{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:23:34Z","timestamp":1776785014970,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,12,1]],"date-time":"2006-12-01T00:00:00Z","timestamp":1164931200000},"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                    We compute the Newton step for the characteristic polynomial and for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix as the reciprocal of the trace of an appropriate matrix. We show that, after the Yule\u2013Walker equations are solved, this trace can be computed in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal O}(n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    additional arithmetic operations, which is in contrast to existing methods, which rely on a recursion, requiring\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis n squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal O}(n^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    additional arithmetic operations.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01796-5","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"817-832","source":"Crossref","is-referenced-by-count":2,"title":["Computation of the Newton step for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix"],"prefix":"10.1090","volume":"75","author":[{"given":"A.","family":"Melman","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,12,1]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BFb0072474","article-title":"The generalized Schur algorithm for the superfast solution of Toeplitz systems","author":"Ammar, Gregory S.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/3540172122"},{"key":"2","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1016\/0024-3795(89)90701-5","article-title":"Numerical experience with a superfast real Toeplitz solver","volume":"121","author":"Ammar, Gregory S.","year":"1989","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/0024-3795(76)90101-4","article-title":"Eigenvalues and eigenvectors of symmetric centrosymmetric matrices","volume":"13","author":"Cantoni, A.","year":"1976","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1137\/0907009","article-title":"Computing the minimum eigenvalue of a symmetric positive definite Toeplitz matrix","volume":"7","author":"Cybenko, George","year":"1986","journal-title":"SIAM J. 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