{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:43:12Z","timestamp":1787330592553,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1703719"],"award-info":[{"award-number":["DMS-1703719"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these rational functions obey a recursion, allowing one to iteratively generate optimal rational approximants of $\\sqrt{z}$ of high degree using compositions and products of low-degree rational functions. The corresponding iterations for the matrix square root converge to $A^{1\/2}$ for any input matrix $A$ having no nonpositive real eigenvalues. In special limiting cases, these iterations reduce to known iterations for the matrix square root: the lowest-order version is an optimally scaled Newton iteration, and for certain parameter choices, the principal family of Pad\u00e9 iterations is recovered. Theoretical results and numerical experiments indicate that the iterations perform especially well on matrices having eigenvalues with widely varying magnitudes.<\/jats:p>","DOI":"10.1137\/18m1178529","type":"journal-article","created":{"date-parts":[[2019,6,4]],"date-time":"2019-06-04T14:36:04Z","timestamp":1559658964000},"page":"696-719","source":"Crossref","is-referenced-by-count":9,"title":["Zolotarev Iterations for the Matrix Square Root"],"prefix":"10.1137","volume":"40","author":[{"given":"Evan S.","family":"Gawlik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,6,4]]},"reference":[{"key":"atypb1","unstructured":"N. I. Akhiezer,\n                      Theory of Approximation\n                      , Frederick Ungar, New York, 1956."},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"N. I. Akhiezer,\n                      Elements of the Theory of Elliptic Functions\n                      , Transl. Math. Managr. 79, AMS, Providence, RI, 1990.","DOI":"10.1090\/mmono\/079"},{"key":"atypb3","unstructured":"B. Beckermann,\n                      Optimally scaled Newton iterations for the matrix square root\n                      , presented at Advances in Matrix Functions and Matrix Equations Workshop, Manchester, UK, 2013."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/16M1096426"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"D. Braess,\n                      Nonlinear Approximation Theory\n                      , Springer Ser. Comput. Math., Springer, New York, 1986.","DOI":"10.1007\/978-3-642-61609-9"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/070699895"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(76)90020-5"},{"key":"atypb9","unstructured":"E. S. Gawlik,\n                      Rational Minimax Iterations for Computing the Matrix $p$th Root\n                      , preprint,arXiv:1903.06268, 2019."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/18M1182747"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/140980090"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/070700607"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0829624-5"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019150005407"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"N. J. Higham,\n                      Functions of Matrices: Theory and Computation\n                      , SIAM, Philadelphia, 2008.","DOI":"10.1137\/1.9780898717778"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479804442218"},{"key":"atypb18","unstructured":"L. Hogben,\n                      Handbook of Linear Algebra\n                      , CRC Press, Boca Raton, FL, 2016."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-003-0079-9"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/16M1087278"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999360688"},{"key":"atypb22","unstructured":"Y. Li and H. Yang,\n                      Spectrum Slicing for Sparse Hermitian Definite Matrices Based on Zolotarev's Functions\n                      , preprint,arXiv:1701.08935, 2017."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803426656"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/140990334"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1970-0273809-4"},{"key":"atypb26","unstructured":"J. Nocedal and S. Wright,\n                      Numerical Optimization\n                      , Springer, New York, 2006."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01295091"},{"key":"atypb28","first-page":"207","author":"Todd J.","year":"1984","journal-title":"New York"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/0516015"},{"key":"atypb30","unstructured":"E. Wachspress,\n                      Positive Definite Square Root of a Positive Definite Square Matrix\n                      , unpublished manuscript, 1962."},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"E. Wachspress,\n                      The ADI Model Problem\n                      , Springer, New York, 2013.","DOI":"10.1007\/978-1-4614-5122-8"},{"key":"atypb32","first-page":"1","volume":"30","author":"Zolotarev E. I.","year":"1877","journal-title":"Zap. S.-Petersburg Akad. Nauk."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1178529","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:14:33Z","timestamp":1787328873000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1178529"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":30,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1178529"],"URL":"https:\/\/doi.org\/10.1137\/18m1178529","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}