{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T02:34:32Z","timestamp":1769308472679,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,1]],"date-time":"2021-10-01T00:00:00Z","timestamp":1633046400000},"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 article presents an inexact optimal hybrid conjugate gradient (CG) method for solving symmetric nonlinear systems. The method is a convex combination of the optimal Dai\u2013Liao (DL) and the extended three-term Polak\u2013Ribi\u00e9re\u2013Polyak (PRP) CG methods. However, two different formulas for selecting the convex parameter are derived by using the conjugacy condition and also by combining the proposed direction with the default Newton direction. The proposed method is again derivative-free, therefore the Jacobian information is not required throughout the iteration process. Furthermore, the global convergence of the proposed method is shown using some appropriate assumptions. Finally, the numerical performance of the method is demonstrated by solving some examples of symmetric nonlinear problems and comparing them with some existing symmetric nonlinear equations CG solvers.<\/jats:p>","DOI":"10.3390\/sym13101829","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1829","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["An Inexact Optimal Hybrid Conjugate Gradient Method for Solving Symmetric Nonlinear Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7262-490X","authenticated-orcid":false,"given":"Jamilu","family":"Sabi\u2019u","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan"},{"name":"Department of Mathematics, Yusuf Maitama Sule University Kano, Kano 700241, Nigeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0975-2623","authenticated-orcid":false,"given":"Kanikar","family":"Muangchoo","sequence":"additional","affiliation":[{"name":"Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381, Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0337-1216","authenticated-orcid":false,"given":"Abdullah","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6142-3694","authenticated-orcid":false,"given":"Auwal Bala","family":"Abubakar","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria"},{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa 0204, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5624-2768","authenticated-orcid":false,"given":"Kazeem Olalekan","family":"Aremu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa 0204, South Africa"},{"name":"Department of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto P.M.B. 2346, Nigeria"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1137\/S0036142998335704","article-title":"A Globally and Superlinearly Convergent Gauss-Newton-Based BFGS Method for Symmetric Nonlinear Equations","volume":"37","author":"Li","year":"1999","journal-title":"SIAM J. 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