{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:09:10Z","timestamp":1760234950633,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T00:00:00Z","timestamp":1626048000000},"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>A variety of strategies are used to construct algorithms for solving equations. However, higher order derivatives are usually assumed to calculate the convergence order. More importantly, bounds on error and uniqueness regions for the solution are also not derived. Therefore, the benefits of these algorithms are limited. We simply use the first derivative to tackle all these issues and study the ball analysis for two sixth order algorithms under the same set of conditions. In addition, we present a calculable ball comparison between these algorithms. In this manner, we enhance the utility of these algorithms. Our idea is very general. That is why it can also be used to extend other algorithms as well in the same way.<\/jats:p>","DOI":"10.3390\/a14070207","type":"journal-article","created":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T21:56:37Z","timestamp":1626126997000},"page":"207","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extended High Order Algorithms for Equations under the Same Set of Conditions"],"prefix":"10.3390","volume":"14","author":[{"given":"Ioannis K.","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8456-6391","authenticated-orcid":false,"given":"Debasis","family":"Sharma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, IIIT Bhubaneswar, Bhubaneswar 751003, Odisha, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Technology, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2489-4080","authenticated-orcid":false,"given":"Sanjaya Kumar","family":"Parhi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Fakir Mohan University, Balasore 756020, Odisha, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shanta Kumari","family":"Sunanda","sequence":"additional","affiliation":[{"name":"Department of Mathematics, IIIT Bhubaneswar, Bhubaneswar 751003, Odisha, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Michael I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oklahoma, Norman, OK 73071, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,12]]},"reference":[{"key":"ref_1","first-page":"516","article-title":"A new time-efficient and convergent nonlinear solver","volume":"355","author":"Abro","year":"2019","journal-title":"Appl. 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