{"id":317,"date":"2017-08-17T18:37:01","date_gmt":"2017-08-17T09:37:01","guid":{"rendered":"http:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/?p=317"},"modified":"2017-08-17T18:37:01","modified_gmt":"2017-08-17T09:37:01","slug":"%e3%80%8c%e6%9c%80%e9%81%a9%e3%83%a2%e3%83%87%e3%83%aa%e3%83%b3%e3%82%b0%e3%80%8d%e3%82%bb%e3%83%9f%e3%83%8a%e3%83%bc%e6%a1%88%e5%86%85-920-2","status":"publish","type":"post","link":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/2017\/08\/17\/%e3%80%8c%e6%9c%80%e9%81%a9%e3%83%a2%e3%83%87%e3%83%aa%e3%83%b3%e3%82%b0%e3%80%8d%e3%82%bb%e3%83%9f%e3%83%8a%e3%83%bc%e6%a1%88%e5%86%85-920-2\/","title":{"rendered":"\u300c\u6700\u9069\u30e2\u30c7\u30ea\u30f3\u30b0\u300d\u30bb\u30df\u30ca\u30fc\u6848\u5185 (9\/20)"},"content":{"rendered":"<p>\u65e5\u6642\uff1a\u30002017\u5e749\u670820\u65e5(\u6c34) 14:30\uff5e15:30<br \/>\n\u5834\u6240\uff1a\u3000\u6771\u4eac\u5927\u5b66\u5de5\u5b66\u90e8 14\u53f7\u9928 5\u968e 534\u53f7\u5ba4<br \/>\n<a href=\"http:\/\/www.u-tokyo.ac.jp\/campusmap\/cam01_04_15_j.html\" target=\"_blank\" rel=\"noopener noreferrer\">http:\/\/www.u-tokyo.ac.jp\/campusmap\/cam01_04_15_j.html<\/a><\/p>\n<p>\u8b1b\u6f14\u8005\uff1a \u4e2d\u52d9 \u4f51\u6cbb (University of Oxford)<\/p>\n<p>\u8b1b\u6f14\u984c\u76ee: Best and near-best rational approximation<br \/>\nvia adaptive barycentric\u00a0 representation<\/p>\n<p>\u8b1b\u6f14\u6982\u8981\uff1a<br \/>\nRational approximation can outperform polynomial approximation by a<br \/>\nlandslide when there are singularities in or near the domain of<br \/>\napproximation. However, its use has been limited relative to<br \/>\npolynomials, primarily due to ill-conditioning (in addition to spurious<br \/>\npoles). In this work we show that the conditioning, and the overall<br \/>\nutility of rational functions, can be improved dramatically by an<br \/>\nadaptive barycentric representation for rational functions, wherein one<br \/>\nchooses a basis depending not only on the domain but also the function<br \/>\nto be approximated.<\/p>\n<p>We first introduce a new algorithm, called AAA (triple A, standing for<br \/>\n&#8220;adaptive Antoulas-Anderson&#8221;) for rational approximation on a real or<br \/>\ncomplex set of points. Even on a disk or interval the algorithm may<br \/>\noutperform existing methods, and on more complicated domains it is<br \/>\nespecially competitive. The core ideas are (1) representation of the<br \/>\nrational approximant in barycentric form with interpolation at certain<br \/>\nsupport points and (2) greedy selection of the support points (which<br \/>\ndetermines the basis) to avoid exponential instabilities.<\/p>\n<p>We next consider computing rational minimax approximations (best<br \/>\nrational approximant) on an real interval. We show that far more robust<br \/>\nalgorithms than previously available can be developed using the<br \/>\nbarycentric representation. Our improved rational Remez algorithm<br \/>\ncalculates approximations up to type (80,80) of |x| on [\uff0d1,1] in<br \/>\nstandard 16-digit floating point arithmetic, a problem for which Varga,<br \/>\nRuttan, and Carpenter required 200-digit extended precision.<\/p>\n<p>Based on joint work with S. Filip, O. Sete, and L. N. Trefethen (Oxford)<br \/>\nand B. Beckermann (Lille).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u65e5\u6642\uff1a\u30002017\u5e749\u670820\u65e5(\u6c34) 14:30\uff5e15:30 \u5834\u6240\uff1a\u3000\u6771\u4eac\u5927\u5b66\u5de5\u5b66\u90e8 14\u53f7\u9928 5\u968e &hellip; <\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-317","post","type-post","status-publish","format-standard","hentry","category-3"],"_links":{"self":[{"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/posts\/317","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/comments?post=317"}],"version-history":[{"count":0,"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/posts\/317\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/media?parent=317"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/categories?post=317"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.opt.mist.i.u-tokyo.ac.jp\/crest-model\/wp-json\/wp\/v2\/tags?post=317"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}