JOURNAL OF NUMERICAL ANALYSIS AND APPROXIMATION THEORY J. Numer. Anal. Approx. Theory, vol. 44 (2015) no. 1, pp. 25–37 ictp.acad.ro/jnaat ANALYTIC AND EMPIRICAL STUDY OF THE RATE OF CONVERGENCE OF SOME ITERATIVE METHODS VASILE BERINDE 1,2 , ABDUL RAHIM KHAN 2 and M ˘ AD ˘ ALINA P ˘ ACURAR 3 Dedicated to prof. I. P˘av˘ aloiu on the occasion of his 75th anniversary Abstract. We study analytically and empirically the rate of convergence of two k-step fixed point iterative methods in the family of methods (1) xn+1 = T (x i 0 +n-k+1 ,x i 1 +n-k+1 ,...,x i k-1 +n-k+1 ),n ≥ k - 1, where T : X k → X is a mapping satisfying some Preˇ si´ c type contraction condi- tions and (i0,i1,...,i k-1 ) is a permutation of (0, 1,...,k - 1). We also consider the Picard iteration associated to the fixed point problem x = T (x,...,x) and compare analytically and empirically the rate and speed of convergence of three iterative methods. Our approach opens a new perspective on the study of the rate of convergence / speed of convergence of fixed point iterative methods and also illustrates the essential difference between them by means of some concrete numerical experiments. MSC 2010. 47H09, 47H10, 54H25. Keywords. metric space, contractive mapping, fixed point, k-step fixed point iterative method, rate of convergence. 1. INTRODUCTION Inthe book [23] (see also[28]), I. P˘av˘aloiu studiedsome multistepiterative methods for solving the scalar equation (2) x = ϕ(x) where ϕ : I → I is a function and I ⊂ R is an interval. In order to solve (2), he considers a function g : I s → I , where s ≥ 1 is an integer, and the restriction of g to the diagonal of I s coincides with ϕ, that is, (3) g(x, x, . . . , x)= ϕ(x), ∀x ∈ I. 1 Department of Mathematics and Computer Science North University of Baia Mare Vic- torie1 76, 430072 Baia Mare, Romania, e-mail: vberinde@ubm.ro. 2 Department of Mathematics and Statistics King Fahd University of Petroleum and Min- erals Dhahran, Saudi Arabia, e-mail: arahim@kfupm.edu.sa. 3 Department of Statistics, Analysis, Forecast and Mathematics Faculty of Economics and Bussiness Administration Babe¸ s-Bolyai University of Cluj-Napoca 56-60 T. Mihali St., 400591 Cluj-Napoca, Romania, e-mail: madalina.pacurar@econ.ubbcluj.ro.