Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations

ABSTRACT

In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.

In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.

KEYWORDS

Nonlinear Equations, Iterative Methods, Multipoint Iterative Methods, Newton’s Method, Traub-Ostrowski’s Method, Predictor-Corrector Methods, Order of Convergence

Nonlinear Equations, Iterative Methods, Multipoint Iterative Methods, Newton’s Method, Traub-Ostrowski’s Method, Predictor-Corrector Methods, Order of Convergence

Cite this paper

nullS. Kumar, V. Kanwar and S. Singh, "Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations,"*Applied Mathematics*, Vol. 1 No. 3, 2010, pp. 153-158. doi: 10.4236/am.2010.13020.

nullS. Kumar, V. Kanwar and S. Singh, "Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations,"

References

[1] A. M. Ostrowski, “Solution of Equations in Euclidean and Banach Space,” 3rd Edition, Academic Press, New York, 1973.

[2] J. F. Traub, “Iterative Methods for the Solution of Equations,” Prentice Hall, Englewood Cliffs, New Jersey, 1964.

[3] V. Kanwar and S. K. Tomar, “Modified Families of Newton, Halley and Chebyshev Methods,” Applied Mathematics and Computation, Vol. 192, No. 1, September 2007, pp. 20-26.

[4] V. Kanwar and S. K. Tomar, “Exponentially Fitted Variants of Newton’s Method with Quadratic and Cubic Convergence,” International Journal of Computer Mathematics, Vol. 86, No. 9, September 2009, pp. 1603-1611.

[5] G. H. Nedzhibov, V. I. Hasanov and M. G. Petkov, “On Some Families of Multi-Point Iterative Methods for Solving Nonlinear Equations,” Numerical Algorithms, Vol. 42, No. 2, June 2006, pp. 127-136.

[6] W. Werner, “Some Improvement of Classical Methods for the Solution of in Nonlinear Equations in Numerical Solution of Nonlinear Equations,” Lecture Notes Mathematics, Vol. 878, 1981, pp. 426-440.

[7] M. Grau and J. L. Díaz-Barrero, “An Improvement to Ostrowski Root-Finding Method,” Applied Mathematics and Computation, Vol. 173, No. 1, February 2006, pp. 369-375, 450-456.

[8] J. R. Sharma and R. K. Guha, “A Family of Modified Ostrowski Methods with July Accelerated Sixth Order Convergence,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 11-115.

[9] N. A. Mir, K. Ayub and A. Rafiq, “A Third-Order Convergent Iterative Method for Solving Non-Linear Equations,” International Journal of Computer Mathematics, Vol. 87, No. 4, March 2010, pp. 849-854.

[10] Mamta, V. Kanwar, V. K. Kukreja and S. Singh, “On a Class of Quadratically Convergent Iteration Formulae,” Applied Mathematics and Computation, Vol. 166, No. 3, July 2005, pp. 633-637.

[11] V. I. Hasnov, I. G. Ivanov, and G. Nedzhibov, “A New Modification of Newton’s Method,” Application of Ma-thematics in Engineering, Heron, Sofia, Vol. 27, 2002, pp. 278-286.

[12] K. C. Gupta, V. Kanwar and Sanjeev Kumar, “A Family of Ellipse Methods for Solving Nonlinear Equations,” International Journal of Mathematical Education in Science and Technology, Vol. 40, No. 4, January 2009, pp. 571-575.

[13] N. A. Mir, K. Ayub and T. Zaman, “Some Quadrature Based Three-Step Iterative Methods for Nonlinear Equations,” Applied Mathematics and Computation, Vol. 193, No. 2, November 2007, pp. 366-373.

[14] G. V. Milovannovi? and A. S. Cvetkovic, “A Note on Three-Step Iterative Methods for Nonlinear Equations,” Studia University “Babes-Bolyai”, Mathematica, Vol. LII, No. 3, 2007, pp. 137-146.

[15] A. Rafiq, S. Hussain, F. Ahmad, M. Awais and F. Zafar, “An Efficient Three-Step Iterative Method with Sixth- Order Convergence for Solving Nonlinear Equations,” International Journal of Computer Mathematics, Vol. 84, No. 3, March 2007, pp. 369-375.

[1] A. M. Ostrowski, “Solution of Equations in Euclidean and Banach Space,” 3rd Edition, Academic Press, New York, 1973.

[2] J. F. Traub, “Iterative Methods for the Solution of Equations,” Prentice Hall, Englewood Cliffs, New Jersey, 1964.

[3] V. Kanwar and S. K. Tomar, “Modified Families of Newton, Halley and Chebyshev Methods,” Applied Mathematics and Computation, Vol. 192, No. 1, September 2007, pp. 20-26.

[4] V. Kanwar and S. K. Tomar, “Exponentially Fitted Variants of Newton’s Method with Quadratic and Cubic Convergence,” International Journal of Computer Mathematics, Vol. 86, No. 9, September 2009, pp. 1603-1611.

[5] G. H. Nedzhibov, V. I. Hasanov and M. G. Petkov, “On Some Families of Multi-Point Iterative Methods for Solving Nonlinear Equations,” Numerical Algorithms, Vol. 42, No. 2, June 2006, pp. 127-136.

[6] W. Werner, “Some Improvement of Classical Methods for the Solution of in Nonlinear Equations in Numerical Solution of Nonlinear Equations,” Lecture Notes Mathematics, Vol. 878, 1981, pp. 426-440.

[7] M. Grau and J. L. Díaz-Barrero, “An Improvement to Ostrowski Root-Finding Method,” Applied Mathematics and Computation, Vol. 173, No. 1, February 2006, pp. 369-375, 450-456.

[8] J. R. Sharma and R. K. Guha, “A Family of Modified Ostrowski Methods with July Accelerated Sixth Order Convergence,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 11-115.

[9] N. A. Mir, K. Ayub and A. Rafiq, “A Third-Order Convergent Iterative Method for Solving Non-Linear Equations,” International Journal of Computer Mathematics, Vol. 87, No. 4, March 2010, pp. 849-854.

[10] Mamta, V. Kanwar, V. K. Kukreja and S. Singh, “On a Class of Quadratically Convergent Iteration Formulae,” Applied Mathematics and Computation, Vol. 166, No. 3, July 2005, pp. 633-637.

[11] V. I. Hasnov, I. G. Ivanov, and G. Nedzhibov, “A New Modification of Newton’s Method,” Application of Ma-thematics in Engineering, Heron, Sofia, Vol. 27, 2002, pp. 278-286.

[12] K. C. Gupta, V. Kanwar and Sanjeev Kumar, “A Family of Ellipse Methods for Solving Nonlinear Equations,” International Journal of Mathematical Education in Science and Technology, Vol. 40, No. 4, January 2009, pp. 571-575.

[13] N. A. Mir, K. Ayub and T. Zaman, “Some Quadrature Based Three-Step Iterative Methods for Nonlinear Equations,” Applied Mathematics and Computation, Vol. 193, No. 2, November 2007, pp. 366-373.

[14] G. V. Milovannovi? and A. S. Cvetkovic, “A Note on Three-Step Iterative Methods for Nonlinear Equations,” Studia University “Babes-Bolyai”, Mathematica, Vol. LII, No. 3, 2007, pp. 137-146.

[15] A. Rafiq, S. Hussain, F. Ahmad, M. Awais and F. Zafar, “An Efficient Three-Step Iterative Method with Sixth- Order Convergence for Solving Nonlinear Equations,” International Journal of Computer Mathematics, Vol. 84, No. 3, March 2007, pp. 369-375.