{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T21:08:29Z","timestamp":1767906509410,"version":"3.49.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T00:00:00Z","timestamp":1762732800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T00:00:00Z","timestamp":1762732800000},"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":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s12190-025-02664-0","type":"journal-article","created":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T15:56:39Z","timestamp":1762790199000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two fixed-point iteration methods for generalized absolute value equations: formulation and analysis"],"prefix":"10.1007","volume":"72","author":[{"given":"Rashid","family":"Ali","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,11,10]]},"reference":[{"key":"2664_CR1","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s11590-006-0005-6","volume":"1","author":"O.L. Mangasarian","year":"2007","unstructured":"Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3\u20138 (2007)","journal-title":"Optim. Lett."},{"key":"2664_CR2","doi-asserted-by":"publisher","first-page":"3381","DOI":"10.1007\/s12190-024-02334-7","volume":"71","author":"D. Hussain","year":"2025","unstructured":"Hussain, D., Pan, K., Kumar, B.: Numerical exploration of two generalized iteration methods for solving nonlinear complementarity problems. J. Appl. Math. Comput. 71, 3381\u20133397 (2025)","journal-title":"J. Appl. Math. Comput."},{"issue":"7","key":"2664_CR3","doi-asserted-by":"publisher","first-page":"1527","DOI":"10.1007\/s11590-011-0347-6","volume":"6","author":"O.L. Mangasarian","year":"2012","unstructured":"Mangasarian, O.L.: Primal-dual bilinear programming solution of the absolute value equation. Optim. Lett. 6(7), 1527\u20131533 (2012)","journal-title":"Optim. Lett."},{"key":"2664_CR4","doi-asserted-by":"publisher","first-page":"3219","DOI":"10.1007\/s12190-024-02093-5","volume":"70","author":"W.H. Luo","year":"2024","unstructured":"Luo, W.H., Guo, J., Yin, L.: A dimension expanded Newton-type method for absolute value equations. J. Appl. Math. Comput. 70, 3219\u20133233 (2024)","journal-title":"J. Appl. Math. Comput."},{"issue":"2","key":"2664_CR5","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/s11590-008-0102-9","volume":"3","author":"O.L. Magasarian","year":"2009","unstructured":"Magasarian, O.L.: Knapsack feasibility as an absolute value equation solvable by successive linear programming. Optim. Lett. 3(2), 161\u2013170 (2009)","journal-title":"Optim. Lett."},{"key":"2664_CR6","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/j.laa.2006.05.004","volume":"419","author":"O.L. Mangasarian","year":"2006","unstructured":"Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359\u2013367 (2006)","journal-title":"Linear Algebra Appl."},{"key":"2664_CR7","doi-asserted-by":"crossref","unstructured":"Rohn, J.: A theorem of the alternatives for the equation $$Ax + B|x| = b$$. Linear Multilinear Algebra 52, 421\u2013426 (2004)","DOI":"10.1080\/0308108042000220686"},{"key":"2664_CR8","first-page":"1","volume":"99","author":"F. Mezzadri","year":"2020","unstructured":"Mezzadri, F.: On the solution of general absolute value equations. Appl. Math. Lett. 99, 1\u20137 (2020)","journal-title":"Appl. Math. Lett."},{"key":"2664_CR9","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/s10589-007-9158-1","volume":"44","author":"O. Prokopyev","year":"2009","unstructured":"Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44, 363\u2013372 (2009)","journal-title":"Comput. Optim. Appl."},{"key":"2664_CR10","doi-asserted-by":"crossref","unstructured":"Li, C.-X.: A preconditioned AOR iterative method for the absolute value equations. Int. J. Comput. Methods 14(2) (2017)","DOI":"10.1142\/S0219876217500165"},{"issue":"5","key":"2664_CR11","doi-asserted-by":"publisher","first-page":"5171","DOI":"10.3934\/math.2020332","volume":"5","author":"S. Wu","year":"2020","unstructured":"Wu, S., Li, C.: A special shift splitting iteration method for absolute value equation. AIMS Math. 5(5), 5171\u20135183 (2020)","journal-title":"AIMS Math."