{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T23:33:47Z","timestamp":1768433627673,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T00:00:00Z","timestamp":1653523200000},"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>In this work, some new inequalities for the numerical radius of block n-by-n matrices are presented. As an application, the bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition method is proved. We highlight several numerical examples showing that our approach to bounding zeros of polynomials could be very effective in comparison with the most famous results as well as some recent results presented in the field. Finally, observations, a discussion, and a conclusion regarding our proposed bound of zeros are considered. Namely, it is proved that our proposed bound is more efficient than any other bound under some conditions; this is supported with many polynomial examples explaining our choice of restrictions.<\/jats:p>","DOI":"10.3390\/a15060184","type":"journal-article","created":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T08:50:22Z","timestamp":1653555022000},"page":"184","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bounding the Zeros of Polynomials Using the Frobenius Companion Matrix Partitioned by the Cartesian Decomposition"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6696-9119","authenticated-orcid":false,"given":"Mohammad W.","family":"Alomari","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, P.O. Box 2600, Irbid 21110, Jordan"}]},{"given":"Christophe","family":"Chesneau","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universit\u00e9 de Caen Basse-Normandie, F-14032 Caen, France"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Borwein, P., and Erd\u00e9lyi, T. (1995). Polynomials and Polynomial Inequalities, Speinger.","DOI":"10.1007\/978-1-4612-0793-1"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Milovanovi\u0107, G.V., Mitrinovixcx, D.S., and Rassias, T.M. (1994). Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific.","DOI":"10.1142\/1284"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (1985). Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511810817"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (1991). Topics in Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511840371"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.laa.2013.09.049","article-title":"Numerical radius inequalities for n \u00d7 n operator matrices","volume":"468","author":"Kittaneh","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11785-021-01161-z","article-title":"Numerical radius inequalities for Hilbert space operators","volume":"15","author":"Alomari","year":"2021","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1208","DOI":"10.1080\/03081087.2019.1624682","article-title":"Refinement of some numerical radius inequalities for Hilbert space operators","volume":"69","author":"Alomari","year":"2021","journal-title":"Linear Multilinear Algebra"},{"key":"ref_8","first-page":"3","article-title":"On the generalized mixed Schwarz inequality","volume":"46","author":"Alomari","year":"2020","journal-title":"Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerbaijan"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1080\/03081080801915792","article-title":"Numerical radius inequalities for operator matrices","volume":"57","author":"Kittaneh","year":"2009","journal-title":"Linear Multilinear Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/BF01378777","article-title":"Norm inequalities of positive operator matrices","volume":"22","author":"Hou","year":"1995","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1002\/nla.1811","article-title":"Tridiagonal Toeplitz matrices: Properties and novel applications","volume":"20","author":"Noschese","year":"2013","journal-title":"Numer. Lin. Algebra"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"B\u00f6ttcher, A., and Grudsky, S.M. (2022, April 20). Spectral Properties of Banded Toeplitz Matrices; SIAM: 2005. Available online: https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/1.9780898717853.fm.","DOI":"10.1137\/1.9780898717853"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1007\/BF01343117","article-title":"Notes on some inequalities for linear operators","volume":"125","author":"Kato","year":"1952","journal-title":"Math. Ann."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s00013-003-0525-6","article-title":"Bounds for the zeros of polynomials from matrix inequalities","volume":"81","author":"Kittaneh","year":"2003","journal-title":"Arch. Math. (Basel)"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s00020-011-1893-0","article-title":"Numerical radius inequalities for certain 2 \u00d7 2 operator matrices","volume":"71","author":"Hirzallah","year":"2011","journal-title":"Integr. Equ. Oper. Theory"},{"key":"ref_16","first-page":"359","article-title":"Buzano\u2019s inequality and bounds for roots of algebraic equations","volume":"117","author":"Fujii","year":"1993","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1070\/SM1988v059n01ABEH003123","article-title":"The geometry of the Hausdorff domain in localization problems for the spectrum of arbitrary matrices","volume":"59","author":"Abdurakhmanov","year":"1988","journal-title":"Math. USSR Sb."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"129132","DOI":"10.1155\/2012\/129132","article-title":"On numerical radius of a matrix and estimation of bounds for zeros of a polynomial","volume":"2012","author":"Paul","year":"2012","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_19","first-page":"127","article-title":"Bounds for zeros of polynomials using traces and determinants","volume":"69","author":"Linden","year":"2000","journal-title":"Semin. Fachbereich Math. Feu Hagen"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"56","DOI":"10.15352\/afa\/1391614569","article-title":"Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polynomials","volume":"5","author":"Kittaneh","year":"2014","journal-title":"Ann. Funct. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1080\/03081087.2020.1756199","article-title":"A novel numerical radius upper bounds for 2 \u00d7 2, operator matrices","volume":"70","author":"Jaradat","year":"2022","journal-title":"Linear Multilinear Algebra"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/6\/184\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:18:52Z","timestamp":1760138332000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/6\/184"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,26]]},"references-count":21,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["a15060184"],"URL":"https:\/\/doi.org\/10.3390\/a15060184","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,26]]}}}