{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T20:55:31Z","timestamp":1766609731026,"version":"3.40.4"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T00:00:00Z","timestamp":1738713600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T00:00:00Z","timestamp":1738713600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s40314-025-03103-3","type":"journal-article","created":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T12:49:06Z","timestamp":1738759746000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An efficient spectral three-term derivative-free projection algorithm for a large-scale system of monotone nonlinear equations"],"prefix":"10.1007","volume":"44","author":[{"given":"Qun","family":"Li","sequence":"first","affiliation":[]},{"given":"Guolin","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Maolin","family":"Liang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,5]]},"reference":[{"issue":"5","key":"3103_CR1","doi-asserted-by":"publisher","first-page":"6760","DOI":"10.1007\/s40314-018-0712-5","volume":"37","author":"AB Abubakar","year":"2018","unstructured":"Abubakar AB, Kuman P (2018) An improved three-term derivative-free method for solving nonlinear equation. Comput Appl Math 37(5):6760\u20136773","journal-title":"Comput Appl Math"},{"key":"3103_CR2","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s12190-020-01461-1","volume":"67","author":"AB Abubakar","year":"2021","unstructured":"Abubakar AB, Sabi\u2019u J, Kumam P, Shah A (2021) Solving nonlinear monotone operator equations via modified SR1 update. J Appl Math Comput 67:343\u2013373","journal-title":"J Appl Math Comput"},{"key":"3103_CR3","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1007\/s10957-023-02281-6","volume":"200","author":"K Ahmed","year":"2024","unstructured":"Ahmed K, Waziri MY, Murtala S, Halilu AS, Sabi\u2019u J (2024) On a scaled symmetric Dai\u2013Liao-type scheme for constrained system of nonlinear equations with applications. J Optim Theory Appl 200:669\u2013702","journal-title":"J Optim Theory Appl"},{"key":"3103_CR4","doi-asserted-by":"publisher","first-page":"1583","DOI":"10.1007\/s11075-023-01678-8","volume":"96","author":"K Ahmed","year":"2024","unstructured":"Ahmed K, Waziri MY, Murtala S, Halilu AS, Sabi\u2019u J (2024) Another Hager\u2013Zhang-type method via singular-value study for constrained monotone equations with application. Numer Algorthim 96:1583\u20131623","journal-title":"Numer Algorthim"},{"key":"3103_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114630","volume":"417","author":"K Amini","year":"2023","unstructured":"Amini K, Faramarzi P (2023) Global convergence of a modified spectral three-term CG algorithm for nonconvex unconstrained optimization problems. J Comput Appl Math 417:114630","journal-title":"J Comput Appl Math"},{"key":"3103_CR6","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/j.apnum.2019.05.012","volume":"145","author":"AB Awwal","year":"2019","unstructured":"Awwal AB, Kumam P, Abubakar AB (2019) A modified conjugate gradient method for monotone nonlinear equations with convex constraints. Appl Numer Math 145:507\u2013520","journal-title":"Appl Numer Math"},{"issue":"2","key":"3103_CR7","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1137\/0801015","volume":"1","author":"Y Bing","year":"1991","unstructured":"Bing Y, Lin G (1991) An efficient implementation of Merrill\u2019s method for sparse or partially separable systems of nonlinear equations. SIAM J Optim 1(2):206\u2013221","journal-title":"SIAM J Optim"},{"key":"3103_CR8","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.mcm.2009.04.007","volume":"50","author":"WY Cheng","year":"2009","unstructured":"Cheng WY (2009) A PRP type method for systems of monotone equations. Math Comput Modell 50:15\u201320","journal-title":"Math Comput Modell"},{"key":"3103_CR9","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1080\/10556789508805619","volume":"5","author":"SP Dirkse","year":"1995","unstructured":"Dirkse SP, Ferris MC (1995) MCPLIB: a collection of nonlinear mixed complementarity problems. Optim Methods Softw 5:319\u2013345","journal-title":"Optim Methods Softw"},{"issue":"2","key":"3103_CR10","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan ED, Mor\u00e9 JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2):201\u2013213","journal-title":"Math Program"},{"issue":"4","key":"3103_CR11","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s10092-018-0291-2","volume":"55","author":"PT Gao","year":"2018","unstructured":"Gao PT, He CJ (2018) An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo 55(4):53","journal-title":"Calcolo"},{"issue":"4","key":"3103_CR12","doi-asserted-by":"publisher","first-page":"1625","DOI":"10.1093\/imanum\/drq015","volume":"31","author":"QN Li","year":"2011","unstructured":"Li QN, Li DH (2011) A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J Numer Anal 31(4):1625\u20131635","journal-title":"IMA J Numer Anal"},{"key":"3103_CR13","doi-asserted-by":"publisher","first-page":"1343","DOI":"10.1007\/s11075-020-01010-8","volume":"87","author":"Q Li","year":"2021","unstructured":"Li Q, Zheng B (2021) Scaled three-term derivative-free methods for solving