{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:30:46Z","timestamp":1772119846855,"version":"3.50.1"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T00:00:00Z","timestamp":1723420800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T00:00:00Z","timestamp":1723420800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100009042","name":"Universidad de Sevilla","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100009042","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica, Matlab, and Python as examples) do not offer a solution, is to find all the real solutions of a system of\n                    <jats:italic>n<\/jats:italic>\n                    nonlinear equations in a certain finite domain of the\n                    <jats:italic>n<\/jats:italic>\n                    -dimensional space of variables. We present two similar algorithms of minimum length and computational weight to solve this problem, in which one resembles a graphical tool of edge detection in an image extended to\n                    <jats:italic>n<\/jats:italic>\n                    dimensions. To do this, we discretize the\n                    <jats:italic>n<\/jats:italic>\n                    -dimensional space sector in which the solutions are sought. Once the discretized hypersurfaces (edges) defined by each nonlinear equation of the\n                    <jats:italic>n<\/jats:italic>\n                    -dimensional system have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in each\n                    <jats:italic>n<\/jats:italic>\n                    -dimensional tile or cell containing at least one solution marks the approximate locations of all the hyperpoints that constitute the solutions. This makes the final Newton-Raphson step rapidly convergent to all the existent solutions in the predefined space sector with the desired degree of accuracy.\n                  <\/jats:p>","DOI":"10.1007\/s11075-024-01908-7","type":"journal-article","created":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T01:02:07Z","timestamp":1723424527000},"page":"1111-1125","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain"],"prefix":"10.1007","volume":"99","author":[{"given":"Fernando","family":"Chueca-D\u00edez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alfonso M.","family":"Ga\u00f1\u00e1n-Calvo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,12]]},"reference":[{"key":"1908_CR1","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1017\/S0004972708000117","volume":"77","author":"I Alolyan","year":"2008","unstructured":"Alolyan, I.: An algorithm for finding all zeros of vector functions. Bull. Austral. Math. Soc. 77, 353\u2013363 (2008)","journal-title":"Bull. Austral. Math. Soc."},{"key":"1908_CR2","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1090\/S0025-5718-1965-0198670-6","volume":"19","author":"CG Broyden","year":"1965","unstructured":"Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comp. 19, 577\u2013593 (1965)","journal-title":"Math. Comp."},{"key":"1908_CR3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511624117","volume-title":"Mathematical explorations with matlab","author":"K Chen","year":"1999","unstructured":"Chen, K., Giblin, P., Irving, A.: Mathematical explorations with matlab. Cambridge University Press, Cambridge (1999)"},{"key":"1908_CR4","first-page":"877","volume":"168","author":"S Effati","year":"2005","unstructured":"Effati, S., Nazemi, A.R.: A new method for solving a system of nonlinear equations. Science Direct 168, 877\u2013894 (2005)","journal-title":"Science Direct"},{"key":"1908_CR5","volume-title":"Recent advances in global optimization for process synthesis, design and control: Enclosure of all solutions","author":"C Floudas","year":"1999","unstructured":"Floudas, C.: Recent advances in global optimization for process synthesis, design and control: Enclosure of all solutions. Elsevier, Amsterdam (1999)"},{"key":"1908_CR6","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1016\/j.ins.2020.06.042","volume":"541","author":"W Gao","year":"2020","unstructured":"Gao, W., Luo, Y., Xu, J., Zhu, S.: Evolutionary algorithm with multiobjective optimization technique for solving nonlinear equation systemss. Inf. Sci. 541, 345\u2013361 (2020)","journal-title":"Inf. Sci."},{"key":"1908_CR7","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1109\/TSMC.2018.2828018","volume":"50","author":"W Gong","year":"2020","unstructured":"Gong, W., Wang, Y., Cai, Z., Wang, L.: Finding multiple roots of nonlinear equation systems via a repulsion-based adaptive differential evolution. IEEE Trans. Syst. Man Cybern. Syst. 50, 1499\u20131513 (2020)","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"1908_CR8","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1090\/qam\/736504","volume":"42","author":"CS Hsu","year":"1984","unstructured":"Hsu, C.S.: A simplicial mapping method for locating the zeros of a function. Q. Appl. Math. 42, 41\u201359 (1984)","journal-title":"Q. Appl. Math."},{"key":"1908_CR9","doi-asserted-by":"publisher","first-page":"164","DOI":"10.1090\/qam\/10666","volume":"2","author":"K Levenberg","year":"1944","unstructured":"Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Quart. Appl. Math. 2, 164\u2013168 (1944)","journal-title":"Quart. Appl. Math."},{"key":"1908_CR10","volume-title":"Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-newton method","author":"YZ Luo","year":"2007","unstructured":"Luo, Y.Z., Tang, G.J., Zhou, L.N.: Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-newton method. Elsevier, Amsterdam (2007)"},{"key":"1908_CR11","doi-asserted-by":"publisher","first-page":"679","DOI":"10.20965\/jaciii.2015.p0697","volume":"19","author":"IA Sidarto","year":"2015","unstructured":"Sidarto, I.A., Kania, A.: Finding all solutions of systems of nonlinear equations using spiral dynamics inspired optimization with clustering. J. Adv. Comput. Intell. Intell. Inform. 19, 679\u2013707 (2015)","journal-title":"J. Adv. Comput. Intell. Intell. Inform."},{"issue":"5","key":"1908_CR12","doi-asserted-by":"publisher","first-page":"697","DOI":"10.20965\/jaciii.2015.p0697","volume":"19","author":"KA Sidarto","year":"2015","unstructured":"Sidarto, K.A., Kania, A.: Finding all solutions of systems of nonlinear equations using spiral dynamics inspired optimization with clustering. Journal of Advanced Computational Intelligence and Intelligent Informatics. 19(5), 697\u2013707 (2015)","journal-title":"Journal of Advanced Computational Intelligence and Intelligent Informatics."},{"key":"1908_CR13","doi-asserted-by":"publisher","first-page":"2465","DOI":"10.1016\/j.nonrwa.2009.08.003","volume":"11","author":"I Tsoulos","year":"2010","unstructured":"Tsoulos, I., Stavrakoudis, A.: On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods. Nonlinear Anal. Real World Appl. 11, 2465\u20132471 (2010)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"1908_CR14","first-page":"134","volume":"275","author":"M Waseem","year":"2016","unstructured":"Waseem, M., Noor, M.A., Noor, K.I.: Efficient method for solving a system of nonlinear equations. Appl. Math. Comput. 275, 134\u2013146 (2016)","journal-title":"Appl. Math. Comput."},{"key":"1908_CR15","doi-asserted-by":"crossref","unstructured":"Zufiria, P.J., Guttalu, R.S.: On an application of dynamical systems theory to determine all the zeros of a vector function. J. Math. Anal. Appl. 152, 269\u2013295 (1990)","DOI":"10.1016\/0022-247X(90)90103-M"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01908-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01908-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01908-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T04:43:17Z","timestamp":1750135397000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01908-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,12]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["1908"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01908-7","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3834704\/v1","asserted-by":"object"}]},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,12]]},"assertion":[{"value":"4 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}}]}}