{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:37:56Z","timestamp":1767832676404,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University Researches Supporting Project","award":["TURSP- 2020\/326"],"award-info":[{"award-number":["TURSP- 2020\/326"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we derived a modified conjugate gradient (CG) parameter by adopting the Birgin and Marti\u00b4nez strategy using the descent three-term CG direction and the Newton direction. The proposed CG parameter is applied and suggests a robust algorithm for solving constrained monotone equations with an application to image restoration problems. The global convergence of this algorithm is established using some proper assumptions. Lastly, the numerical comparison with some existing algorithms shows that the proposed algorithm is a robust approach for solving large-scale systems of monotone equations. Additionally, the proposed CG parameter can be used to solve the symmetric system of nonlinear equations as well as other relevant classes of nonlinear equations.<\/jats:p>","DOI":"10.3390\/sym14050936","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T02:19:48Z","timestamp":1651717188000},"page":"936","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Scaled Three-Term Conjugate Gradient Methods for Solving Monotone Equations with Application"],"prefix":"10.3390","volume":"14","author":[{"given":"Jamilu","family":"Sabi\u2019u","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Yusuf Maitama Sule University, Kano P.M.B. 3099, Nigeria"},{"name":"Numerical Optimization Research Group, Bayero University, Gwarzo Road, Kano P.M.B. 3011, Nigeria"}]},{"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 P.M.B. 2346, Nigeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6076-625X","authenticated-orcid":false,"given":"Ali","family":"Althobaiti","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]},{"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 Park Road, Islamabad 45550, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1080\/02331939708844339","article-title":"Newton-type methods with generalized distances for constrained optimization","volume":"41","author":"Iusem","year":"1997","journal-title":"Optimization"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"101174","DOI":"10.1016\/j.najef.2020.101174","article-title":"Efficient predictability of stock return volatility: The role of stock market implied volatility","volume":"52","author":"Dai","year":"2020","journal-title":"N. Am. J. Econ. 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