{"id":155,"date":"2014-11-13T09:16:28","date_gmt":"2014-11-13T08:16:28","guid":{"rendered":"http:\/\/www.ens-lyon.fr\/LIP\/AriC\/?page_id=155"},"modified":"2014-11-27T12:27:21","modified_gmt":"2014-11-27T11:27:21","slug":"asna","status":"publish","type":"page","link":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/asna","title":{"rendered":"Approximations: from symbolic to numerical computation, and applications"},"content":{"rendered":"<p><a href=\"http:\/\/perso.ens-lyon.fr\/stephan.thomasse\/cours\/MASTER2.html\" target=\"_blank\">Master d&#8217;informatique fondamentale<\/a> of <a href=\"http:\/\/www.ens-lyon.eu\" target=\"_blank\">\u00c9cole Normale Sup\u00e9rieure de Lyon<\/a>, Fall-Winter 2015.<\/p>\n<p>This course covers part of approximation theory from the point of view of effective computation. Computation of polynomial or rational approximants, efficient computation of Taylor series expansions or series expansions based on families of orthogonal polynomial, use of these expansions to produce approximations to a prescribed accuracy (Remez&#8217; algorithm, Taylor and Chebyshev models). The power of these techniques is illustrated with three applications from different scientific domains: irrationality proofs in number theory, efficient evaluation of numerical functions, computational issues related to Near-Earth Objects.<\/p>\n<table border=\"0\">\n<tbody>\n<tr align=\"left\"><!--\n\n\n<td><a href=\"#transparents\">Transparents<\/a>   --><\/p>\n<td><a href=\"#lecturenotes\">Lecture notes of the course<\/a><\/td>\n<td><a href=\"#organisation\">Organisation<\/a><\/td>\n<td><a href=\"#evaluation\">Evaluation<\/a><\/td>\n<td><a href=\"#references\">References<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>Informations<\/h3>\n<ul>\n<li>Next meeting: ?? ??:00 ??m, room ?? (4th floor of the GN1 building)<\/li>\n<li>The schedule will be the following: lectures on the ??<br \/>\n<!-- 9\/17,\n9\/24, 10\/3, 10\/8, 10\/15, 10\/22, 11\/5, 11\/12, 11\/19, 11\/26,\n12\/3, 12\/17.  Take-home test to turn in on the 11\/21. Exam on the 11\/28. --><\/li>\n<\/ul>\n<hr \/>\n<h3><a name=\"lecturenotes\"><\/a>Lecture notes of the course<\/h3>\n<p>You can have a look at a draft version of the <a href=\"http:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-content\/uploads\/2014\/11\/asna-notes.pdf\">lecture notes<\/a>.<\/p>\n<ul>\n<\/ul>\n<ul><!--\n\t\n\n<li><a href=\".\/2013\/chap1.pdf\" target=\"_blank\"><\/a>Class n.1, 9\/17\/2013, pages 3-10 of the lecture notes\n\t\n\n<li><a href=\".\/2013\/cours2.pdf\" target=\"_blank\"><\/a>Class n.2, 9\/24\/2013, pages 11-16 of the lecture notes (to be completed!)<\/li>\n\n\n\t\n\n<li><a href=\".\/2013\/cours3.pdf\" target=\"_blank\"><\/a>Class n.3, 10\/3\/2013, pages 17-22 of the lecture notes. Note by NB: I apologize for being late. Here are the <a href=\".\/2013\/slides2-3.pdf\" target=\"_blank\">slides<\/a> of classes 2 and 3 and the Maple <a href=\".\/2013\/cours3.mw\" target=\"_blank\">session<\/a> (Remez' algorithm and Runge's phenomenon)<\/li>\n\n\n\t\n\n<li><a href=\".\/2013\/chap3.pdf\"><\/a>Class n.4, 10\/08\/2013, pages 23-31 of the lecture notes<\/li>\n\n\n\t\n\n<li><a href=\".