772168(86j:47088). Zbl 0556.65049.

  9. F. A. Potra, Sharp error bounds for a class of Newton--like methods, Libertas Math. 5 (1985), 71-84. MR0816258(87f:65073). Zbl 0581.47050.

Ioannis K. Argyros Saïd Hilout
Cameron University, Poitiers University,
Department of Mathematics Sciences, Laboratoire de Mathématiques et Applications,
Lawton, OK 73505, USA. Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179,
e-mail: 86962 Futuroscope Chasseneuil Cedex, France.