{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T08:44:56Z","timestamp":1791189896753,"version":"4.1.0"},"reference-count":16,"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":[[2003,1]]},"abstract":"<jats:p>\n                    This paper describes a new O(n log\n                    <jats:sup>3<\/jats:sup>\n                    (n)) solver for the positive definite Toeplitz system Tx=b. Instead of computing generators for the inverse of T, the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The generators of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well-known numerical instability inherent in explicit inversion. Experiments suggest that the algorithm is backward stable in most cases.\n                  <\/jats:p>","DOI":"10.1137\/s089547980241791x","type":"journal-article","created":{"date-parts":[[2003,12,22]],"date-time":"2003-12-22T21:00:49Z","timestamp":1072126849000},"page":"669-693","source":"Crossref","is-referenced-by-count":36,"title":["A Superfast Toeplitz Solver with Improved Numerical Stability"],"prefix":"10.1137","volume":"25","author":[{"given":"Michael","family":"Stewart","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","unstructured":"Gregory Ammar, William Gragg, The implementation and use of the generalized Schur algorithm, North\u2010Holland, Amsterdam, 1986, 265\u201327989b:65112"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0609005"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90161-5"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479891221563"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(80)90013-9"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479895287419"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90107-8"},{"key":"R8","unstructured":"S. Haykin,\n                      Adaptive Filter Theory\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1991."},{"key":"R9","volume-title":"Accuracy and stability of numerical algorithms","author":"Higham Nicholas","year":"1996"},{"key":"R10","unstructured":"B. R. Musicus,\n                      Levinson and Fast Choleski Algorithms for Toeplitz and Almost Toeplitz Matrices\n                      , Research report, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, 1984."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0609022"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1980-0572859-4"},{"key":"R13","doi-asserted-by":"publisher","unstructured":"Michael Stewart, Cholesky factorization of semidefinite Toeplitz matrices, Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), Vol. 254, 1997, 497\u201352510.1016\/S0024-3795(96)00517-497m:65059","DOI":"10.1016\/S0024-3795(96)00517-4"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897319581"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S089547989528692X"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899362302"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S089547980241791X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:37:56Z","timestamp":1787333876000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S089547980241791X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S089547980241791X"],"URL":"https:\/\/doi.org\/10.1137\/s089547980241791x","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}