{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:29:48Z","timestamp":1787336988689,"version":"3.56.0"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>\n                    It is well known that preconditioned conjugate gradient (PCG) methods are widely used to solve ill-conditioned Toeplitz linear systems T\n                    <jats:sub>n<\/jats:sub>\n                    (f)x=b. In this paper we present a new preconditioning technique for the solution of symmetric Toeplitz systems generated by nonnegative functions f with zeros of even order. More specifically, f is divided by the appropriate trigonometric polynomial g of the smallest degree, with zeros the zeros of f to eliminate its zeros. Using rational approximation we approximate $\\sqrt{f\/g}$ by $\\frac{p}{q}$, $p,q$ trigonometric polynomials and consider $\\frac{p^2g}{q^2}$ as a very satisfactory approximation of f. We propose the matrix $M_n=B^{-1}_n(q)B_n(p^2g)B^{-1}_n(q)$, where $B(\\cdot)$ denotes the associated band Toeplitz matrix, as a preconditioner whence a good clustering of the spectrum of its preconditioned matrix is obtained. We also show that the proposed technique can be very flexible, a fact that is confirmed by various numerical experiments so that in many cases it constitutes a much more efficient strategy than the existing ones.\n                  <\/jats:p>","DOI":"10.1137\/s0895479800376314","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"728-743","source":"Crossref","is-referenced-by-count":19,"title":["New Band Toeplitz Preconditioners for Ill-Conditioned Symmetric Positive Definite Toeplitz Systems"],"prefix":"10.1137","volume":"23","author":[{"given":"D.","family":"Noutsos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.","family":"Vassalos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389448"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"D. Bini and F. Di Benedetto,\n                      A new preconditioner for parallel solution of positive definite Toeplitz systems\n                      , in Proceedings of the Second Annual Symposium on Parallel Algorithms and Architectures, Crete, Greece, 1990, pp. 220\u2013223.","DOI":"10.1145\/97444.97688"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0614035"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897324585"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1426-7"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0906025"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/11.3.333"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90226-E"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0915011"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0916041"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(93)90297-9"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050411"},{"key":"R13","volume-title":"Toeplitz forms and their applications","author":"Grenander Ulf","year":"1984"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01935646"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0913083"},{"key":"R16","volume-title":"Approximation of functions","author":"Lorentz G.","year":"1986"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171502"},{"key":"R18","volume-title":"An introduction to the approximation of functions","author":"Rivlin Theodore","year":"1981"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02575833"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01740550"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00833-8"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897316904"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050413"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1986742171"},{"key":"R25","unstructured":"H. Widom,\n                      Toeplitz matrices\n                      , in Studies in Real and Complex Analysis, Stud. Math. 3, Math. Assoc. Amer., Buffalo, NY, 1965, pp. 179\u2013209."},{"key":"R26","volume-title":"The algebraic eigenvalue problem","author":"Wilkinson J.","year":"1965"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479800376314","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:39:39Z","timestamp":1787333979000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479800376314"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S0895479800376314"],"URL":"https:\/\/doi.org\/10.1137\/s0895479800376314","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}