\/2013\/chap4.pdf\"><\/a>Class n.5, 10\/15\/2013, pages 33-42 of the lecture notes<\/li>\n\n\n\t\n\n<li><a href=\".\/2013\/chap5.pdf\"><\/a>Class n.6, 10\/22\/2013, pages 43-58 of the lecture notes<\/li>\n\n\n\t\n\n<li><a href=\".\/2013\/chap6.pdf\"><\/a>Class n.7, 11\/5\/2013, pages 59-66 of the lecture notes<\/li>\n\n\n--><\/ul>\n<p>&nbsp;<\/p>\n<ul><!--\n\t\n\n<li>Class n.1, 9\/17\/2013, pages 3-10 of the lecture notes\n\t\n\n<li>Class n.2, 9\/24\/2013, pages 11-16 of the lecture notes (to be completed!)<\/li>\n\n\n\t\n\n<li>Class n.3, 10\/3\/2013, pages 17-22 of the lecture notes. Here are the <a href=\".\/2013\/slides2-3.pdf\" target=\"_blank\">slides<\/a> of classes 2 and 3 and the Maple <a href=\".\/2013\/cours3.mw\" target=\"_blank\">session<\/a> (Remez' algorithm and Runge's phenomenon)<\/li>\n\n\n\t\n\n<li>Class n.4, 10\/08\/2013, pages 23-31 of the lecture notes<\/li>\n\n\n\t\n\n<li>Class n.5, 10\/15\/2013, pages 33-42 of the lecture notes<\/li>\n\n\n\t\n\n<li>Class n.6, 10\/22\/2013, pages 43-58 of the lecture notes<\/li>\n\n\n\t\n\n<li>Class n.7, 11\/5\/2013, pages 59-65 of the lecture notes<\/li>\n\n\n\t\n\n<li>Class n.8, 11\/12\/2013, pages 65-71 of the lecture notes<\/li>\n\n\n\t\n\n<li><a href=\".\/2013\/\" target=\"_blank\"><\/a>Class n.9, 11\/19\/2013, lecture notes to be completed. Here are the corresponding <a href=\".\/2013\/slides9.pdf\" target=\"_blank\">slides<\/a>.<\/li>\n\n\n\t\n\n<li>Class n.10, 11\/26\/2013, pages 81-89 of the lecture notes<\/li>\n\n\n\t\n\n<li>Class n.11, 12\/3\/2013, lecture notes to be completed. Here are the corresponding <a href=\".\/2013\/slides11.pdf\" target=\"_blank\">slides<\/a><\/li>\n\n\n\t\n\n<li>Class n.12, 12\/17\/2013, lecture notes to be completed. Here are the corresponding <a href=\".\/2013\/slides12.pdf\" target=\"_blank\">slides<\/a><\/li>\n\n\n--><\/ul>\n<hr \/>\n<h3><a name=\"organisation\"><\/a>Organisation<\/h3>\n<ul>\n<li>Hours : ?? <!-- Tuesdays from 8:00 am to 10:00 am, <room B2 (4th floor of the main building of ENS Lyon, Monod part).<\/li>\n\n\n--><\/li>\n<li>Person in charge: <a href=\"http:\/\/perso.ens-lyon.fr\/nicolas.brisebarre\" target=\"_blank\">Nicolas Brisebarre<\/a>, Office n.175, <a href=\"http:\/\/www.ens-lyon.fr\/LIP\/AriC\/accessLUG.html\" target=\"_blank\">LUG<\/a>.<\/li>\n<li>Teachers: <a href=\"http:\/\/perso.ens-lyon.fr\/nicolas.brisebarre\" target=\"_blank\">Nicolas Brisebarre<\/a> and <a href=\"http:\/\/perso.ens-lyon.fr\/bruno.salvy\" target=\"_blank\">Bruno Salvy<\/a>.<\/li>\n<li>The lectures will be delivered in French or English on request.<\/li>\n<\/ul>\n<hr \/>\n<h3><a name=\"evaluation\"><\/a>Evaluation<\/h3>\n<ul><!--\n\t\n\n<li>Evaluation: there will be a <a href=\".\/2013\/practical_exam_asna_2013.pdf\" target=\"_blank\">take-home exam<\/a> (consisting of a practical session on a computer) to turn in on the 17th of January (before midnight) and an exam on Tuesday, the 7th of January, from 8am to 10am, room B2 (3rd floor of the GN1 building).\n\n--><\/p>\n<li>Evaluation: there will be a take-home exam (consisting of a practical session on a computer) to turn in and an exam.