{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T02:31:01Z","timestamp":1774924261831,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,26]],"date-time":"2022-12-26T00:00:00Z","timestamp":1672012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The main research object of this paper is to present a systematic computational procedure for computing the inverse of a generalized Vandermonde matrix. Short and rigorous proofs for the formulas of the determinant and the inverse of a generalized Vandermonde matrix are proposed. The computational cost of this method is O(n2). The proposed method can be used efficiently for hand calculation as well as for computer programming. Some examples are given for the sake of illustration. Furthermore, we present a simulation study to compare the time spent to calculate the inverse using the proposed algorithm and the inverse function in Maple.<\/jats:p>","DOI":"10.3390\/axioms12010027","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T04:35:43Z","timestamp":1672115743000},"page":"27","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Fast Novel Recursive Algorithm for Computing the Inverse of a Generalized Vandermonde Matrix"],"prefix":"10.3390","volume":"12","author":[{"given":"Ahmed","family":"Arafat","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Technology and Applied Sciences-ALRustaq, Rustaq 329, Oman"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Moawwad","family":"El-Mikkawy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1093\/imanum\/7.2.235","article-title":"Families of Runge-Kutta-Nystrom formulae","volume":"7","author":"Dormand","year":"1987","journal-title":"IMA J. Numer. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1093\/imanum\/7.4.423","article-title":"High-order embedded Runge-Kutta-Nystrom formulae","volume":"7","author":"Dormand","year":"1987","journal-title":"IMA J. Numer. Anal."},{"key":"ref_3","first-page":"33","article-title":"A new optimized non-FSAL embedded Runge\u2013Kutta\u2013Nystrom algorithm of orders 6 and 4 in six stages","volume":"145","author":"Rahmo","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1080\/0025570X.1984.11977069","article-title":"The generalized Vandermonde matrix","volume":"57","author":"Kalman","year":"1984","journal-title":"Math. Mag."},{"key":"ref_5","unstructured":"Vein, R., and Dale, P. (2006). Determinants and Their Applications in Mathematical Physics, Springer Science & Business Media."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1109\/TAC.1964.1105716","article-title":"Determination of the inverse Vandermonde matrix","volume":"9","author":"Tou","year":"1964","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1109\/97.489066","article-title":"Inversion of the Van der Monde matrix","volume":"3","author":"Neagoe","year":"1996","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1090\/S0025-5718-1970-0290541-1","article-title":"Solution of Vandermonde systems of linear equations","volume":"24","author":"Pereyra","year":"1970","journal-title":"Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/BF01432880","article-title":"On inverses of Vandermonde and confluent Vandermonde matrices III","volume":"29","author":"Gautschi","year":"1978","journal-title":"Numer. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/S1474-6670(17)63534-1","article-title":"On the inversion of Vandermonde matrix","volume":"14","author":"Eisinberg","year":"1981","journal-title":"IFAC Proc. Vol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/S0024-3795(00)00124-5","article-title":"Explicit factorization of the Vandermonde matrix","volume":"315","author":"Phillips","year":"2000","journal-title":"Linear Algebra Its Appl."},{"key":"ref_12","first-page":"1384","article-title":"On the inversion of the Vandermonde matrix","volume":"174","author":"Eisinberg","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"982","DOI":"10.1016\/j.aml.2006.10.003","article-title":"LU factorization of the Vandermonde matrix and its applications","volume":"20","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"ref_14","unstructured":"Moya-Cessa, H., and Soto-Eguibar, F. (2012). Inverse of the Vandermonde and Vandermonde confluent matrices. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"924757","DOI":"10.1155\/2015\/924757","article-title":"A new derivation and recursive algorithm based on Wronskian matrix for Vandermonde inverse matrix","volume":"2015","author":"Zhou","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1175061","DOI":"10.1080\/23311916.2016.1175061","article-title":"A recursive algorithm for computing the inverse of the Vandermonde matrix","volume":"3","author":"Ghassabeh","year":"2016","journal-title":"Cogent Eng."},{"key":"ref_17","first-page":"15","article-title":"On computing the inverse of Vandermonde matrix","volume":"13","author":"Man","year":"2018","journal-title":"Adv. Theor. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1080\/00029890.2018.1427393","article-title":"On One Type of Generalized Vandermonde Determinants","volume":"125","year":"2018","journal-title":"Am. Math. Mon."},{"key":"ref_19","unstructured":"Hosseini, M.S., Chen, A., and Plataniotis, K.N. (2019). On the Closed Form Expression of Elementary Symmetric Polynomials and the Inverse of Vandermonde Matrix. arXiv."},{"key":"ref_20","first-page":"207","article-title":"A simple method for finding the inverse matrix of Vandermonde matrix","volume":"71","author":"Rawashdeh","year":"2019","journal-title":"Mat. Vesn."},{"key":"ref_21","first-page":"643","article-title":"Explicit inverse of a generalized Vandermonde matrix","volume":"146","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.cam.2004.01.032","article-title":"Symmetric functions and the Vandermonde matrix","volume":"172","author":"Akmaz","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","first-page":"8770","article-title":"Remarks on two symmetric polynomials and some matrices","volume":"219","author":"Atlan","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Respondek, J.S. (2020). Recursive Matrix Calculation Paradigm by the Example of Structured Matrix. Information, 11.","DOI":"10.3390\/info11010042"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Knutson, D. (1973). \u03bb-Rings and the Representation Theory of the Symmetric Group, Springer.","DOI":"10.1007\/BFb0069217"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Stanley, R.P. (1999). Enumerative Combinatorics, Springer. Cambridge Studies in Advanced Mathematics.","DOI":"10.1017\/CBO9780511609589"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bergeron, F. (2009). Algebraic Combinatorics and Coinvariant Spaces, AK Peters\/CRC Press.","DOI":"10.1201\/b10583"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Macdonald, I.G. (1998). Symmetric Functions and Hall Polynomials, Oxford University Press.","DOI":"10.1090\/ulect\/012"},{"key":"ref_29","first-page":"3311","article-title":"Notes on particular symmetric polynomials with applications","volume":"215","author":"Sogabe","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/00207390500226093","article-title":"An application of the Vandermonde determinant","volume":"37","author":"Xu","year":"2006","journal-title":"Int. J. Math. Educ. Sci. Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4139728","DOI":"10.1155\/2022\/4139728","article-title":"On the Elementary Symmetric Polynomials and the Zeros of Legendre Polynomials","volume":"2022","author":"Alatawi","year":"2022","journal-title":"J. Math."},{"key":"ref_32","unstructured":"Hou, S.H., and Hou, E. (2008, January 19\u201321). Triangular factors of the inverse of Vandermonde matrices. Proceedings of the 2008 International MultiConference of Engineers and Computer Scientists (Volume II IMECS 2008), Hong Kong, China."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:51:40Z","timestamp":1760147500000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,26]]},"references-count":32,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010027"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010027","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,26]]}}}