{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T00:47:41Z","timestamp":1791938861488,"version":"4.3.4"},"reference-count":39,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>An algorithm for computing matrix functions is presented. It employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of the nontrivial diagonal blocks evaluated via a Taylor series. A parameter is used to balance the conflicting requirements of producing small diagonal blocks and keeping the separations of the blocks large. The algorithm is intended primarily for functions having a Taylor series with an infinite radius of convergence, but it can be adapted for certain other functions, such as the logarithm. Novel features introduced here include a convergence test that avoids premature termination of the Taylor series evaluation and an algorithm for reordering and blocking the Schur form. Numerical experiments show that the algorithm is competitive with existing special-purpose algorithms for the matrix exponential, logarithm, and cosine. Nevertheless, the algorithm can be numerically unstable with the default choice of its blocking parameter (or in certain cases for all choices), and we explain why determining the optimal parameter appears to be a very difficult problem. A MATLAB implementation is available that is much more reliable than the function funm in MATLAB 6.5 (R13).<\/jats:p>","DOI":"10.1137\/s0895479802410815","type":"journal-article","created":{"date-parts":[[2003,9,19]],"date-time":"2003-09-19T21:40:15Z","timestamp":1064007615000},"page":"464-485","source":"Crossref","is-referenced-by-count":127,"title":["A Schur-Parlett Algorithm for Computing Matrix Functions"],"prefix":"10.1137","volume":"25","author":[{"given":"Philip I.","family":"Davies","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicholas J.","family":"Higham","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen,\n                      LAPACK Users\u2019 Guide\n                      , 3rd ed., SIAM, Philadelphia, PA, 1999.","DOI":"10.1137\/1.9780898719604"},{"key":"R2","unstructured":"K. J. \u00c5str\u00f6m and B. Wittenmark,\n                      Computer\u2010Controlled Systems: Theory and Design\n                      , 3rd ed., Prentice\u2013Hall, Englewood Cliffs, NJ, 1997."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90286-W"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0716028"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(83)90010-1"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0079-6565(90)80015-A"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1984.1103389"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899364015"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.6995"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894273614"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00042-2"},{"key":"R12","unstructured":"F. R. Gantmacher,\n                      The Theory of Matrices\n                      , Vol. 1, Chelsea, New York, 1959."},{"key":"R13","volume-title":"Matrix computations","author":"Golub Gene","year":"1996"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1006\/jcom.1995.1019"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.91.17.7962"},{"key":"R16","unstructured":"N. J. 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K\u00e5gstr\u00f6m,\n                      Numerical computation of matrix functions\n                      , Report UMINF\u201058.77, Department of Information Processing, University of Ume\u00e5, Sweden, 1977."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0613009"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1145\/355900.355912"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0610014"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"P.\u2010F. Lavall\u00e9e, A. Malyshev, M. Sadkane, Spectral portrait of matrices by block diagonalization, Lecture Notes in Comput. Sci., Vol. 1196, Springer, Berlin, 1997, 266\u20132731475974","DOI":"10.1007\/3-540-62598-4_103"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/0614070"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/S00361445024180"},{"key":"R31","unstructured":"K. C. 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Ng,\n                      Development of an accurate algorithm for exp(Bt)\n                      , Technical report PAM\u2010294, Center for Pure and Applied Mathematics, University of California, Berkeley, CA, 1985."},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/0901013"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801392697"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/0714065"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/0716016"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1137\/0714039"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479802410815","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:38:05Z","timestamp":1787333885000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479802410815"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":39,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479802410815"],"URL":"https:\/\/doi.org\/10.1137\/s0895479802410815","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}