<\/li>\n<\/ul>\n<hr \/>\n<h3><a name=\"references\"><\/a>References<\/h3>\n<h4>Lecture notes<\/h4>\n<ul>\n<li><a href=\"http:\/\/perso.ens-lyon.fr\/bruno.salvy\/M1ENS\/poly.pdf\" target=\"_blank\">Math\u00e9matiques Exp\u00e9rimentales<\/a> (in French), lecture notes by A. Bostan and B. Salvy. <a href=\"http:\/\/perso.ens-lyon.fr\/bruno.salvy\/M1ENS\/\">ENS Paris first year of Master<\/a>.<\/li>\n<li><a href=\"http:\/\/perso.ens-lyon.fr\/bruno.salvy\/mpri\/poly.pdf\" target=\"_blank\">Algorithmes efficaces en calcul formel<\/a> (in French), lecture notes by A. Bostan, F. Chyzak, G.Lecerf and B. Salvy. <a href=\"https:\/\/wikimpri.dptinfo.ens-cachan.fr\/doku.php?id=cours:c-2-22\" target=\"_blank\">Parisian Master of Research in Computer Science<\/a>.<\/li>\n<\/ul>\n<h4>Books<\/h4>\n<ul>\n<li>E. B. Burger and R. Tubbs. Making Transcendence Transparent: An intuitive approach to classical transcendental number theory. Springer.<\/li>\n<li>J.-P. Demailly. Analyse num\u00e9rique et \u00e9quations diff\u00e9rentielles. EDP Sciences.<\/li>\n<li>J. von zur Gathen and J. Gerhard. Modern Computer Algebra, Cambridge University Press.<\/li>\n<li>J. C. Mason and D. C. Handscomb. Chebyshev Polynomials. Chapman &amp; Hall\/CRC.<\/li>\n<li>M. Schatzman. Analyse num\u00e9rique, une approche math\u00e9matique. Dunod. And in English: Numerical Analysis: A Mathematical Introduction. Oxford University Press.<\/li>\n<li>L. N. Trefethen. <a href=\"http:\/\/www2.maths.ox.ac.uk\/chebfun\/ATAP\/\" target=\"_blank\">Approximation Theory and Approximation Practice<\/a>.<\/li>\n<li>W. Tucker. Validated Numerics, a short introduction to rigorous computations. Princeton University Press.<\/li>\n<\/ul>\n<p>More advanced level.<\/p>\n<ul>\n<li>C. M. Bender and S. A. Orszag. Advanced mathematical methods for scientists and engineers, McGraw-Hill Book Co..<\/li>\n<li>D. Bini and V. Y. Pan. Polynomial and Matrix Computations, Volume 1: Fundamental Algorithms, Birkh\u00e4user.<\/li>\n<li>J. P. Boyd. <a href=\"http:\/\/www-personal.umich.edu\/~jpboyd\/BOOK_Spectral2000.html\" target=\"_blank\">Chebyshev and Fourier Spectral Methods<\/a>. Dover.<\/li>\n<li>E. W. Cheney. Introduction to Approximation Theory. AMS Chelsea Pub.<\/li>\n<li>A. Gil, J. Segura and N. M. Temme, Numerical methods for special functions, Society for Industrial and Applied Mathematics (SIAM).<\/li>\n<li>E. W. Kaucher et W. L. Miranker. Self-validating numerics for function space problems. Academic Press, 1984.<\/li>\n<li>R. Moore and M. J. Cloud. Computational functional analysis. Horwood Pub.<\/li>\n<li>M. Powell. Approximation theory and methods. Cambridge University Press.<\/li>\n<li>L. B. Rall. Computational Solution of Nonlinear Operator Equations. John Wiley &amp; Sons Ltd.<\/li>\n<\/ul>\n<h4>Master theses<\/h4>\n<h4>PhD theses<\/h4>\n<ul>\n<li>A. Benoit, <a href=\"http:\/\/hal.inria.fr\/pastel-00726487\/\" target=\"_blank\">Algorithmique semi-num\u00e9rique rapide des s\u00e9ries de Tchebychev<\/a>. Th\u00e8se de doctorat, \u00c9cole Polytechnique, juillet 2012<\/li>\n<li>S. Chevillard, <a href=\"http:\/\/www-sop.inria.fr\/members\/Sylvain.Chevillard\/download\/papers\/TheseSylvainChevillard.pdf\" target=\"_blank\">\u00c9valuation efficace de