{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T20:49:46Z","timestamp":1781556586181,"version":"3.54.5"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,7,21]],"date-time":"2020-07-21T00:00:00Z","timestamp":1595289600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,21]],"date-time":"2020-07-21T00:00:00Z","timestamp":1595289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s11075-020-00983-w","type":"journal-article","created":{"date-parts":[[2020,7,21]],"date-time":"2020-07-21T13:04:59Z","timestamp":1595336699000},"page":"667-687","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9779-7647","authenticated-orcid":false,"given":"Francesco","family":"Mezzadri","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emanuele","family":"Galligani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,7,21]]},"reference":[{"issue":"1","key":"983_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10092-018-0244-9","volume":"55","author":"F Mezzadri","year":"2018","unstructured":"Mezzadri, F., Galligani, E.: An intexact Newton methods for solving complementarity problems in hydrodynamic lubrication. Calcolo 55(1), 1\u201328 (2018)","journal-title":"Calcolo"},{"key":"983_CR2","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1002\/fld.1650040407","volume":"4","author":"M Kostreva","year":"1984","unstructured":"Kostreva, M.: Elasto-hydrodynamic lubrication: a nonlinear complementarity problem, Internat. J. Numer. Methods Fluids 4, 377\u2013397 (1984)","journal-title":"J. Numer. Methods Fluids"},{"key":"983_CR3","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1115\/1.3260872","volume":"106","author":"K Oh","year":"1984","unstructured":"Oh, K.: The numerical solution of dynamically loaded elastohydrodynamic contact as a nonlinear complementarity problem. ASME J. Tribology. 106, 88\u201395 (1984)","journal-title":"ASME J. Tribology."},{"key":"983_CR4","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/BF01390185","volume":"43","author":"GH Meyer","year":"1984","unstructured":"Meyer, G.H.: Free boundary problems with nonlinear source terms. Numer. Math. 43, 463\u2013482 (1984)","journal-title":"Numer. Math."},{"key":"983_CR5","doi-asserted-by":"crossref","unstructured":"Cottle, R.W., Pang, J.-S., Stone, R.E.: The linear complementarity problem, classics in applied mathematics, society for industrial and applied mathematics philadelphia (U.S.A.) (2009)","DOI":"10.1137\/1.9780898719000"},{"key":"983_CR6","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/s11075-010-9439-0","volume":"57","author":"R Andreani","year":"2011","unstructured":"Andreani, R., J\u00fadice, J.J., Mart\u00ednez, J.M., Patr\u00edcio, J.: A projected-gradient interior-point algorithm for complementarity problems. Numer. Algorithms 57, 457\u2013485 (2011)","journal-title":"Numer. Algorithms"},{"key":"983_CR7","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1007\/s11075-018-0480-8","volume":"79","author":"CA Arias","year":"2018","unstructured":"Arias, C.A., Mart\u00ednez, J.M.: Fast convergence of an inexact interior point method for horizontal complementarity problems. Numer. Algorithms 79, 1187\u20131210 (2018)","journal-title":"Numer. Algorithms"},{"key":"983_CR8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0898-1221(96)00172-1","volume":"32","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: The convergence of parallel iteration algorithms for linear complementarity problems. Comput. Math. Appl. 32, 1\u201317 (1996)","journal-title":"Comput. Math. Appl."},{"key":"983_CR9","first-page":"125","volume":"13","author":"Z-Z Bai","year":"2001","unstructured":"Bai, Z.-Z., Evans, D.J.: Matrix multisplitting methods with applications to linear complementarity problems: parallel synchronous and chaotic methods. R\u00e9seaux et Syst\u00e8mes R\u00e9partis: Calculateurs Parall\u00e8les 13, 125\u2013154 (2001)","journal-title":"R\u00e9seaux et Syst\u00e8mes R\u00e9partis: Calculateurs Parall\u00e8les"},{"key":"983_CR10","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1080\/00207160211927","volume":"79","author":"Z-Z Bai","year":"2002","unstructured":"Bai, Z.-Z., Evans, D.J.: Matrix multisplitting methods with applications to linear complementarity problems: parallel asynchronous methods. Int. J. Comput. Math. 79, 205\u2013232 (2002)","journal-title":"Int. J. Comput. Math."},{"key":"983_CR11","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1002\/nla.680","volume":"17","author":"Z-Z Bai","year":"2010","unstructured":"Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917\u2013933 (2010)","journal-title":"Numer. Linear Algebra Appl."},{"key":"983_CR12","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/BF01268170","volume":"22","author":"O Mangasarian","year":"1977","unstructured":"Mangasarian, O.: Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465\u2013485 (1977)","journal-title":"J. Optim. Theory Appl."},{"key":"983_CR13","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1137\/0309028","volume":"9","author":"C Cryer","year":"1971","unstructured":"Cryer, C.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control 9, 385\u2013392 (1971)","journal-title":"SIAM J. Control"},{"key":"983_CR14","first-page":"228","volume":"5","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: On the monotone convergence of the projected iteration methods for linear complementarity problem. Numer. Math. J. Chinese Univ. (English Ser.) 5, 228\u2013233 (1996)","journal-title":"Numer. Math. J. Chinese Univ. (English Ser.)"