{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T09:47:36Z","timestamp":1775382456259,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,11,3]],"date-time":"2015-11-03T00:00:00Z","timestamp":1446508800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>An application of iterative methods for computing the Moore\u2013Penrose inverse in balancing chemical equations is considered. With the aim to illustrate proposed algorithms, an improved high order hyper-power matrix iterative method for computing generalized inverses is introduced and applied. The improvements of the hyper-power iterative scheme are based on its proper factorization, as well as on the possibility to accelerate the iterations in the initial phase of the convergence. Although the effectiveness of our approach is confirmed on the basis of the theoretical point of view, some numerical comparisons in balancing chemical equations, as well as on randomly-generated matrices are furnished.<\/jats:p>","DOI":"10.3390\/a8040982","type":"journal-article","created":{"date-parts":[[2015,11,3]],"date-time":"2015-11-03T11:17:07Z","timestamp":1446549427000},"page":"982-998","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Some Matrix Iterations for Computing Generalized Inverses and Balancing Chemical Equations"],"prefix":"10.3390","volume":"8","author":[{"given":"Farahnaz","family":"Soleimani","sequence":"first","affiliation":[{"name":"Department of Chemistry, Roudehen Branch, Islamic Azad University, 39731 Roudehen, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0655-3741","authenticated-orcid":false,"given":"Predrag","family":"Stanimirovi\u00b4c","sequence":"additional","affiliation":[{"name":"Faculty of Sciences and Mathematics, University of Ni\u0161, Vi\u0161egradska 33, 18000 Ni\u0161, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fazlollah","family":"Soleymani","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Ferdowsi University of Mashhad, 91779 Mashhad, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,11,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/S0097-8485(97)00066-1","article-title":"Algebraic constructs for the graphical and computational solution to balancing chemical equations","volume":"22","author":"Phillips","year":"1998","journal-title":"Comput. Chem."},{"key":"ref_2","first-page":"104","article-title":"A new generalized matrix inverse method for balancing chemical equations and their stability","volume":"2","author":"Risteski","year":"2008","journal-title":"Bol. Soc. Qum. Mex."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"223","DOI":"10.5012\/jkcs.2008.52.3.223","article-title":"A new pseudoinverse matrix method for balancing chemical equations and their stability","volume":"52","author":"Risteski","year":"2008","journal-title":"J. Korean Chem. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/0020739780090310","article-title":"Generalized matrix inverse approach for automatic balancing of chemical equations","volume":"9","author":"Krishnamurthy","year":"1978","journal-title":"Int. J. Math. Educ. Sci. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1137\/0713017","article-title":"Residue arithmetic algorithms for exact computation of g-inverses of matrices","volume":"13","author":"Mahadeva","year":"1976","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_6","first-page":"215","article-title":"A new generalized algebra for the balancing of \u2135 chemical reactions","volume":"48","author":"Risteski","year":"2014","journal-title":"Mater. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1021\/ed074p1369","article-title":"Using Mathematica and Maple to obtain chemical equations","volume":"74","author":"Smith","year":"1997","journal-title":"J. Chem. Educ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1162\/NECO_a_00549","article-title":"A novel iterative method for computing generalized inverse","volume":"26","author":"Xia","year":"2014","journal-title":"Neural Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1002\/wics.164","article-title":"Gaussian elimination","volume":"332\u2013334","author":"Higham","year":"2011","journal-title":"WIREs Comp. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1016\/j.mcm.2006.02.004","article-title":"Chemical equation balancing: An integer programming approach","volume":"44","author":"Sen","year":"2006","journal-title":"Math. Comput. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1023\/A:1017979821326","article-title":"Linear variational Diophantine techniques in mass balance of chemical reactions","volume":"30","author":"Balasubramanian","year":"2001","journal-title":"J. Math. Chem."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1090\/S0025-5718-1965-0179915-5","article-title":"An iterative method for computing the generalized inverse of an arbitrary matrix","volume":"19","year":"1965","journal-title":"Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/S0024-3795(01)00309-3","article-title":"A geometrical approach on generalized inverses by Neumann-type series","volume":"332\u2013334","author":"Climent","year":"2001","journal-title":"Linear Algebra Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1016\/j.laa.2013.05.005","article-title":"Higher-order convergent iterative method for computing the generalized inverse and its application to Toeplitz matrices","volume":"439","author":"Liu","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1016\/j.laa.2015.07.010","article-title":"On hyper-power family of iterations for computing outer inverses possessing high efficiencies","volume":"484","author":"Soleymani","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1002\/zamm.19330130111","article-title":"Iterative Berechnung der Reziproken matrix","volume":"13","author":"Schulz","year":"1933","journal-title":"Z. Angew. Math. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s12190-014-0759-4","article-title":"Finding the Moore\u2013Penrose inverse by a new matrix iteration","volume":"47","author":"Soleymani","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s11075-014-9913-1","article-title":"An efficient and stable Newton-type iterative method for computing generalized inverse A T , S ( 2 )","volume":"69","author":"Soleymani","year":"2015","journal-title":"Numer. Algorithms"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1023\/A:1025150130276","article-title":"A concise description of an old problem: Application of matrices to obtain the balancing coefficients of chemical equations","volume":"34","author":"Ramasami","year":"2003","journal-title":"J. Math. Chem."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.cam.2014.11.009","article-title":"Finding generalized inverses by a fast and efficient numerical method","volume":"279","author":"Sharifi","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.cam.2013.12.033","article-title":"A class of numerical algorithms for computing outer inverses","volume":"263","author":"Soleymani","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.amc.2014.01.021","article-title":"On the extension of Householder\u2019s method for weighted Moore\u2013Penrose inverse","volume":"231","author":"Soleimani","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.cam.2014.01.034","article-title":"Two improvements of the iterative method for computing Moore\u2013Penrose inverse based on Penrose equations","volume":"267","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_24","unstructured":"Wolfram, S. (2003). The Mathematica Book, Wolfram Media. [5th ed.]."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"560931","DOI":"10.1155\/2014\/560931","article-title":"An inversion-free method for finding positive definite solution of a rational matrix equation","volume":"2014","author":"Soleymani","year":"2014","journal-title":"Sci. World J."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/4\/982\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:51:22Z","timestamp":1760215882000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/4\/982"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,3]]},"references-count":25,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["a8040982"],"URL":"https:\/\/doi.org\/10.3390\/a8040982","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,11,3]]}}}