fonctions num\u00e9riques &#8211; Outils et exemples<\/a>. Th\u00e8se de doctorat, \u00c9cole Normale Sup\u00e9rieure de Lyon, septembre 2011<\/li>\n<li>P. Di Lizia, <a href=\"http:\/\/www.astrodynamics.eu\/Astrodynamics.eu\/PhD_Dissertations_files\/DiLiziaPhDThesis.pdf\" target=\"_blank\">Robust Space Trajectory and Space System Design using Differential Algebra<\/a>, Politecnico di Milano, 2008<\/li>\n<li>M. Jolde\u015f, <a href=\"http:\/\/perso.ens-lyon.fr\/mioara.joldes\/these\/theseJoldes.pdf\" target=\"_blank\">Rigorous Polynomial Approximations and Applications<\/a>. Th\u00e8se de doctorat, \u00c9cole Normale Sup\u00e9rieure de Lyon, septembre 2011<\/li>\n<li>M. Mezzarobba, <a href=\"http:\/\/www.marc.mezzarobba.net\/these\/these-mezzarobba.pdf\" target=\"_blank\">Autour de l&#8217;\u00e9valuation num\u00e9rique des fonctions D-finies<\/a>, Th\u00e8se de doctorat, \u00c9cole Polytechnique, octobre 2011<\/li>\n<\/ul>\n<h4>Softwares<\/h4>\n<ul>\n<li><a href=\"http:\/\/www2.maths.ox.ac.uk\/chebfun\/\" target=\"_blank\">Chebfun<\/a>, numerical computing with functions.<\/li>\n<li><a href=\"http:\/\/www.maplesoft.com\/\" target=\"_blank\">Maple<\/a><\/li>\n<li><a href=\"http:\/\/www.mathemagix.org\" target=\"_blank\">Mathemagix<\/a><\/li>\n<li><a href=\"http:\/\/pari.math.u-bordeaux.fr\/\" target=\"_blank\">Pari\/GP<\/a><\/li>\n<li><a href=\"http:\/\/www.sagemath.org\/\" target=\"_blank\">Sage<\/a>, a free open-source mathematics software. A nice presentation (in French) is available<br \/>\n<a href=\"http:\/\/sagebook.gforge.inria.fr\/\" target=\"_blank\">here<\/a>.<\/li>\n<li><a href=\"https:\/\/www.scilab.org\" target=\"_blank\">Scilab<\/a><\/li>\n<\/ul>\n<h4>Others<\/h4>\n<ul>\n<li><a href=\"http:\/\/www.math.technion.ac.il\/hat\/\" target=\"_blank\">History of Approximation Theory<\/a><\/li>\n<li><a href=\"http:\/\/www.math.technion.ac.il\/sat\/\" target=\"_blank\">Surveys in Approximation Theory<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Master d&#8217;informatique fondamentale of \u00c9cole Normale Sup\u00e9rieure de Lyon, Fall-Winter 2015. This course covers part of approximation theory from the point of view of effective computation. Computation of polynomial or rational approximants, efficient computation of Taylor series expansions or series expansions based on families of orthogonal polynomial, use of these expansions to produce approximations to &hellip; <a href=\"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/asna\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Approximations: from symbolic to numerical computation, and applications<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-155","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/pages\/155","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/comments?post=155"}],"version-history":[{"count":5,"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/pages\/155\/revisions"}],"predecessor-version":[{"id":184,"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/pages\/155\/revisions\/184"}],"wp:attachment":[{"href":"https:\/\/www.ens-lyon.fr\/LIP\/AriC\/wp-json\/wp\/v2\/media?parent=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}