{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T02:40:07Z","timestamp":1777344007597,"version":"3.51.4"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471354"],"award-info":[{"award-number":["12471354"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018557","name":"Science and Technology Project of Nantong City","doi-asserted-by":"publisher","award":["JC2024043"],"award-info":[{"award-number":["JC2024043"]}],"id":[{"id":"10.13039\/501100018557","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s40314-025-03544-w","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:29:17Z","timestamp":1765780157000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two-step modulus-based matrix splitting iteration method for horizontal implicit complementarity problems"],"prefix":"10.1007","volume":"45","author":[{"given":"Yang","family":"Cao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2795-8409","authenticated-orcid":false,"given":"Qin-Qin","family":"Shen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,12,15]]},"reference":[{"issue":"6","key":"3544_CR1","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1002\/nla.680","volume":"17","author":"Z-Z Bai","year":"2010","unstructured":"Bai Z-Z (2010) Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl 17(6):917\u2013933","journal-title":"Numer Linear Algebra Appl"},{"issue":"3","key":"3544_CR2","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2510","volume":"31","author":"Z-Z Bai","year":"2024","unstructured":"Bai Z-Z (2024) A two-step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations. Numer Linear Algebra Appl 31(3):e2510","journal-title":"Numer Linear Algebra Appl"},{"key":"3544_CR3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611976632","volume-title":"Matrix analysis and computations","author":"Z-Z Bai","year":"2021","unstructured":"Bai Z-Z, Pan J-Y (2021) Matrix analysis and computations. SIAM, Philadelphia"},{"key":"3544_CR4","doi-asserted-by":"publisher","first-page":"2923","DOI":"10.1016\/j.apnum.2009.06.005","volume":"59","author":"Z-Z Bai","year":"2009","unstructured":"Bai Z-Z, Yang X (2009) On HSS-based iteration methods for weakly nonlinear systems. Appl Numer Math 59:2923\u20132936","journal-title":"Appl Numer Math"},{"issue":"3","key":"3544_CR5","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1002\/nla.1835","volume":"20","author":"Z-Z Bai","year":"2013","unstructured":"Bai Z-Z, Zhang L-L (2013) Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl 20(3):425\u2013439","journal-title":"Numer Linear Algebra Appl"},{"issue":"1","key":"3544_CR6","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s11075-012-9566-x","volume":"62","author":"Z-Z Bai","year":"2013","unstructured":"Bai Z-Z, Zhang L-L (2013) Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer Algorithms 62(1):59\u201377","journal-title":"Numer Algorithms"},{"issue":"6","key":"3544_CR7","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2105","volume":"24","author":"Z-Z Bai","year":"2017","unstructured":"Bai Z-Z, Zhang L-L (2017) Modulus-based multigrid methods for linear complementarity problems. Numer Linear Algebra Appl 24(6):e2105","journal-title":"Numer Linear Algebra Appl"},{"issue":"3","key":"3544_CR8","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1137\/S0895479801395458","volume":"24","author":"Z-Z Bai","year":"2003","unstructured":"Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24(3):603\u2013626","journal-title":"SIAM J Matrix Anal Appl"},{"issue":"1\u20132","key":"3544_CR9","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/S0377-0427(00)00432-5","volume":"124","author":"SC Billups","year":"2000","unstructured":"Billups SC, Murty KG (2000) Complementarity problems. J Comput Appl Math 124(1\u20132):303\u2013318","journal-title":"J Comput Appl Math"},{"issue":"4","key":"3544_CR10","doi-asserted-by":"publisher","first-page":"1377","DOI":"10.1007\/s11075-019-00660-7","volume":"82","author":"Y Cao","year":"2019","unstructured":"Cao Y, Wang A (2019) Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems. Numer Algorithms 82(4):1377\u20131394","journal-title":"Numer Algorithms"},{"key":"3544_CR11","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.cam.2014.05.017","volume":"272","author":"Y Cao","year":"2014","unstructured":"Cao Y, Du J, Niu Q (2014) Shift-splitting preconditioners for saddle point problems. J Comput Appl Math 272:239\u2013250","journal-title":"J