large-scale nonlinear monotone equations. Numer Algorthim 87:1343\u20131367","journal-title":"Numer Algorthim"},{"key":"3103_CR14","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1016\/j.cam.2018.10.035","volume":"350","author":"XL Li","year":"2019","unstructured":"Li XL, Shi JJ, Dong XL, Yu JL (2019) A new conjugate gradient method based on quasi-Newton equation for unconstrained optimization. J Comput Appl Math 350:372\u2013379","journal-title":"J Comput Appl Math"},{"issue":"1","key":"3103_CR15","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s11075-018-0603-2","volume":"82","author":"JK Liu","year":"2019","unstructured":"Liu JK, Feng YM (2019) A derivative-free iterative method for nonlinear monotone equations with convex constrains. Numer Algorthim 82(1):245\u2013262","journal-title":"Numer Algorthim"},{"key":"3103_CR16","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1016\/j.apnum.2022.02.001","volume":"175","author":"PJ Liu","year":"2022","unstructured":"Liu PJ, Shao H, Wang Y, Wu XY (2022) A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations without applications. Appl Numer Math 175:98\u2013107","journal-title":"Appl Numer Math"},{"key":"3103_CR17","first-page":"333","volume":"22","author":"K Meintjes","year":"1987","unstructured":"Meintjes K, Morgan AP (1987) A methodology for solving chemical equilibrium systems. Appl Math Comput 22:333\u2013361","journal-title":"Appl Math Comput"},{"key":"3103_CR18","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1007\/s10898-022-01215-2","volume":"87","author":"JY Rao","year":"2023","unstructured":"Rao JY, Huang N (2023) A derivative-free scaling memoryless DFP method for solving large scale nonlinear equations. J Glob Optim 87:641\u2013677","journal-title":"J Glob Optim"},{"key":"3103_CR19","doi-asserted-by":"publisher","first-page":"3645","DOI":"10.1007\/s12190-023-01897-1","volume":"69","author":"MU Rehman","year":"2023","unstructured":"Rehman MU, Sabi\u2019u J, Sohaib M, Shah A (2023) A projection-based derivative free DFP approach for solving system of nonlinear convex constrained monotone equations with image restoration applications. J Appl Math Comput 69:3645\u20133673","journal-title":"J Appl Math Comput"},{"key":"3103_CR20","doi-asserted-by":"publisher","first-page":"S1113","DOI":"10.1051\/ro\/2020061","volume":"55","author":"J Sabi\u2019u","year":"2021","unstructured":"Sabi\u2019u J, Shah A (2021) An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations. RAIRO Oper. Res. 55:S1113\u2013S1127","journal-title":"RAIRO Oper. Res."},{"key":"3103_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/s12190-024-02123-2","author":"J Sabi\u2019u","year":"2024","unstructured":"Sabi\u2019u J, Sirisubtawee S (2024) An inertial Dai-Liao conjugate method for convex constrained monotone equations that avoids the direction of maximum magnification. J Appl Math Comput. https:\/\/doi.org\/10.1007\/s12190-024-02123-2","journal-title":"J Appl Math Comput"},{"issue":"2","key":"3103_CR22","doi-asserted-by":"publisher","first-page":"4843","DOI":"10.3934\/math.2023241","volume":"8","author":"J Sabi\u2019u","year":"2023","unstructured":"Sabi\u2019u J, Althobaiti A, Althobaiti S, Sahoo SK, Botmart T (2023) A scaled Polak\u2013Ribi\u00e8re\u2013Polyak conjugate gradient algorithm for constrained nonlinear systems and motion control. AIMS Math. 8(2):4843\u20134861","journal-title":"AIMS Math."},{"issue":"6","key":"3103_CR23","doi-asserted-by":"publisher","first-page":"e0300547","DOI":"10.1371\/journal.pone.0300547","volume":"19","author":"J Sabi\u2019u","year":"2024","unstructured":"Sabi\u2019u J, Balili A, Emadifar H (2024) An efficient Dai\u2013Yuan projection-based method with application in signal recovery. PLoS ONE 19(6):e0300547","journal-title":"PLoS ONE"},{"key":"3103_CR24","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/978-1-4757-6388-1_18","volume-title":"Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods","author":"MV Solodov","year":"1998","unstructured":"Solodov MV, Svaiter BF (1998) A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima M, Qi L (eds) Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods. Kluwer Academic Publishers, Dordrecht, pp 355\u2013369"},{"issue":"2","key":"3103_CR25","doi-asserted-by":"publisher","first-page":"64","DOI":"10.3390\/a16020064","volume":"16","author":"PS Stanimirovi\u0107","year":"2023","unstructured":"Stanimirovi\u0107 PS, Shaini BI, Sabi\u2019u J, Shah A, Petrovi\u0107 MJ, Ivanov B, Cao XW, Stupina A, Li S (2023) Improved gradient descent iterations for solving systems of nonlinear equations. Algorithms 16(2):64","journal-title":"Algorithms"},{"issue":"5","key":"3103_CR26","doi-asserted-by":"publisher","first-page":"e2374","DOI":"10.1002\/nla.2374","volume":"28","author":"N Ullah","year":"2021","unstructured":"Ullah N, Sabi\u2019u J, Shah A (2021) A derivative-free scaling memoryless Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno method for solving a system of monotone nonlinear equations. Numer Linear Algebra Appl 28(5):e2374","journal-title":"Numer Linear Algebra Appl"},{"issue":"5","key":"3103_CR27","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.3390\/math11051221","volume":"11","author":"N