},{"key":"2664_CR12","doi-asserted-by":"publisher","first-page":"742","DOI":"10.1016\/j.cam.2017.07.009","volume":"330","author":"N. Zainali","year":"2018","unstructured":"Zainali, N., Lotfi, T.: On developing a stable and quadratic convergent method for solving absolute value equation. J. Comput. Appl. Math. 330, 742\u2013747 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"2664_CR13","doi-asserted-by":"publisher","first-page":"1042","DOI":"10.1186\/s40064-016-2720-5","volume":"5","author":"J. Feng","year":"2016","unstructured":"Feng, J., Liu, S.: An improved generalized Newton method for absolute value equations. SpringerPlus 5, 1042 (2016)","journal-title":"SpringerPlus"},{"key":"2664_CR14","unstructured":"Fakharzadeh, A.J., Shams, N.N.: An efficient algorithm for solving absolute value equations. J. Math. Ext. 15(3), Paper no. 4 (2021)"},{"key":"2664_CR15","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s12190-016-1065-0","volume":"56","author":"B. Saheya","year":"2018","unstructured":"Saheya, B., Yu, C.-H., Chen, J.-S.: Numerical comparisons based on four smoothing functions for absolute value equation. J. Appl. Math. Comput. 56, 131\u2013149 (2018)","journal-title":"J. Appl. Math. Comput."},{"key":"2664_CR16","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.amc.2017.05.035","volume":"311","author":"Y.-F. Ke","year":"2017","unstructured":"Ke, Y.-F., Ma, C.-F.: SOR-like iteration method for solving absolute value equations. Appl. Math. Comput. 311, 195\u2013202 (2017)","journal-title":"Appl. Math. Comput."},{"key":"2664_CR17","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.aml.2019.03.033","volume":"97","author":"P. Guo","year":"2019","unstructured":"Guo, P., Wu, S.-L., Li, C.-X.: On the SOR-like iteration method for solving absolute value equations. Appl. Math. Lett. 97, 107\u2013113 (2019)","journal-title":"Appl. Math. Lett."},{"key":"2664_CR18","doi-asserted-by":"publisher","first-page":"109479","DOI":"10.1016\/j.aml.2025.109479","volume":"164","author":"T. Luo","year":"2025","unstructured":"Luo, T., Liu, J., Chen, C., Wang, Q.: A monotone block coordinate descent method for solving absolute value equations. Appl. Math. Lett. 164, 109479 (2025)","journal-title":"Appl. Math. Lett."},{"key":"2664_CR19","doi-asserted-by":"publisher","first-page":"113578","DOI":"10.1016\/j.cam.2021.113578","volume":"394","author":"H.-Y. Zhou","year":"2021","unstructured":"Zhou, H.-Y., Wu, S.-L., Li, C.-X.: Newton-based matrix splitting method for generalized absolute value equation. J. Comput. Appl. Math. 394, 113578 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"2664_CR20","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1007\/s10957-018-1439-6","volume":"181","author":"A. Wang","year":"2019","unstructured":"Wang, A., Cao, Y., Chen, J.-X.: Modified Newton-type iteration methods for generalized absolute value equations. J. Optim. Theory Appl. 181, 216\u2013230 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"2664_CR21","first-page":"1","volume":"20","author":"R. Ali","year":"2021","unstructured":"Ali, R., Pan, K.: The new iteration methods for solving absolute value equations. Appl. Math. 20, 1\u201314 (2021)","journal-title":"Appl. Math."},{"key":"2664_CR22","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1007\/s10957-023-02357-3","volume":"200","author":"S. Kumar","year":"2024","unstructured":"Kumar, S.: Correction to: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 200, 891\u2013893 (2024)","journal-title":"J. Optim. Theory Appl."},{"key":"2664_CR23","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1016\/j.apnum.2020.08.001","volume":"158","author":"M. Dehghan","year":"2020","unstructured":"Dehghan, M., Shirilord, A.: Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation. Appl. Numer. Math. 158, 425\u2013438 (2020)","journal-title":"Appl. Numer. Math."},{"key":"2664_CR24","doi-asserted-by":"publisher","first-page":"2191","DOI":"10.1007\/s11590-014-0727-9","volume":"8","author":"D.K. Salkuyeh","year":"2014","unstructured":"Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191\u20132202 (2014)","journal-title":"Optim. Lett."},{"key":"2664_CR25","doi-asserted-by":"publisher","first-page":"e34505","DOI":"10.1016\/j.heliyon.2024.e34505","volume":"10","author":"R. Ali","year":"2024","unstructured":"Ali, R., Zhang, Z., Awwad, F.: The study of new fixed-point iteration schemes for solving absolute value equations. Heliyon. 10, e34505 (2024)","journal-title":"Heliyon."},{"key":"2664_CR26","doi-asserted-by":"publisher","first-page":"116887","DOI":"10.1016\/j.cam.2025.116887","volume":"473","author":"D. Yu","year":"2026","unstructured":"Yu, D., Ma, T., Zhang, G., Chen, C., Han, D.: Predefined-time robust gradient neural network for solving absolute value equations. J. Comput. Appl. Math. 473, 116887 (2026)","journal-title":"J. Comput. Appl. Math."