Comput Appl Math"},{"issue":"4","key":"3544_CR12","doi-asserted-by":"publisher","first-page":"1989","DOI":"10.1007\/s11075-021-01103-y","volume":"88","author":"F Chen","year":"2021","unstructured":"Chen F, Zhu Y, Muratova GV (2021) Two-step modulus-based matrix splitting iteration methods for retinex problem. Numer Algorithms 88(4):1989\u20132005","journal-title":"Numer Algorithms"},{"key":"3544_CR13","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719000","volume-title":"The linear complementarity problem","author":"RW Cottle","year":"2009","unstructured":"Cottle RW, Pang J-S, Stone RE (2009) The linear complementarity problem. SIAM, Philadelphia"},{"issue":"2","key":"3544_CR14","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1002\/nla.609","volume":"16","author":"J-L Dong","year":"2009","unstructured":"Dong J-L, Jiang M-Q (2009) A modified modulus method for symmetric positive-definite linear complementarity problems. Numer Linear Algebra Appl 16(2):129\u2013143","journal-title":"Numer Linear Algebra Appl"},{"issue":"4","key":"3544_CR15","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1137\/S0036144595285963","volume":"39","author":"MC Ferris","year":"1997","unstructured":"Ferris MC, Pang J-S (1997) Engineering and economic applications of complementarity problems. SIAM Rev 39(4):669\u2013713","journal-title":"SIAM Rev"},{"issue":"1","key":"3544_CR16","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF01385865","volume":"63","author":"A Frommer","year":"1992","unstructured":"Frommer A, Szyld DB (1992) H-splittings and two-stage iterative methods. Numer Math 63(1):345\u2013356","journal-title":"Numer Math"},{"key":"3544_CR17","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.apnum.2022.07.011","volume":"182","author":"J He","year":"2022","unstructured":"He J, Zheng H, Vong S (2022) Fast modulus-based matrix splitting iteration methods for implicit complementarity problems. Appl Numer Math 182:28\u201341","journal-title":"Appl Numer Math"},{"issue":"14","key":"3544_CR18","doi-asserted-by":"publisher","first-page":"2392","DOI":"10.1080\/03081087.2022.2104789","volume":"71","author":"J He","year":"2023","unstructured":"He J, Zheng H, Vong S (2023) Modulus-based matrix splitting iteration methods with new splitting scheme for horizontal implicit complementarity problems. Linear Multilinear Algebra 71(14):2392-C2408","journal-title":"Linear Multilinear Algebra"},{"issue":"4","key":"3544_CR19","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1002\/nla.2044","volume":"23","author":"J-T Hong","year":"2016","unstructured":"Hong J-T, Li C-L (2016) Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems. Numer Linear Algebra Appl 23(4):629\u2013641","journal-title":"Numer Linear Algebra Appl"},{"issue":"3","key":"3544_CR20","first-page":"272","volume":"4","author":"J-G Hu","year":"1982","unstructured":"Hu J-G (1982) Estimates of $$\\Vert B^{-1}A\\Vert _{\\infty }$$ and their applications (in Chinese). Math Numer Sin 4(3):272\u2013282","journal-title":"Math Numer Sin"},{"key":"3544_CR21","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1007\/s10092-024-00622-z","volume":"61","author":"Z-G Huang","year":"2024","unstructured":"Huang Z-G (2024) Improved modulus-based matrix splitting iteration method for horizontal implicit complementarity problems. Calcolo 61:70","journal-title":"Calcolo"},{"key":"3544_CR22","volume-title":"Dold A","author":"G Isac","year":"1992","unstructured":"Isac G (1992) The implicit complementarity problem. In: Eckmann B, Takens F (eds) Dold A. Complementarity problems. Lecture notes in mathematics. Springer, Heidelberg"},{"issue":"2","key":"3544_CR23","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1017\/S000497270000993X","volume":"32","author":"G Isac","year":"1985","unstructured":"Isac G (1985) On the implicit complementarity problem in Hilbert spaces. Bull Aust Math Soc 32(2):251\u2013260","journal-title":"Bull Aust Math Soc"},{"issue":"15","key":"3544_CR24","doi-asserted-by":"publisher","first-page":"4875","DOI":"10.2298\/FIL1915875J","volume":"33","author":"L Jia","year":"2019","unstructured":"Jia L, Wang X (2019) A generalized two-step modulus-based matrix splitting iteration method for implicit complementarity problems of H+-matrices. Filomat 33(15):4875\u20134888","journal-title":"Filomat"},{"issue":"2","key":"3544_CR25","first-page":"817","volume":"7","author":"NI