{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:40:22Z","timestamp":1760229622469,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,20]],"date-time":"2022-06-20T00:00:00Z","timestamp":1655683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"],"award-info":[{"award-number":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Jiangsu Province","award":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"],"award-info":[{"award-number":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"]}]},{"name":"Natural Science Foundation of Jiangsu Higher Education Institutions of China","award":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"],"award-info":[{"award-number":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"]}]},{"name":"China Postdoctoral Science Foundation","award":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"],"award-info":[{"award-number":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"]}]},{"name":"Qing Lan Project of the Jiangsu Higher Education Institutions","award":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"],"award-info":[{"award-number":["12001396","BK20170591","BK20200268","21KJB110017","20KJB110005","2018M642130"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Rational eigenvalue problems (REPs) have important applications in engineering applications and have attracted more and more attention in recent years. Based on the theory of low-rank modification, we discuss the spectral properties and distribution of the symmetric rational eigenvalue problems, and present two numerical iteration methods for the above REPs. Numerical experiments demonstrate the effectiveness of our proposed methods.<\/jats:p>","DOI":"10.3390\/sym14061270","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T23:11:19Z","timestamp":1655939479000},"page":"1270","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Spectral Distribution and Numerical Methods for Rational Eigenvalue Problems"],"prefix":"10.3390","volume":"14","author":[{"given":"Xiaoping","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"},{"name":"Department of Mathematics, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xueying","family":"Shi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"An","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taizhou University, Taizhou 225300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1137\/0111049","article-title":"The degree of a rational matrix function","volume":"11","author":"Duffin","year":"1963","journal-title":"J. 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Solids.","DOI":"10.1177\/1081286517726368"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.jcp.2003.10.026","article-title":"Numerical simulation of a three dimensional quantum dot","volume":"196","author":"Hwang","year":"2004","journal-title":"J. Comput. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1016\/j.jcp.2006.01.034","article-title":"Iterative projection methods for computing relevant energy states of a quantum dot","volume":"217","author":"Voss","year":"2006","journal-title":"J. Computat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1137\/090777542","article-title":"Solving rational eigenvalue problems via linearization","volume":"32","author":"Su","year":"2011","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"e2214","DOI":"10.1002\/nla.2214","article-title":"A compact rational Krylov method for large-scale rational eigenvalue problems","volume":"26","author":"Dopico","year":"2019","journal-title":"Numer. Linear Algebr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1137\/0707043","article-title":"On an approach to the solution of the generalized latent value problem for \u03bb-matrices","volume":"7","author":"Kublanovskaya","year":"1970","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.1137\/0720075","article-title":"On Kublanovskaya\u2019s approach to the solution of the generalized latent value problem for functional \u03bb-matrices","volume":"20","author":"Jain","year":"1983","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_13","first-page":"374","article-title":"QR decomposition and nonlinear eigenvalue problems","volume":"11","author":"Li","year":"1989","journal-title":"Math. Numer. Sinica"},{"key":"ref_14","first-page":"1","article-title":"Compute multiply nonlinear eigenvalues","volume":"10","author":"Li","year":"1992","journal-title":"J. Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"745","DOI":"10.4208\/jcm.1004-m0009","article-title":"On smooth LU decompositions with applications to solutions of nonlinear eigenvalue problems","volume":"28","author":"Dai","year":"2010","journal-title":"J. Comput. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0024-3795(95)90004-7","article-title":"Nonequivalence deflation for the solution of matrix talent value problems","volume":"231","author":"Guo","year":"1995","journal-title":"Linear Algebra Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1002\/mma.1670040126","article-title":"A minmax principle for nonlinear eigenvalue problems with application to nonoverdamped systems","volume":"4","author":"Voss","year":"1982","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3991","DOI":"10.1016\/j.laa.2010.06.048","article-title":"Nonlinear Rayleigh functionals","volume":"436","author":"Schwetlick","year":"2012","journal-title":"Linear Algebra Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1023\/B:BITN.0000039424.56697.8b","article-title":"An Arnoldi method for nonlinear eigenvalue problems","volume":"44","author":"Voss","year":"2004","journal-title":"BIT"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1284","DOI":"10.1016\/j.compstruc.2006.08.088","article-title":"A Jacobi Davidson method for nonlinear and nonsymmetric eigenproblems","volume":"85","author":"Voss","year":"2007","journal-title":"Comput. Struct."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"109678","DOI":"10.1016\/j.jcp.2020.109678","article-title":"A Riesz-projection-based method for nonlinear eigenvalue problems","volume":"419","author":"Binkowski","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"139","DOI":"10.4208\/eajam.100916.061117a","article-title":"A Modified Newton Method for Nonlinear Eigenvalue Problems","volume":"8","author":"Chen","year":"2018","journal-title":"East Asian J. Appl. Math."},{"key":"ref_23","first-page":"124663","article-title":"Successive linear Newton interpolation methods for solving the large-scale nonlinear eigenvalue problems","volume":"387","author":"Chen","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.apnum.2020.11.007","article-title":"Modified successive approximation methods for the nonlinear eigenvalue problems","volume":"164","author":"Chen","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1137\/070779528","article-title":"Nonlinear low rank modification of a symmetric eigenvalue problem","volume":"32","author":"Voss","year":"2011","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_26","unstructured":"Wilkinson, J.H. (1965). The Algebraic Eigenvalue Problem, Clarendon Press."},{"key":"ref_27","first-page":"116","article-title":"Partially orthogonal projection method and its variations for solving polynomial eigenvalue problem","volume":"38","author":"Wei","year":"2016","journal-title":"Numer. Math. J. Chin. Univ."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1270\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:35:22Z","timestamp":1760139322000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1270"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,20]]},"references-count":27,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061270"],"URL":"https:\/\/doi.org\/10.3390\/sym14061270","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,20]]}}}