{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T21:57:40Z","timestamp":1778363860002,"version":"3.51.4"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,30]],"date-time":"2021-01-30T00:00:00Z","timestamp":1611964800000},"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>Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak\u2013Ribi\u00e9re\u2013Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations.<\/jats:p>","DOI":"10.3390\/sym13020234","type":"journal-article","created":{"date-parts":[[2021,1,30]],"date-time":"2021-01-30T08:56:01Z","timestamp":1611996961000},"page":"234","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric 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 12000, Pakistan"},{"name":"Department of Mathematics, Yusuf Maitama Sule University Kano, Kano 700241, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0975-2623","authenticated-orcid":false,"given":"Kanikar","family":"Muangchoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381, Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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 12000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4838-7465","authenticated-orcid":false,"given":"Lateef Olakunle","family":"Jolaoso","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa 0204, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,30]]},"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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