Kalashnykova","year":"2011","unstructured":"Kalashnykova NI, Kalashnikov VV, Franco AA (2011) Inexact Newton method to solve nonlinear implicit complementarity problems. Int J Innov Comput Inf Control 7(2):817\u2013825","journal-title":"Int J Innov Comput Inf Control"},{"key":"3544_CR26","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719451","volume-title":"An introduction to variational inequalities and their applications","author":"D Kinderlehrer","year":"2000","unstructured":"Kinderlehrer D, Stampacchia G (2000) An introduction to variational inequalities and their applications. SIAM, Philadelphia"},{"issue":"2","key":"3544_CR27","doi-asserted-by":"publisher","first-page":"363","DOI":"10.4208\/eajam.261215.220217a","volume":"7","author":"C-L Li","year":"2017","unstructured":"Li C-L, Hong J-T (2017) Modulus-based synchronous multisplitting iteration methods for an implicit complementarity problem. East Asian J Appl Math 7(2):363\u2013375","journal-title":"East Asian J Appl Math"},{"key":"3544_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114370","volume":"413","author":"N Li","year":"2022","unstructured":"Li N, Ding J, Yin J-F (2022) Modified relaxation two-sweep modulus-based matrix splitting iteration method for solving a class of implicit complementarity problems. J Comput Appl Math 413:114370","journal-title":"J Comput Appl Math"},{"issue":"1","key":"3544_CR29","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s11075-019-00677-y","volume":"83","author":"F Mezzadri","year":"2020","unstructured":"Mezzadri F, Galligani E (2020) Modulus-based matrix splitting methods for horizontal linear complementarity problems. Numer Algorithms 83(1):201\u2013219","journal-title":"Numer Algorithms"},{"key":"3544_CR30","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1016\/0022-247X(88)90413-1","volume":"133","author":"MA Noor","year":"1988","unstructured":"Noor MA (1988) Fixed point approach for complementarity problems. J Math Anal Appl 133:437\u2013448","journal-title":"J Math Anal Appl"},{"issue":"2","key":"3544_CR31","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/BF00934765","volume":"37","author":"J-S Pang","year":"1982","unstructured":"Pang J-S (1982) On the convergence of a basic iterative method for the implicit complementarity problem. J Optim Theory Appl 37(2):149\u2013162","journal-title":"J Optim Theory Appl"},{"key":"3544_CR32","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.orl.2003.11.004","volume":"32","author":"U Sch\u00e4fer","year":"2004","unstructured":"Sch\u00e4fer U (2004) On the modulus algorithm for the linear complementarity problem. Oper Res Lett 32:350\u2013354","journal-title":"Oper Res Lett"},{"issue":"1","key":"3544_CR33","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/s40314-019-0984-4","volume":"39","author":"Q Shi","year":"2020","unstructured":"Shi Q, Shen Q-Q, Tang T-P (2020) A class of two-step modulus-based matrix splitting iteration methods for quasi-complementarity problems. Comput Appl Math 39(1):11","journal-title":"Comput Appl Math"},{"key":"3544_CR34","first-page":"11","volume":"2","author":"A Wang","year":"2018","unstructured":"Wang A, Cao Y, Shi Q (2018) Convergence analysis of modulus-based matrix splitting iterative methods for implicit complementarity problems. J Inequal Appl 2:11","journal-title":"J Inequal Appl"},{"issue":"3","key":"3544_CR35","doi-asserted-by":"publisher","first-page":"867","DOI":"10.4208\/nmtma.OA-2018-0034","volume":"12","author":"Y Wang","year":"2019","unstructured":"Wang Y, Yin J-F, Dou Q-Y, Li R (2019) Two-step modulus-based matrix splitting iteration methods for a class of implicit complementarity problems. Numer Math Theory Methods Appl 12(3):867\u2013883","journal-title":"Numer Math Theory Methods Appl"},{"issue":"6","key":"3544_CR36","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s40314-024-02860-x","volume":"43","author":"L-X Wang","year":"2024","unstructured":"Wang L-X, Cao Y, Shen Q-Q, Zhou C-C (2024) Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems. Comput Appl Math 43(6):337","journal-title":"Comput Appl Math"},{"issue":"4","key":"3544_CR37","doi-asserted-by":"publisher","first-page":"1308","DOI":"10.1007\/s42967-023-00311-8","volume":"7","author":"L-X Wang","year":"2025","unstructured":"Wang L-X, Shen Q-Q, Cao Y (2025) Modulus-based matrix splitting iteration method for horizontal quasi-complementarity problem. Commun Appl Math Comput 