{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:36:25Z","timestamp":1776728185295,"version":"3.51.2"},"reference-count":5,"publisher":"American Mathematical Society (AMS)","issue":"235","license":[{"start":{"date-parts":[[2001,3,6]],"date-time":"2001-03-06T00:00:00Z","timestamp":983836800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a quadratic polynomial over a splitting field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be the set of zeros of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We define an associative and commutative binary relation on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G identical-to upper K union StartSet normal infinity EndSet minus upper S\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo>\n                              \u2261\n                              \n                            <\/mml:mo>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mo>\n                              \u222a\n                              \n                            <\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G\\equiv K\\cup \\{\\infty \\}-S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    so that every M\u00f6bius transformation with fixed point set\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u201cplus\u201d\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for some\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton\u2019s method, the secant method, Halley\u2019s method, and higher order methods. If\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a M\u00f6bius transformation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are distinct and the iterates of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    converge, then Newton\u2019s method converges with order 2, and higher order generalizations converge accordingly.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01242-4","type":"journal-article","created":{"date-parts":[[2002,11,6]],"date-time":"2002-11-06T14:02:22Z","timestamp":1036591342000},"page":"1305-1310","source":"Crossref","is-referenced-by-count":2,"title":["On iterates of M\u00f6bius transformations on fields"],"prefix":"10.1090","volume":"70","author":[{"given":"Sam","family":"Northshield","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,3,6]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/00150517.1986.12429787","article-title":"Newton\u2019s method and simple continued fractions","volume":"24","author":"Filaseta, Michael","year":"1986","journal-title":"Fibonacci Quart.","ISSN":"https:\/\/id.crossref.org\/issn\/0015-0517","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00150517.1981.12430114","article-title":"Newton\u2019s method and ratios of Fibonacci numbers","volume":"19","author":"Gill, John","year":"1981","journal-title":"Fibonacci Quart.","ISSN":"https:\/\/id.crossref.org\/issn\/0015-0517","issn-type":"print"},{"issue":"172","key":"3","doi-asserted-by":"publisher","first-page":"553","DOI":"10.2307\/2008145","article-title":"Aitken sequences and generalized Fibonacci numbers","volume":"45","author":"McCabe, J. H.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"203","key":"4","doi-asserted-by":"publisher","first-page":"365","DOI":"10.2307\/2152961","article-title":"Generalized Fibonacci and Lucas sequences and rootfinding methods","volume":"61","author":"Muskat, Joseph B.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"354","DOI":"10.2307\/2322139","article-title":"Aitken sequences and Fibonacci numbers","volume":"91","author":"Phillips, G. M.","year":"1984","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-235\/S0025-5718-00-01242-4\/S0025-5718-00-01242-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-235\/S0025-5718-00-01242-4\/S0025-5718-00-01242-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:42:16Z","timestamp":1776724936000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-235\/S0025-5718-00-01242-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3,6]]},"references-count":5,"journal-issue":{"issue":"235","published-print":{"date-parts":[[2001,7]]}},"alternative-id":["S0025-5718-00-01242-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01242-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2000,3,6]]}}}