{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T03:26:26Z","timestamp":1769311586975,"version":"3.49.0"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2015,12,24]],"date-time":"2015-12-24T00:00:00Z","timestamp":1450915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Kung-Traub\u2019s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of                                        2 d\u22121. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached by a method with three function evaluations is four, even for quadratic functions. In this paper, we show that the conjecture fails for quadratic functions. In fact, we can find a 2-point method with three function evaluations reaching fifth order convergence. We also develop 2-point 3rd to 8th order methods with one function and two first derivative evaluations using weight functions. Furthermore, we show that with the same number of function evaluations we can develop higher order 2-point methods of order                                        r               +               2                                 , where r is a positive integer,                                        \u2265               1                                 . We also show that we can develop a higher order method with the same number of function evaluations if we know the asymptotic error constant of the previous method. We prove the local convergence of these methods which we term as Babajee\u2019s Quadratic Iterative Methods and we extend these methods to systems involving quadratic equations. We test our methods with some numerical experiments including an application to Chandrasekhar\u2019s integral equation arising in radiative heat transfer theory.<\/jats:p>","DOI":"10.3390\/a9010001","type":"journal-article","created":{"date-parts":[[2015,12,23]],"date-time":"2015-12-23T07:10:20Z","timestamp":1450854620000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations"],"prefix":"10.3390","volume":"9","author":[{"given":"Diyashvir","family":"Babajee","sequence":"first","affiliation":[{"name":"Independent Scholar, 65, Captain Pontre Street, Sainte Croix, Port Louis 11708, Mauritius"}]}],"member":"1968","published-online":{"date-parts":[[2015,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7958","DOI":"10.1016\/j.amc.2012.02.016","article-title":"Several Improvements of the 2-point third order midpoint iterative method using weight functions","volume":"218","author":"Babajee","year":"2012","journal-title":"Appl. Math. Comp."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1007\/s13370-014-0237-z","article-title":"On a two-parameter Chebyshev-Halley-like family of optimal two-point fourth order methods free from second derivatives","volume":"26","author":"Babajee","year":"2015","journal-title":"Afr. Mat."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Babajee, D.K.R., and Jaunky, V.C. (2013). Applications of Higher-Order Optimal Newton Secant Iterative Methods in Ocean Acidification and Investigation of Long-Run Implications of CO2 Emissions on Alkalinity of Seawater. ISRN Appl. Math.","DOI":"10.1155\/2013\/785287"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Babajee, D.K.R., and Thukral, R. (2012). On a 4-point sixteenth-order King family of iterative methods for solving nonlinear equations. Int. J. Math. Math. Sci.","DOI":"10.1155\/2012\/979245"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1016\/j.camwa.2011.11.040","article-title":"Finding solution of nonlinear equations by a class of optimal methods","volume":"63","author":"Sharifi","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1016\/j.aml.2011.10.030","article-title":"Two new classes of optimal Jarratt-type fourth-order methods","volume":"25","author":"Soleymani","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_7","unstructured":"Wait, R. (1979). The Numerical Solution of Algebraic Equations, John Wiley & Sons."},{"key":"ref_8","unstructured":"Ostrowski, A.M. (1960). Solutions of Equations and System of Equations, Academic Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1145\/321850.321860","article-title":"Optimal Order of one-point and multipoint iteration","volume":"21","author":"Kung","year":"1974","journal-title":"J. Assoc. Comput. Mach."},{"key":"ref_10","unstructured":"Jarratt, P. (1970). A Review of Methods for Solving Nonlinear Algebraic Equations in One Variable, Gordon & Breach Science Publishers."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11075-009-9359-z","article-title":"A modified Newton-Jarratt\u2019s composition","volume":"55","author":"Cordero","year":"2010","journal-title":"Numer. Algor."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kalantari, B. (2009). Polynomial Root-Finding and Polynomiography, World Scientific Publishing Co. Pte. Ltd.","DOI":"10.1142\/9789812811837"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.amc.2007.01.062","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comp."},{"key":"ref_14","first-page":"485","article-title":"Accelerated methods of order 2p for systems of nonlinear equations","volume":"53","author":"Cordero","year":"2010","journal-title":"J. Comp. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1678","DOI":"10.1016\/j.amc.2006.11.022","article-title":"Super cubic iterative methods to solve systems of nonlinear equations","volume":"188","author":"Darvishi","year":"2007","journal-title":"Appl. Math. Comp."},{"key":"ref_16","unstructured":"Chandrasekhar, D. (1960). Radiative Transfer, Dover Publications."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1017\/S0004972700009953","article-title":"Quadratic equations and applications to Chandrasekhar\u2019s and related equations","volume":"32","author":"Argyros","year":"1985","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01837974","article-title":"On a class of nonlinear integral equations arising in neutron transport","volume":"35","author":"Argyros","year":"1988","journal-title":"Aequ. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1080\/01630569908816917","article-title":"Solving nonlinear integral equations arising in radiative transfer","volume":"20","author":"Ezquerro","year":"1999","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Soleymani, F., Babajee, D.K.R., Shateyi, S., and Motsa, S.S. (2012). Construction of Optimal Derivative-Free Techniques without Memory. J. Appl. Math.","DOI":"10.1155\/2012\/497023"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/9\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:54:36Z","timestamp":1760216076000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/9\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,24]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,3]]}},"alternative-id":["a9010001"],"URL":"https:\/\/doi.org\/10.3390\/a9010001","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,24]]}}}