7(4):1308\u20131332","journal-title":"Commun Appl Math Comput"},{"key":"3544_CR38","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2024.116183","volume":"454","author":"L-X Wang","year":"2025","unstructured":"Wang L-X, Cao Y, Shen Q-Q (2025) Improved modulus-based matrix splitting iteration methods for a class of horizontal implicit complementarity problems. J Comput Appl Math 454:116183","journal-title":"J Comput Appl Math"},{"key":"3544_CR39","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.apnum.2018.05.017","volume":"132","author":"S-L Wu","year":"2018","unstructured":"Wu S-L, Guo P (2018) Modulus-based matrix splitting algorithms for quasi-complementarity problems. Appl Numer Math 132:127\u2013137","journal-title":"Appl Numer Math"},{"issue":"1","key":"3544_CR40","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s11075-010-9416-7","volume":"57","author":"L-L Zhang","year":"2011","unstructured":"Zhang L-L (2011) Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer Algorithms 57(1):83\u201399","journal-title":"Numer Algorithms"},{"key":"3544_CR41","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.apnum.2020.09.008","volume":"164","author":"L-L Zhang","year":"2021","unstructured":"Zhang L-L, Ren Z-R (2021) A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems. Appl Numer Math 164:89\u2013100","journal-title":"Appl Numer Math"},{"issue":"8","key":"3544_CR42","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1007\/s40314-025-03373-x","volume":"44","author":"Y-X Zhang","year":"2025","unstructured":"Zhang Y-X, Zheng H, Vong S (2025) An inner-outer modulus-based method for solving horizontal linear complementarity problems. Comput Appl Math 44(8):420","journal-title":"Comput Appl Math"},{"issue":"2","key":"3544_CR43","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s13160-020-00405-3","volume":"37","author":"H Zheng","year":"2020","unstructured":"Zheng H, Qu W (2020) Superlinearly convergent methods for solving a class of implicit complementarity problems based on sign analysis. Jpn J Ind Appl Math 37(2):433\u2013447","journal-title":"Jpn J Ind Appl Math"},{"issue":"2","key":"3544_CR44","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s11075-018-0614-z","volume":"82","author":"H Zheng","year":"2019","unstructured":"Zheng H, Vong S (2019) A modified modulus-based matrix splitting iteration method for solving implicit complementarity problems. Numer Algorithms 82(2):573\u2013592","journal-title":"Numer Algorithms"},{"issue":"2","key":"3544_CR45","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s11075-012-9664-9","volume":"64","author":"N Zheng","year":"2013","unstructured":"Zheng N, Yin J-F (2013) Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem. Numer Algorithms 64(2):245\u2013262","journal-title":"Numer Algorithms"},{"issue":"8","key":"3544_CR46","doi-asserted-by":"publisher","first-page":"2350129","DOI":"10.1142\/S0218126623501293","volume":"32","author":"C-C Zhou","year":"2023","unstructured":"Zhou C-C, Cao Y, Shi Q (2023) A robust two-step modulus-based matrix splitting iteration method for mixed-size cell circuit legalization problem. J Circuit Syst Comput 32(8):2350129","journal-title":"J Circuit Syst Comput"},{"key":"3544_CR47","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.vlsi.2022.08.010","volume":"88","author":"C-C Zhou","year":"2023","unstructured":"Zhou C-C, Qiu J, Cao Y, Yang G-C, Shen Q-Q, Shi Q (2023) An accelerated modulus-based matrix splitting iteration method for mixed-size cell circuits legalization. Integration 88:20\u201331","journal-title":"Integration"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03544-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03544-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03544-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T02:02:06Z","timestamp":1777341726000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03544-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,15]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["3544"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03544-w","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,15]]},"assertion":[{"value":"26 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 November 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 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 author declares no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"116"}}