{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T19:52:34Z","timestamp":1791229954010,"version":"4.1.0"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>New methods are proposed for the numerical evaluation of $f(\\mathbf{A})$ or $f(\\mathbf{A}) b$, where $f(\\mathbf{A})$ is a function such as $\\mathbf{A}^{1\/2}$ or $\\log (\\mathbf{A})$ with singularities in $(-\\infty,0]$ and $\\mathbf{A}$ is a matrix with eigenvalues on or near $(0,\\infty)$. The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of $f(\\mathbf{A})b$ is typically reduced to one or two dozen linear system solves, which can be carried out in parallel.<\/jats:p>","DOI":"10.1137\/070700607","type":"journal-article","created":{"date-parts":[[2008,6,11]],"date-time":"2008-06-11T18:02:42Z","timestamp":1213207362000},"page":"2505-2523","source":"Crossref","is-referenced-by-count":155,"title":["Computing $A^\\alpha, \\log(A)$, and Related Matrix Functions by Contour Integrals"],"prefix":"10.1137","volume":"46","author":[{"given":"Nicholas","family":"Hale","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicholas J.","family":"Higham","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lloyd N.","family":"Trefethen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,6,11]]},"reference":[{"key":"R1","unstructured":"M. Abramowitz and I. A. Stegun,\n                      Handbook of Mathematical Functions\n                      , Dover, New York, 1972."},{"key":"R2","unstructured":"N. I. Achieser,\n                      Theory of Approximation\n                      , Frederick Ungar, New York, 1956."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022362401426"},{"key":"R4","unstructured":"P. I. Davies and N. J. Higham,\n                      \n                        Computing $f(A)b$ for matrix functions\n                        f\n                      \n                      , in QCD and Numerical Analysis III, A. Bori\u00e7i, A. Frommer, B. Jo\u00f3, A. Kennedy, and B. Pendleton, eds., Lect. Notes Computat. Sci. Engrg. 47, Springer-Verlag, Berlin, 2005, pp. 15\u201324."},{"key":"R5","unstructured":"P. J. Davis,\n                      On the numerical integration of periodic analytic functions\n                      , in On Numerical Integration: Proceedings of a Symposium, R. E. Langer, ed., Madison, WI, 1958, Math. Res. Ctr., U. of Wisconsin, Madison, WI, 1959, pp. 45\u201359."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"P. J. Davis and P. Rabinowitz,\n                      Methods of Numerical Integration\n                      , 2nd ed., Academic Press, Orlando, FL, 1984.","DOI":"10.1016\/B978-0-12-206360-2.50012-1"},{"key":"R7","unstructured":"T. A. Driscoll,\n                      The Schwarz\u2013Christoffel toolbox\n                      , available online at http:\/\/www.math. udel.edu\/~driscoll\/software\/SC\/."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1145\/1067967.1067971"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"T. A. Driscoll and L. N. Trefethen,\n                      Schwarz\u2013Christoffel Mapping\n                      , Cambridge University Press, Cambridge, UK, 2002.","DOI":"10.1017\/CBO9780511546808"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.2478\/cmam-2001-0022"},{"key":"R11","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":"R12","first-page":"3","volume":"17","author":"Iri M.","year":"1987","journal-title":"Kokyuroku Res. Inst. Math. Sci., Kyoto Univ., 91 (1970), pp. 82\u2013118 (in Japanese, translated into English in J. Comput. Appl. Math."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0920-5632(03)02466-6"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1109\/9.402226"},{"key":"R15","first-page":"T83","volume":"48","author":"Martensen E.","year":"1968","journal-title":"Z. Angew. Math. Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-2267","issn-type":"print"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/24.3.439"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.2977\/prims\/1145474600"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1028999029"},{"key":"R19","first-page":"571","volume":"4","author":"Poisson S.-D.","year":"1827","journal-title":"M\u00e9moires de L'Acad\u00e9mie Royale des Sciences de L'Institut de France"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1964-0165689-X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/050646342"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(69)90037-0"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01098-4"},{"key":"R24","unstructured":"K.G. Steffens,\n                      The History of Approximation Theory: From Euler to Bernstein\n                      , Birkh\u00e4user Boston, Boston, 2006."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/12.1.103"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/1023037"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"F. Stenger,\n                      Numerical Methods Based on Sinc and Analytic Functions\n                      , Springer-Verlag, New York, 1993.","DOI":"10.1007\/978-1-4612-2706-9"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436624"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.2977\/prims\/1195192451"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/23.1.97"},{"key":"R31","doi-asserted-by":"crossref","unstructured":"N. M. Temme,\n                      Special Functions\n                      , John Wiley and Sons, New York, 1996.","DOI":"10.1002\/9781118032572"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/0720030"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-006-0077-9"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4655(02)00455-1"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/050625837"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-01945-X"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070700607","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:23Z","timestamp":1787325263000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070700607"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070700607"],"URL":"https:\/\/doi.org\/10.1137\/070700607","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}