{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:02Z","timestamp":1787337242405,"version":"build-2736575974"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>\n                    In this paper, we are interested in the iterative solution of ill-conditioned Toeplitz systems generated by continuous nonnegative real-valued functions f with a finite number of zeros. We construct new w-circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f * K\n                    <jats:sub>N<\/jats:sub>\n                    with a suitable positive reproducing kernel K\n                    <jats:sub>N<\/jats:sub>\n                    . By the restriction to positive kernels we obtain positive definite preconditioners. Moreover, if f has only zeros of even order $\\le 2s$, then we can prove that the property $ \\int_{-\\pi}^{\\pi} t^{2k} K_N (t) \\, \\mbox{d} t \\le C N^{-2k} $ $(k=0,\\hspace*{1.5pt}\\ldots,s)$ of the kernel is necessary and sufficient to ensure the convergence of the PCG method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were confirmed by numerical tests.\n                  <\/jats:p>","DOI":"10.1137\/s1064827599351428","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1741-1761","source":"Crossref","is-referenced-by-count":17,"title":["Preconditioners for Ill-Conditioned Toeplitz Systems Constructed from Positive Kernels"],"prefix":"10.1137","volume":"22","author":[{"given":"Daniel","family":"Potts","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gabriele","family":"Steidl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":"O. 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Ng,\n                      Circulant preconditioners for ill\u2010conditioned Hermitian Toeplitz matrices\n                      , in Proceedings of the International Congress Chin. Math., Beijing, 1998, Chinese University, Hong Kong, 1998, pp. 165\u2013174."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999354083"},{"key":"R18","unstructured":"R. H. Chan, A. M. Yip, and M. K. Ng,\n                      The Best Circulant Preconditioners for Hermitian Toeplitz Systems II: The Multiple\u2010Zero Case\n                      , Chinese University, Hong Kong, 1999, preprint."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0909051"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01235-7"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"R. A. 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