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For matrices whose fields of values are a circular disk centered at the origin, we show that the cost of Uhlig\u2019s method blows up with respect to the desired relative accuracy. More generally, we also analyze the local behavior of Uhlig\u2019s cutting procedure at outermost points in the field of values, showing that it often has a fast Q-linear rate of convergence and is Q-superlinear at corners. Finally, we identify and address inefficiencies in both the level-set and cutting-plane approaches and propose refined versions of these techniques.<\/jats:p>","DOI":"10.1137\/21m1455826","type":"journal-article","created":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T04:56:38Z","timestamp":1682571398000},"page":"A753-A780","source":"Crossref","is-referenced-by-count":2,"title":["Convergence Rate Analysis and Improved Iterations for Numerical Radius Computation"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8426-0242","authenticated-orcid":true,"given":"Tim","family":"Mitchell","sequence":"first","affiliation":[{"name":"Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr 1, 39106 Magdeburg, Germany. Current address: Department of Computer Science, Queens College\/CUNY, Flushing, NY 11367 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(90)90037-U"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479800367439"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02419030"},{"key":"ref4","first-page":"590","volume":"12","author":"Berger C. A.","year":"1965","journal-title":"Notices Amer. Math. Soc."},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/23.3.359"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/17M1137966"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/19M1246213"},{"key":"ref8","doi-asserted-by":"crossref","unstructured":"P. Benner , \nT. Mitchell , and \nM. L. 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