{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T17:18:06Z","timestamp":1772644686051,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we revisit the two-step Jarratt method from the perspective of numerical stability. While high-order iterative schemes are often examined in terms of convergence rate and computational efficiency, their backward stability properties have received comparatively less attention. We begin by establishing the method\u2019s strong consistency. Next, we provide a quantitative backward stability assessment within the standard floating-point arithmetic framework, deriving explicit perturbation bounds that show that the iteration errors remain proportional to machine precision. To support the theoretical findings, we present numerical experiments\u2014including tests under finite-precision perturbations\u2014as well as Python implementations and visualizations of the numerical examples. The results illustrate that the two-step Jarratt method not only achieves a high convergence order but also remains numerically robust for well-conditioned nonlinear systems.<\/jats:p>","DOI":"10.3390\/axioms15030186","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T13:11:36Z","timestamp":1772629896000},"page":"186","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Consistency and Quantitative Backward Stability Analysis of the Two-Step Jarratt Method for Nonlinear Systems"],"prefix":"10.3390","volume":"15","author":[{"given":"Vahideh","family":"Rasouli","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ha.C., Islamic Azad University, Hamedan 1477893855, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taher","family":"Lotfi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ha.C., Islamic Azad University, Hamedan 1477893855, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9893-0761","authenticated-orcid":false,"given":"Juan R.","family":"Torregrosa","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"ref_1","unstructured":"Ortega, J.M., and Rheinboldt, W.G. 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