{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T20:07:09Z","timestamp":1767989229055,"version":"3.49.0"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,9,7]],"date-time":"2019-09-07T00:00:00Z","timestamp":1567814400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,9,7]],"date-time":"2019-09-07T00:00:00Z","timestamp":1567814400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["BK20181405"],"award-info":[{"award-number":["BK20181405"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of china","doi-asserted-by":"crossref","award":["U1533202"],"award-info":[{"award-number":["U1533202"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"Natural science foundation of china","doi-asserted-by":"crossref","award":["U1811464"],"award-info":[{"award-number":["U1811464"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"Natural science foundation of china","doi-asserted-by":"crossref","award":["11571124"],"award-info":[{"award-number":["11571124"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s11075-019-00799-3","type":"journal-article","created":{"date-parts":[[2019,9,7]],"date-time":"2019-09-07T05:55:03Z","timestamp":1567835703000},"page":"1-21","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["A class of modified modulus-based synchronous multisplitting iteration methods for linear complementarity problems"],"prefix":"10.1007","volume":"85","author":[{"given":"Weiwei","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lei","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8953-1478","authenticated-orcid":false,"given":"Xiaofei","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junfeng","family":"Yin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,9,7]]},"reference":[{"key":"799_CR1","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. Numerical Linear Algebra with Applications 17, 917\u2013933 (2010)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"799_CR2","first-page":"228","volume":"5","author":"ZZ Bai","year":"1996","unstructured":"Bai, Z.Z.: On the monotone convergence of the projected iteration methods for linear complementarity problem. Numerical Mathematics-A Journal of Chinese Universities (English Series) 5, 228\u2013233 (1996)","journal-title":"Numerical Mathematics-A Journal of Chinese Universities (English Series)"},{"key":"799_CR3","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. Numerical Linear Algebra with Applications 20, 425\u2013439 (2013)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"799_CR4","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":"799_CR5","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":"799_CR6","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1080\/00207169708804569","volume":"63","author":"Z-Z Bai","year":"1997","unstructured":"Bai, Z.-Z., Evans, D.J.: Matrix multisplitting relaxation methods for linear complementarity problems. Int. J. Comput. Math. 63, 309\u2013326 (1997)","journal-title":"Int. J. Comput. Math."},{"key":"799_CR7","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":"799_CR8","volume-title":"Nonnegative Matrix in the Mathematical Sciences","author":"A Berman","year":"1979","unstructured":"Berman, A., Plemmons, R.J.: Nonnegative Matrix in the Mathematical Sciences. Academic Press, New York (1979)"},{"key":"799_CR9","volume-title":"The Linear Complementarity Problem","author":"RW Cottle","year":"1992","unstructured":"Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)"},{"key":"799_CR10","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1137\/0309028","volume":"9","author":"CW Cryer","year":"1971","unstructured":"Cryer, C. W.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM Journal on Control 9, 385\u2013392 (1971)","journal-title":"SIAM Journal on Control"},{"key":"799_CR11","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1002\/nla.1896","volume":"21","author":"Lj Cvetkovi\u0107","year":"2014","unstructured":"Cvetkovi\u0107, Lj., Kosti\u0107, V.: A note on the convergence of the MSMAOR method for linear complementarity problems. Numerical Linear Algebra with Applications 21, 534\u2013539 (2014)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"799_CR12","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1002\/nla.609","volume":"16","author":"J-L Dong","year":"2009","unstructured":"Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numerical Linear Algebra with Applications 16, 129\u2013143 (2009)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"799_CR13","doi-asserted-by":"crossref","first-page":"1226","DOI":"10.1137\/15M1045892","volume":"9","author":"J-L Dong","year":"2016","unstructured":"Dong, J.-L., Gao, J.-B., Ju, F.-J., Shen, J.H.: Modulus methods for nonnegatively constrained image restoration. SIAM J. Imag. Sci. 9, 1226\u20131246 (2016)","journal-title":"SIAM J. Imag. Sci."},{"key":"799_CR14","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1137\/S0036144595285963","volume":"39","author":"MC Ferris","year":"1997","unstructured":"Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669\u2013713 (1997)","journal-title":"SIAM Rev."},{"key":"799_CR15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.ins.2017.10.047","volume":"429","author":"Y-M Huang","year":"2018","unstructured":"Huang, Y.-M., Yan, H.-Y., Wen, Y.W., Yang, X.: Rank minimization with applications to image noise removal. Inform. Sci. 429, 147\u2013163 (2018)","journal-title":"Inform. Sci."},{"key":"799_CR16","doi-asserted-by":"crossref","first-page":"4534","DOI":"10.1109\/TIP.2012.2205007","volume":"21","author":"Y-M Huang","year":"2012","unstructured":"Huang, Y.-M., Moisan, L., Ng, M. K., Zeng, T.-Y.: Multiplicative noise removal via a learned dictionary. IEEE Trans. Image Process. 