Ullah","year":"2023","unstructured":"Ullah N, Shah A, Sabi\u2019u J, Jiao XM, Awwal AM, Pakkaranang N, Shah SK, Panyanak B (2023) A one-parameter memoryless DFP algorithm for solving system of monotone nonlinear equations with application in image processing. Mathematics 11(5):1221","journal-title":"Mathematics"},{"key":"3103_CR28","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s00186-006-0140-y","volume":"66","author":"CW Wang","year":"2007","unstructured":"Wang CW, Wang YJ, Xu CL (2007) A projection method for a system of nonlinear monotone equations with convex constraints. Math Methods Oper Res 66:33\u201346","journal-title":"Math Methods Oper Res"},{"key":"3103_CR29","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/s10092-015-0140-5","volume":"53","author":"XY Wang","year":"2016","unstructured":"Wang XY, Li SJ, Kou XP (2016) A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints. Calcolo 53:133\u2013145","journal-title":"Calcolo"},{"key":"3103_CR30","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1051\/ro\/2021190","volume":"56","author":"MY Waziri","year":"2022","unstructured":"Waziri MY, Ahmed K, Halilu AS, Sabi\u2019u J (2022) Two new Hager\u2013Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing. RAIRO Oper Res 56:239\u2013273","journal-title":"RAIRO Oper Res"},{"key":"3103_CR31","volume-title":"Power generations, operations, and control","author":"AJ Wood","year":"1996","unstructured":"Wood AJ, Wollenberg BF (1996) Power generations, operations, and control. Wiley, New York"},{"issue":"1","key":"3103_CR32","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1016\/j.jmaa.2013.04.017","volume":"405","author":"YH Xiao","year":"2013","unstructured":"Xiao YH, Zhu H (2013) A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J Math Anal Appl 405(1):310\u2013319","journal-title":"J Math Anal Appl"},{"issue":"3","key":"3103_CR33","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1016\/j.cam.2010.01.001","volume":"234","author":"QR Yan","year":"2010","unstructured":"Yan QR, Peng XZ, Li DH (2010) A globally convergent derivative-free method for solving large scale nonlinear monotone equations. J Comput Appl Math 234(3):649\u2013657","journal-title":"J Comput Appl Math"},{"issue":"1","key":"3103_CR34","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s11075-020-01043-z","volume":"88","author":"JH Yin","year":"2021","unstructured":"Yin JH, Jian JB, Jiang XZ, Liu MX, Wang LZ (2021) A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications. Numer. Algorthim 88(1):389\u2013418","journal-title":"Numer. Algorthim"},{"issue":"4","key":"3103_CR35","doi-asserted-by":"publisher","first-page":"962","DOI":"10.1137\/S1052623499357957","volume":"11","author":"YB Zhao","year":"2001","unstructured":"Zhao YB, Li D (2001) Monotonicity of fixed point and normal mappings associated with variational inequality and its application. SIAM J Optim 11(4):962\u2013973","journal-title":"SIAM J Optim"},{"key":"3103_CR36","doi-asserted-by":"publisher","first-page":"112781","DOI":"10.1016\/j.cam.2020.112781","volume":"375","author":"L Zheng","year":"2020","unstructured":"Zheng L, Yang L, Liang Y (2020) A conjugate gradient projection method for solving equations with convex constraints. J Comput Appl Math 375:112781","journal-title":"J Comput Appl Math"},{"key":"3103_CR37","first-page":"92677","volume":"8","author":"L Zheng","year":"2020","unstructured":"Zheng L, Yang L, Liang Y (2020) A modified spectral gradient projection method for solving non-linear monotone equations with convex constraints and its application. IEEE Access 8:92677\u201392686","journal-title":"IEEE Access"},{"issue":"1","key":"3103_CR38","first-page":"89","volume":"25","author":"WJ Zhou","year":"2007","unstructured":"Zhou WJ, Li DH (2007) Limited memory BFGS method for nonlinear monotone equations. J Comput Math 25(1):89\u201396","journal-title":"J Comput Math"},{"issue":"1","key":"3103_CR39","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10957-014-0547-1","volume":"164","author":"WJ Zhou","year":"2015","unstructured":"Zhou WJ, Shen DM (2015) Convergence properties of an iterative method for solving symmetric nonlinear equations. J Optim Theory Appl 164(1):277\u2013289","journal-title":"J Optim Theory Appl"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03103-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03103-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03103-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T04:26:14Z","timestamp":1744604774000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03103-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,5]]},"references-count":39,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["3103"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03103-3","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,2,5]]},"assertion":[{"value":"17 March 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 September 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 February 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The corresponding author, on behalf of all the co-authors, declare that there are no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"160"}}