On an iterative algorithm of order 1.839... for solving the nonlinear least squares problems S.M. Shakhno * , O.P. Gnatyshyn Department of Applied Mathematics and Computer Science, Ivan Franko National University of LÕviv, Universitetska 1, LÕviv 79602, Ukraine Abstract Inthepresentpaperforsolvingthenonlinearleast-squaresproblemweproposenew free-derivative iteration method with the order of convergence 1.839... We have elab- oratedtheschemeofproof,bywhichfromonepointofviewthetheoreticalresearchof oneclassofdifferenceiterativemethodswithdifferentordersofconvergencehavebeen conducted. By the results of numerical experiment, the new method is compared with the known difference methods and with the classic Gauss–Newton method. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Nonlinear least squares problem; Free-derivative iteration method; Order convergence; Residual 1. Introduction Nonlinear least squares problem often arise in scientific and engineering problems. This problem can be solved by the best known Gauss–Newton methodanditÕsmodifications[1].Howeverthesemethodsusederivativesfrom function F ðxÞ. As on practice frequently the function itself is derived from an experimentasaresultofcalculationsoncomputer,therewillbenoanalytically given derivatives. If the derivative of function F ðxÞ is given in the form of a * Corresponding author. E-mail address: shakhno@is.lviv.ua (S.M. Shakhno). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.025 Applied Mathematics and Computation 161 (2005) 253–264 www.elsevier.com/locate/amc