21, 4534\u20134543 (2012)","journal-title":"IEEE Trans. Image Process."},{"key":"799_CR17","doi-asserted-by":"crossref","first-page":"A2856","DOI":"10.1137\/120898693","volume":"35","author":"Y-M Huang","year":"2013","unstructured":"Huang, Y.-M., Lu, D.-Y., Zeng, T.-Y.: Two-step approach for the restoration of images corrupted by multiplicative noise. SIAM J. Sci. Comput. 35, A2856\u2013A2873 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"799_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 iteration method for linear complementarity problems of H-matrices. Appl. Math. Lett. 26, 1159\u20131164 (2013)","journal-title":"Appl. Math. Lett."},{"key":"799_CR19","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1080\/03081087.2015.1087457","volume":"64","author":"W Li","year":"2016","unstructured":"Li, W., Zheng, H.: A preconditioned modulus-based iteration method for solving linear complementarity problems of H-matrices. Linear and Multilinear Algebra 64, 1390\u20131403 (2016)","journal-title":"Linear and Multilinear Algebra"},{"key":"799_CR20","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10092-015-0143-2","volume":"53","author":"S-M Liu","year":"2016","unstructured":"Liu, S.-M., Zheng, H, Li, W.: A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems. CALCOLO 53, 189\u2013199 (2016)","journal-title":"CALCOLO"},{"key":"799_CR21","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/0377-0427(94)00103-2","volume":"62","author":"N Machida","year":"1995","unstructured":"Machida, N., Fukushima, M., Ibaraki, T.: A multisplitting method for symmetric linear complementarity problems. J. Comput. Appl. Math. 62, 217\u2013227 (1995)","journal-title":"J. Comput. Appl. Math."},{"key":"799_CR22","doi-asserted-by":"crossref","first-page":"102","DOI":"10.4208\/eajam.020318.220618","volume":"9","author":"X-F Peng","year":"2019","unstructured":"Peng, X.-F., Wang, M., Li, W.: A relaxation two-sweep modulus-based matrix splitting iteration method for linear complementarity problems. East Asian Journal on Applied Mathematics 9, 102\u2013121 (2019)","journal-title":"East Asian Journal on Applied Mathematics"},{"key":"799_CR23","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s002110100333","volume":"92","author":"Y-Z Song","year":"2002","unstructured":"Song, Y.-Z.: Comparison theorems for splittings of matrices. Numerische Mathematik 92, 563\u2013591 (2002)","journal-title":"Numerische Mathematik"},{"key":"799_CR24","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. Springer, Berlin (2000)"},{"key":"799_CR25","first-page":"349","volume":"321","author":"B-L Wen","year":"2018","unstructured":"Wen, B.-L., Zheng, H., Li, W., Peng, X.-F.: The relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems of positive definite matrices. Appl. Math. Comput. 321, 349\u2013357 (2018)","journal-title":"Appl. Math. Comput."},{"key":"799_CR26","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1007\/s11075-018-0477-3","volume":"79","author":"X-P Wu","year":"2018","unstructured":"Wu, X.-P., Peng, X.-F., Li, W.: A preconditioned general modulus-based matrix splitting iteration method for linear complementarity problems of H-matrices. Numer. Algorithms 79, 1131\u20131146 (2018)","journal-title":"Numer. Algorithms"},{"key":"799_CR27","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1002\/nla.1985","volume":"5","author":"W-W Xu","year":"2015","unstructured":"Xu, W.-W.: Modified modulus-based matrix splitting iteration methods for linear complementarity problems. Numerical Linear Algebra with Applications 5, 748\u2013760 (2015)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"799_CR28","doi-asserted-by":"crossref","first-page":"100","DOI":"10.4208\/jcm.1403-m4195","volume":"33","author":"L-L Zhang","year":"2015","unstructured":"Zhang, L.-L.: Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems. J. Comput. Math. 33, 100\u2013112 (2015)","journal-title":"J. Comput. Math."},{"key":"799_CR29","first-page":"638","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. Numerical Linear Algebra with Applications 26, 638\u2013642 (2013)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"799_CR30","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 methods for linear complementarity problem. Numer. Algorithms 64, 245\u2013262 (2013)","journal-title":"Numer. Algorithms"},{"key":"799_CR31","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.-W.: A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems. Numer. Algorithms 74, 137\u2013152 (2017)","journal-title":"Numer. Algorithms"},{"key":"799_CR32","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1080\/03081087.2018.1470602","volume":"67","author":"H Zheng","year":"2018","unstructured":"Zheng, H., Vong, S.-W.: Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems. Linear and Multilinear Algebra 67, 1773\u20131784 (2018)","journal-title":"Linear and Multilinear Algebra"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00799-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00799-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00799-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,6]],"date-time":"2020-09-06T00:35:10Z","timestamp":1599352510000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00799-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,7]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["799"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00799-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,7]]},"assertion":[{"value":"20 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 August 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 September 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}