},{"issue":"41","key":"983_CR15","first-page":"1","volume":"56","author":"F Mezzadri","year":"2019","unstructured":"Mezzadri, F.: On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems. Calcolo 56(41), 1\u201320 (2019)","journal-title":"Calcolo"},{"key":"983_CR16","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s11075-012-9664-9","volume":"64","author":"N Zheng","year":"2013","unstructured":"Zheng, N., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Algorithms 64, 245\u2013262 (2013)","journal-title":"Numer. Algorithms"},{"issue":"1","key":"983_CR17","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s11075-016-0142-7","volume":"74","author":"H Zheng","year":"2017","unstructured":"Zheng, H., Li, W., Vong, S.: A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems. Numer. Algorithms 74(1), 137\u2013152 (2017)","journal-title":"Numer. Algorithms"},{"key":"983_CR18","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/j.aml.2013.06.015","volume":"26","author":"W Li","year":"2013","unstructured":"Li, W.: A general modulus-based matrix splitting method for linear complementarity problems of H-matrices. Appl. Math. Lett. 26, 1159\u20131164 (2013)","journal-title":"Appl. Math. Lett."},{"key":"983_CR19","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1002\/nla.1835","volume":"20","author":"Z-Z Bai","year":"2013","unstructured":"Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425\u2013439 (2013)","journal-title":"Numer. Linear Algebra Appl."},{"key":"983_CR20","doi-asserted-by":"crossref","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.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algorithms 62, 59\u201377 (2013)","journal-title":"Numer. Algorithms"},{"key":"983_CR21","first-page":"1","volume":"e2105","author":"Z-Z Bai","year":"2017","unstructured":"Bai, Z.-Z., Zhang, L.-L.: Modulus-based multigrid methods for linear complementarity problems. Numer. Linear Algebra Appl. 24 e2105, 1\u201315 (2017)","journal-title":"Numer. Linear Algebra Appl. 24"},{"issue":"6","key":"983_CR22","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1016\/j.aml.2013.01.001","volume":"26","author":"L-L Zhang","year":"2013","unstructured":"Zhang, L.-L., Ren, Z.-R.: Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems. Appl. Math. Lett. 26(6), 638\u2013642 (2013)","journal-title":"Appl. Math. Lett."},{"issue":"9","key":"983_CR23","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1080\/03081087.2018.1470602","volume":"67","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Vong, S.: Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems. Linear Multilinear Algebra 67 (9), 1773\u20131784 (2019)","journal-title":"Linear Multilinear Algebra"},{"key":"983_CR24","doi-asserted-by":"publisher","unstructured":"Xu, W.-W., Zhu, L., Peng, X.-F., Liu, H., Yin, J.-F.: A class of modified modulus-based synchronous multisplitting iteration methods for linear complementarity problems Numer. Algorithms (in press) https:\/\/doi.org\/10.1007\/s11075-019-00799-3(2019)","DOI":"10.1007\/s11075-019-00799-3"},{"key":"983_CR25","first-page":"542","volume":"348","author":"P-F Dai","year":"2019","unstructured":"Dai, P.-F., Li, J., Bai, J., Qiu, J.: A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem. Appl. Math. Comput. 348, 542\u2013551 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"983_CR26","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1016\/j.camwa.2018.10.040","volume":"77","author":"H Ren","year":"2019","unstructured":"Ren, H., Wang, X., Tang, X.-B., Wang, T.: The general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems. Comput. Math. Appl. 77(4), 1071\u20131081 (2019)","journal-title":"Comput. Math. Appl."},{"issue":"12","key":"983_CR27","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0898-1221(96)00073-9","volume":"31","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems. Comput. Math. Appl. 31(12), 17\u201333 (1996)","journal-title":"Comput. Math. Appl."},{"key":"983_CR28","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0898-1221(96)00169-1","volume":"32","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z., Wang, D.-R.: A class of parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems. Comput. Math. Appl. 32, 79\u201395 (1996)","journal-title":"Comput. Math. Appl."},{"key":"983_CR29","first-page":"34","volume":"271","author":"Z Xia","year":"2015","unstructured":"Xia, Z., Li, C.-L.: Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem. Appl. Math. Comput. 271, 34\u201342 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"983_CR30","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1002\/nla.2039","volume":"23","author":"N Huang","year":"2016","unstructured":"Huang, N., Ma, C.-F.: The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems. Numer. Linear Algebra Appl. 23(3), 558\u2013569 (2016)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"4","key":"983_CR31","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1007\/s10092-017-0236-1","volume":"54","author":"H Zheng","year":"2017","unstructured":"Zheng, H.: Improved convergence theorems of modulus-based matrix splitting iteration method for nonlinear complementarity problems of H-matrices. Calcolo 54(4), 1481\u20131490 (2017)","journal-title":"Calcolo"},{"issue":"2","key":"983_CR32","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s11075-016-0243-3","volume":"75","author":"R Li","year":"2017","unstructured":"Li, R., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems. Numer. Algorithms 75(2), 339\u2013358 (2017)","journal-title":"Numer. Algorithms"},{"key":"983_CR33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.laa.2016.01.002","volume":"494","author":"S-L Xie","year":"2016","unstructured":"Xie, S.