{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:28:08Z","timestamp":1750220888267,"version":"3.41.0"},"reference-count":26,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2019,9,18]],"date-time":"2019-09-18T00:00:00Z","timestamp":1568764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2019,9,18]]},"abstract":"<jats:p>\n            In the paper \"A Chaotic Search for\n            <jats:italic>i<\/jats:italic>\n            \" ([25]), Strang completely explained the behaviour of Newton's method when using real initial guesses on\n            <jats:italic>f<\/jats:italic>\n            (\n            <jats:italic>x<\/jats:italic>\n            ) =\n            <jats:italic>x<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            + 1, which has only the pair of complex roots \u00b1\n            <jats:italic>i<\/jats:italic>\n            . He explored an exact symbolic formula for the iteration, namely\n            <jats:italic>\n              x\n              <jats:sub>n<\/jats:sub>\n            <\/jats:italic>\n            = cot (2\n            <jats:sup>\n              <jats:italic>n<\/jats:italic>\n            <\/jats:sup>\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>0<\/jats:sub>\n            ), which is valid in exact arithmetic. In this paper, we extend this to to\n            <jats:italic>\n              k\n              <jats:sup>th<\/jats:sup>\n            <\/jats:italic>\n            order Householder methods, which include Halley's method, and to the secant method. Two formulae,\n            <jats:italic>\n              x\n              <jats:sub>n<\/jats:sub>\n            <\/jats:italic>\n            = cot (\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -1\n            <\/jats:sub>\n            +\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -2\n            <\/jats:sub>\n            ) with\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -1\n            <\/jats:sub>\n            = arccot (\n            <jats:italic>x<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -1\n            <\/jats:sub>\n            ) and\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -2\n            <\/jats:sub>\n            = arccot (\n            <jats:italic>x<\/jats:italic>\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n              -2\n            <\/jats:sub>\n            ), and\n            <jats:italic>\n              x\n              <jats:sub>n<\/jats:sub>\n            <\/jats:italic>\n            = cot ((\n            <jats:italic>k<\/jats:italic>\n            + 1)\n            <jats:sup>\n              <jats:italic>n<\/jats:italic>\n            <\/jats:sup>\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>0<\/jats:sub>\n            ) with\n            <jats:italic>\u03b8<\/jats:italic>\n            <jats:sub>0<\/jats:sub>\n            = arccot(\n            <jats:italic>x<\/jats:italic>\n            <jats:sub>0<\/jats:sub>\n            ), are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schr\u00f6der iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's\n            <jats:italic>Fractals[Newton]<\/jats:italic>\n            package to visualize general one-step iterations by disguising them as Newton iterations.\n          <\/jats:p>","DOI":"10.1145\/3363520.3363521","type":"journal-article","created":{"date-parts":[[2019,9,18]],"date-time":"2019-09-18T18:35:06Z","timestamp":1568831706000},"page":"1-22","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Revisiting Gilbert Strang's \"a chaotic search for\n            <i>i<\/i>\n            \""],"prefix":"10.1145","volume":"53","author":[{"given":"Ao","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert M.","family":"Corless","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2019,9,18]]},"reference":[{"key":"e_1_2_1_1_1","volume-title":"Handbook of mathematical functions: with formulas, graphs, and mathematical tables","author":"Abramowitz M.","year":"1967","unstructured":"Abramowitz , M. and Stegun , I. 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(2018) Inverse, shifted inverse, and Rayleigh quotient iteration as Newton's method, SIAM Review, <b>60<\/b>(1), 3--55."}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3363520.3363521","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3363520.3363521","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:44:25Z","timestamp":1750203865000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3363520.3363521"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,18]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,9,18]]}},"alternative-id":["10.1145\/3363520.3363521"],"URL":"https:\/\/doi.org\/10.1145\/3363520.3363521","relation":{},"ISSN":["1932-2240"],"issn-type":[{"type":"print","value":"1932-2240"}],"subject":[],"published":{"date-parts":[[2019,9,18]]},"assertion":[{"value":"2019-09-18","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}