### Converging Newton’s Method With An Inflection Point of A Function

#### Abstract

*For long periods of time, mathematics researchers struggled in obtaining the appropriate starting point when implementing root finding methods, and one of the most famous and applicable is Newton’s method. This iterative method produces sequence that converges to a desired solution with the assumption that the starting point is close enough to a solution. The word “close enough” indicates that we actually do not have any idea how close the initial point needed so that this point can bring into a convergent iteration. This paper comes to answer that question through analyzing the relationship between inflection points of one-dimensional non-linear function with the convergence of Newton’s method. Our purpose is to illustrate that the neighborhood of an inflection point of a function never fails to bring the Newton’s method convergent to a desired solution*

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Fang, L., Sun, L., He, G., 2008, An efficient Newton-type method with fifth-order convergence for solving nonlinear equations, Computational and Applied Mathematics Volume 27, Pages 269 – 274.

Lightstone, A. H., 1965, Concepts of Calculus I, Harper & Row.

Pandiya, R., An efficient method for determining all the extreme points of function with one variable. M.Sc thesis, Universiti Malaysia Terengganu, 2013.

Purcell, E. J., and Varberg, D., 1984, Calculus with Analytic Geometry, Prentice-Hall, Inc., Englewood Cliffs, NJ

Sharma, J. R., Guha, R. K., Sharma, R., 2011, Some modified newton’s methods with fourth-order convergence, Advances in Applied Science Research Volume 2, Pages 240 – 247.

Süli, E. and Mayers, D., 2003, An Introduction to Numerical Analysis, Cambridge University Press, UK.

Wen, G. K., Mamat, M. B., Mohd, I. B., Dasril, Y. B., 2012, Global optimization with nonparametric filled function, Far East Journal of Mathematical Sciences Volume 61, Pages 51 -64.

Wood, A., 1999, Introduction to Numerical Analysis, Addison Wesley Longman, New York.

Fang, L., Sun, L., He, G., 2008, An efficient Newton-type method with fifth-order convergence for solving nonlinear equations, Computational and Applied Mathematics Volume 27, Pages 269 – 274.

Lightstone, A. H., 1965, Concepts of Calculus I, Harper & Row.

Pandiya, R., An efficient method for determining all the extreme points of function with one variable. M.Sc thesis, Universiti Malaysia Terengganu, 2013.

Purcell, E. J., and Varberg, D., 1984, Calculus with Analytic Geometry, Prentice-Hall, Inc., Englewood Cliffs, NJ

Sharma, J. R., Guha, R. K., Sharma, R., 2011, Some modified newton’s methods with fourth-order convergence, Advances in Applied Science Research Volume 2, Pages 240 – 247.

Süli, E. and Mayers, D., 2003, An Introduction to Numerical Analysis, Cambridge University Press, UK.

Wen, G. K., Mamat, M. B., Mohd, I. B., Dasril, Y. B., 2012, Global optimization with nonparametric filled function, Far East Journal of Mathematical Sciences Volume 61, Pages 51 -64.

Wood, A., 1999, Introduction to Numerical Analysis, Addison Wesley Longman, New York.

DOI: https://doi.org/10.24198/jmi.v13.n2.11785.73-81

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