-L., Xu, H.-R., Zeng, J.-P.: Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems. Linear Algebra Appl. 494, 1\u201310 (2016)","journal-title":"Linear Algebra Appl."},{"key":"983_CR34","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1007\/s10957-018-1361-y","volume":"180","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Liu, L.: The sign-based methods for solving a class of nonlinear complementarity problems. J. Optim. Theory Appl. 180, 480\u2013499 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"37","key":"983_CR35","first-page":"1","volume":"55","author":"H Zheng","year":"2018","unstructured":"Zheng, H., Vong, S.: The modulus-based nonsmooth Newton\u2019s method for solving a class of nonlinear complementarity problems of P-matrices. Calcolo 55(37), 1\u201317 (2018)","journal-title":"Calcolo"},{"key":"983_CR36","first-page":"396","volume":"353","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Vong, S., Liu, L.: A direct preconditioned modulus-based iteration method for solving nonlinear complementarity problems of H-matrices. Appl. Math. Comput. 353, 396\u2013405 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"8","key":"983_CR37","doi-asserted-by":"crossref","first-page":"1648","DOI":"10.1080\/00207160.2018.1504928","volume":"96","author":"H Zheng","year":"2018","unstructured":"Zheng, H., Vong, S., Liu, L.: The relaxation modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems. Int. J. Comput. Math. 96(8), 1648\u20131667 (2018)","journal-title":"Int. J. Comput. Math."},{"issue":"2","key":"983_CR38","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1007\/s10957-018-1395-1","volume":"180","author":"F Mezzadri","year":"2019","unstructured":"Mezzadri, F., Galligani, E.: Splitting methods for a class of horizontal linear complementarity problems. J. Optim. Theory Appl. 180(2), 500\u2013517 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"983_CR39","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s11075-019-00677-y","volume":"83","author":"F Mezzadri","year":"2020","unstructured":"Mezzadri, F., Galligani, E.: Modulus-based matrix splitting methods for horizontal linear complementarity problems. Numer. Algorithms 83, 201\u2013219 (2020)","journal-title":"Numer. Algorithms"},{"key":"983_CR40","first-page":"1","volume":"369","author":"H Zheng","year":"2020","unstructured":"Zheng, H., Vong, S.: On convergence of the modulus-based matrix splitting iteartion method for horizontal linear complementarity problem of H+-matrices. Appl. Math. Comput. 369, 1\u20136 (2020)","journal-title":"Appl. Math. Comput."},{"key":"983_CR41","doi-asserted-by":"publisher","unstructured":"F. Mezzadri, F.: Modulus-based synchronous multisplitting methods for solving horizontal linear complementarity problems on parallel computers, Numer. Linear Algebra Appl. (in press). https:\/\/doi.org\/10.1002\/nla.2319 (2020)","DOI":"10.1002\/nla.2319"},{"key":"983_CR42","doi-asserted-by":"publisher","unstructured":"Mezzadri, F., Galligani, E.: A modulus-based nonsmooth Newton\u2019s method for solving horizontal linear complementarity problems, Optim. Lett. (in press) https:\/\/doi.org\/10.1007\/s11590-019-01515-9 (2019)","DOI":"10.1007\/s11590-019-01515-9"},{"key":"983_CR43","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-05156-2","volume-title":"Matrix iterative analysis","author":"RS Varga","year":"2000","unstructured":"Varga, R.S.: Matrix iterative analysis, 2nd edn. Springer, Berlin (2000)","edition":"2nd edn."},{"key":"983_CR44","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1137\/S0895479897324032","volume":"21","author":"Z-Z Bai","year":"1999","unstructured":"Bai, Z.-Z.: On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J. Matrix Anal. Appl. 21, 67\u201378 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"983_CR45","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/BF01385865","volume":"63","author":"A Frommer","year":"1992","unstructured":"Frommer, A., Szyld, D.B.: H-Splittings and two-stage iterative methods. Numer. Math. 63, 345\u2013356 (1992)","journal-title":"Numer. Math."},{"key":"983_CR46","volume-title":"Numerical analysis: a second course, computer science and applied mathematics","author":"JM Ortega","year":"1972","unstructured":"Ortega, J.M.: Numerical analysis: a second course, computer science and applied mathematics. Academic Press, New York (1972)"},{"key":"983_CR47","volume-title":"Iterative solution of nonlinear equations in several variables","author":"JM Ortega","year":"1970","unstructured":"Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, Cambridge (1970)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00983-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00983-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00983-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,20]],"date-time":"2021-07-20T23:15:27Z","timestamp":1626822927000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00983-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,21]]},"references-count":47,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["983"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00983-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,21]]},"assertion":[{"value":"22 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}