Sadananda, R.George, S.Kunnarath, A.Padikkal, J.Argyros, I.K.2026-02-042023Journal of Applied Mathematics and Computing, 2023, 69, 4, pp. 3359-338915985865https://doi.org/10.1007/s12190-023-01886-4https://idr.nitk.ac.in/handle/123456789/21795The new Newton-type iterative method developed by Khirallah et al. (Bull Math Sci Appl 2:01–14, 2012), is shown to be of the convergence order three, without the application of Taylor series expansion. Our analysis is based on the assumptions on second order derivative of the involved operator, unlike the earlier studies. Moreover, this technique is extended to methods of higher order of convergence, five and six. This paper also verifies the theoretical approach using numerical examples and comparisons, in addition to the visualization of Julia and Fatou sets of the corresponding methods. © 2023, The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics.Banach spacesNumerical methodsTaylor seriesConvergence orderFrechet derivativeHigh-orderHigher-orderNewton Cotes methodNewton-CotesOrder of convergenceSecond-order derivativeTaylor's expansionTaylor's series expansionIterative methods