{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:37:34Z","timestamp":1760240254030,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T00:00:00Z","timestamp":1555977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases where the calculations of the fourth derivative are expensive, or the result is unbounded, or even does not exist. We only use the first order derivative of function    \u03a9    in the proposed convergence analysis. Hence, we expand the utilization of the earlier scheme, and we study the computable radii of convergence and error bounds based on the Lipschitz constants. Furthermore, the range of starting points is also explored to know how close the initial guess should be considered for assuring convergence. Several numerical examples where earlier studies cannot be applied illustrate the new technique.<\/jats:p>","DOI":"10.3390\/sym11040586","type":"journal-article","created":{"date-parts":[[2019,4,24]],"date-time":"2019-04-24T03:14:28Z","timestamp":1556075668000},"page":"586","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines"],"prefix":"10.3390","volume":"11","author":[{"given":"Ramandeep","family":"Behl","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fouad Othman","family":"Mallawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"J. A.","family":"Tenreiro Machado","sequence":"additional","affiliation":[{"name":"ISEP-Institute of Engineering, Polytechnic of Porto Department of Electrical Engineering, 431 4294-015 Porto, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,23]]},"reference":[{"key":"ref_1","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_2","unstructured":"Petkovic, M.S., Neta, B., Petkovic, L., and D\u017euni\u010d, J. (2013). Multipoint Methods For Solving Nonlinear Equations, Elsevier."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2081","DOI":"10.1080\/00207160.2012.702897","article-title":"A family of fast derivative-free fourth order multipoint optimal methods for nonlinear equations","volume":"89","author":"Lee","year":"2012","journal-title":"Int. J. Comput. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1007\/s00010-004-2733-y","article-title":"Dynamics of the King and Jarratt iterations","volume":"69","author":"Amat","year":"2005","journal-title":"Aequ. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.jmaa.2010.01.047","article-title":"Chaotic dynamics of a third-order Newton-type method","volume":"366","author":"Amat","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.amc.2008.08.050","article-title":"A modified Chebyshev\u2019s iterative method with at least sixth order of convergence","volume":"206","author":"Amat","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_7","unstructured":"Argyros, I.K. (2008). Convergence and Application of Newton-Type Iterations, Springer."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Numerical Methods in Nonlinear Analysis, World Scientific Publ. Comp.","DOI":"10.1142\/8475"},{"key":"ref_9","first-page":"614612","article-title":"Geometric construction of eighth-order optimal families of Ostrowski\u2019s method","volume":"2015","author":"Behl","year":"2015","journal-title":"Recent Theor. Appl. Approx. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10543-009-0226-z","article-title":"New iterations of R-order four with reduced computational cost","volume":"49","author":"Ezquerro","year":"2009","journal-title":"BIT Numer. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4021","DOI":"10.1016\/j.camwa.2011.09.039","article-title":"Simply constructed family of a Ostrowski\u2019s method with optimal order of convergence","volume":"62","author":"Kanwar","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.amc.2014.01.037","article-title":"Different anomalies in a Jarratt family of iterative root-finding methods","volume":"233","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.amc.2014.09.061","article-title":"A new tool to study real dynamics: The convergence plane","volume":"248","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"129","DOI":"10.4064\/-3-1-129-142","article-title":"An adaptive continuation process for solving systems of nonlinear equations","volume":"3","author":"Rheinboldt","year":"1978","journal-title":"Pol. Acad. Sci. Banach Cent. Publ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0893-9659(00)00100-2","article-title":"A variant of Newton\u2019s method with accelerated third order convergence","volume":"13","author":"Weerakoon","year":"2000","journal-title":"Appl. Math. 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