{"paper":{"title":"Revisiting Gilbert Strang's \"A Chaotic Search for $i$\"","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Ao Li, Robert M. Corless","submitted_at":"2018-08-08T17:48:19Z","abstract_excerpt":"In the paper \"A Chaotic Search for $i$\"~(\\cite{strang1991chaotic}), Strang completely explained the behaviour of Newton's method when using real initial guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\\pm i$. He explored an exact symbolic formula for the iteration, namely $x_{n}=\\cot{ \\left( 2^{n} \\theta_{0} \\right) }$, which is valid in exact arithmetic. In this paper, we extend this to to $k^{th}$ order Householder methods, which include Halley's method, and to the secant method. Two formulae, $x_{n}=\\cot{ \\left( \\theta_{n-1}+\\theta_{n-2} \\right) }$ with $\\theta_{n-1}=\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03229","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}