{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:15:33Z","timestamp":1774437333594,"version":"3.50.1"},"reference-count":13,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T00:00:00Z","timestamp":1448928000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634\u2013651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m-1 . Furthermore, the family of Steffensen-type methods of super convergence is suggested by using arithmetic expressions for the parameters with memory but no additional new evaluation of the function. Their error equations, asymptotic convergence constants and convergence orders are obtained. Finally, they are compared with related root-finding methods in the numerical examples.<\/jats:p>","DOI":"10.3390\/a8041111","type":"journal-article","created":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T10:26:58Z","timestamp":1448965618000},"page":"1111-1120","update-policy":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations"],"prefix":"10.3390","volume":"8","author":[{"given":"Quan","family":"Zheng","sequence":"first","affiliation":[{"name":"College of Sciences, North China University of Technology, Beijing 100144, China"}]},{"given":"Xin","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Sciences, North China University of Technology, Beijing 100144, China"}]},{"given":"Yufeng","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Sciences, North China University of Technology, Beijing 100144, China"}]}],"member":"1968","published-online":{"date-parts":[[2015,12,1]]},"reference":[{"key":"ref_1","unstructured":"Ortega, J.M., and Rheinboldt, W.G. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"634","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. Math."},{"key":"ref_3","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.amc.2009.03.053","article-title":"A Steffensen-like method and its higher-order variants","volume":"214","author":"Zheng","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3486","DOI":"10.1016\/j.amc.2010.04.058","article-title":"Variants of Steffensen-secant method and applications","volume":"216","author":"Zheng","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_6","first-page":"142","article-title":"A Steffensen\u2019s type method in Banach spaces with applications on boundary-value problems","volume":"216","author":"Amat","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.cam.2013.05.013","article-title":"On generalized biparametric multipoint root finding methods with memory","volume":"255","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.amc.2008.12.039","article-title":"A class of two-step Steffensen type methods with fourth-order convergence","volume":"209","author":"Ren","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"9592","DOI":"10.1016\/j.amc.2011.04.035","article-title":"An optimal Steffensen-type family for solving nonlinear equations","volume":"217","author":"Zheng","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2909","DOI":"10.1016\/j.cam.2012.01.035","article-title":"On generalized multipoint root-solvers with memory","volume":"236","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","first-page":"886","article-title":"Doubly-accelerated Steffensen\u2019s methods with memory and their applications on solving nonlinear ODEs","volume":"15","author":"Zheng","year":"2013","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1870","DOI":"10.1016\/j.procs.2014.05.171","article-title":"A one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations","volume":"29","author":"Liu","year":"2014","journal-title":"Procedia Comput. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s11075-014-9951-8","article-title":"Efficient n-point iterative methods with memory for solving nonlinear equations","volume":"70","author":"Wang","year":"2015","journal-title":"Numer. Algorithms"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mdpi.com\/1999-4893\/8\/4\/1111\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:53:07Z","timestamp":1760215987000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mdpi.com\/1999-4893\/8\/4\/1111"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,1]]},"references-count":13,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["a8041111"],"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.3390\/a8041111","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,1]]}}}