},{"key":"2664_CR27","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/j.orl.2009.01.013","volume":"37","author":"M. Dehghan","year":"2009","unstructured":"Dehghan, M., Hajarian, M.: Convergence of SSOR methods for linear complementarity problems. Oper. Res. Lett. 37, 219\u2013223 (2009)","journal-title":"Oper. Res. Lett."},{"key":"2664_CR28","doi-asserted-by":"publisher","first-page":"910","DOI":"10.1108\/EC-10-2011-0131","volume":"30","author":"N.H. Sberi","year":"2013","unstructured":"Sberi, N.H., Edalatpandh, S.A.: Modification of iterative methods for solving linear complementarity problems. Eng. Comput. 30, 910\u2013923 (2013)","journal-title":"Eng. Comput."},{"key":"2664_CR29","first-page":"12","volume":"195","author":"S.-X. Miao","year":"2018","unstructured":"Miao, S.-X., Zhang, D.: On the preconditioned GAOR method for a linear complementarity problem with an M-matrix. J. Inequal. Appl. 195, 12 (2018)","journal-title":"J. Inequal. Appl."},{"key":"2664_CR30","doi-asserted-by":"publisher","first-page":"128378","DOI":"10.1016\/j.amc.2023.128378","volume":"464","author":"R. Ali","year":"2024","unstructured":"Ali, R., Akgul, A.: A new matrix splitting generalized iteration method for linear complementarity problems. Appl. Math. Comput. 464, 128378 (2024)","journal-title":"Appl. Math. Comput."},{"key":"2664_CR31","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/BF00935545","volume":"33","author":"B.H. Ahn","year":"1981","unstructured":"Ahn, B.H.: Solution of nonsymmetric, linear complementarity problems by iterative methods. J. Optim. Theory Appl. 33, 175\u2013187 (1981)","journal-title":"J. Optim. Theory Appl."},{"key":"2664_CR32","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/BF01268170","volume":"22","author":"O.L. Mangasarian","year":"1977","unstructured":"Mangasarian, O.L.: Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465\u2013485 (1977)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"2664_CR33","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s13160-022-00526-x","volume":"40","author":"R. Ali","year":"2023","unstructured":"Ali, R., Pan, K.: Two new fixed point iterative schemes for absolute value equations. Jpn. J. Ind. Appl. Math. 40(1), 303\u2013314 (2023)","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"2664_CR34","doi-asserted-by":"publisher","first-page":"105990","DOI":"10.1016\/j.aml.2019.07.021","volume":"99","author":"Y. Ke","year":"2020","unstructured":"Ke, Y.: The new iteration algorithm for absolute value equation. Appl. Math. Lett. 99, 105990 (2020)","journal-title":"Appl. Math. Lett."},{"key":"2664_CR35","doi-asserted-by":"publisher","first-page":"799","DOI":"10.1007\/s11075-023-01666-y","volume":"96","author":"C. Chen","year":"2024","unstructured":"Chen, C., Huang, B., Yu, D. et al.: Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations. Numer Algor 96, 799\u2013826 (2024)","journal-title":"Numer Algor"},{"key":"2664_CR36","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s11590-012-0560-y","volume":"8","author":"J. Rohn","year":"2014","unstructured":"Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8, 35\u201344 (2014)","journal-title":"Optim. Lett."},{"key":"2664_CR37","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1016\/j.amc.2016.08.020","volume":"293","author":"V. Edalatpour","year":"2017","unstructured":"Edalatpour, V., Hezari, D., Salkuyeh, D.K.: A generalization of the Gauss-Seidel iteration method for solving absolute value equations. Appl. Math. Comput. 293, 156\u2013167 (2017)","journal-title":"Appl. Math. Comput."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02664-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02664-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02664-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T16:46:38Z","timestamp":1767890798000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02664-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,10]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2664"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02664-0","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,10]]},"assertion":[{"value":"25 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 November 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 reports no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interests"}}],"article-number":"3"}}