{"id":279,"date":"2011-06-21T09:50:31","date_gmt":"2011-06-21T09:50:31","guid":{"rendered":"https:\/\/blogs.mathworks.com\/loren\/2011\/06\/21\/update-on-the-chebfun-project\/"},"modified":"2016-07-31T14:14:09","modified_gmt":"2016-07-31T19:14:09","slug":"update-on-the-chebfun-project","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/loren\/2011\/06\/21\/update-on-the-chebfun-project\/","title":{"rendered":"Update on the Chebfun Project"},"content":{"rendered":"<div class=\"content\">\n<p>A while ago, <a href=\"https:\/\/blogs.mathworks.com\/loren\/2010\/01\/12\/the-chebfun-project\/\">I wrote<\/a> about the Chebfun Project. At the time, version 3 had recently come out. I recently visited Nick Trefethen and Nick Hale, part of the Chebfun team, and they showed me the latest and greatest version 4.<\/p>\n<p>As I said in my earlier post, the package is designed to have syntax very much like regular vector notation in MATLAB, though<br \/>\nthe entities represent functions.<\/p>\n<p>Here's a description from the Chebfun website:<\/p>\n<blockquote><p>Chebfun is a collection of algorithms and a software<br \/>\nsystem in object-oriented MATLAB which extends familiar powerful methods<br \/>\nof numerical computation involving numbers to continuous or<br \/>\npiecewise-continuous functions. It also implements continuous analogues<br \/>\nof linear algebra notions like the QR decomposition and the SVD, and<br \/>\nsolves ordinary differential equations. The mathematical basis of the<br \/>\nsystem combines tools of Chebyshev expansions, fast Fourier transform,<br \/>\nbarycentric interpolation, recursive zerofinding, and automatic<br \/>\ndifferentiation.<\/p><\/blockquote>\n<p>&nbsp;<\/p>\n<h3>Contents<\/h3>\n<div>\n<ul>\n<li><a href=\"#1\">Nonlinear Operators<\/a><\/li>\n<li><a href=\"#2\">Graphical User Interface<\/a><\/li>\n<li><a href=\"#3\">Rich Set of Examples<\/a><\/li>\n<li><a href=\"#5\">What Basis Functions Do You Use When Required?<\/a><\/li>\n<\/ul>\n<\/div>\n<h3>Nonlinear Operators<a name=\"1\"><\/a><\/h3>\n<p>One recent addition includes being able to use nonlinear operators more naturally. So, to solve a nonlinear system, guess<br \/>\nwhat?! You get to use the MATLAB operator <a href=\"https:\/\/www.mathworks.com\/help\/releases\/R2011a\/techdoc\/\/ref\/mldivide.html\"><tt>\\<\/tt><\/a>! See Chapter 7 of the Chebfun Guide for a nice example.<\/p>\n<h3>Graphical User Interface<a name=\"2\"><\/a><\/h3>\n<p>Nick and Nick ran the GUI for Chebfun (and have since). You can use it to formulate your problems. In addition, you can<br \/>\nuse it to access the demos and examples. Find a problem similar to on your want to solve, select it in the GUI where it will<br \/>\npopulate all the required fields, and you are ready to run the code or tinker.<\/p>\n<h3>Rich Set of Examples<a name=\"3\"><\/a><\/h3>\n<p><a href=\"http:\/\/www.chebfun.org\/examples\/\">Here is a<\/a> full list of examples. You can get the code, or the example in PDF or HTML form.<\/p>\n<p>I'll mention just a few of my favorites to show you the breadth of possible problems that Chebfun can solve.<\/p>\n<div>\n<ul>\n<li>Orr-Somerfield eigenvalues (hydrodynamic stability)<\/li>\n<li>Carrier equation (ode: boundary value problem)<\/li>\n<li>Double well Schrodinger eigenstates (quantum mechanics)<\/li>\n<li>Coupled system of reaction-diffusion equations<\/li>\n<\/ul>\n<\/div>\n<p>And that's just the tip of the iceberg.<\/p>\n<h3>What Basis Functions Do You Use When Required?<a name=\"5\"><\/a><\/h3>\n<p>Many people use splines as basis functions for approximations, solving differential equations, and so on. Some people use<br \/>\nwavelets for similar purposes. The package discussed here relies on Chebyshev functions. I hope from the examples you can<br \/>\nsee the broad applicability of the Chebfun project. What do you use? And why? Let me know <a href=\"https:\/\/blogs.mathworks.com\/loren\/?p=279#respond\">here<\/a>.<\/p>\n<p><script>\/\/ <![CDATA[\nfunction grabCode_e85e1a810aa74c19b4d62655f2a1c423() {\n        \/\/ Remember the title so we can use it in the new page\n        title = document.title;\n\n        \/\/ Break up these strings so that their presence\n        \/\/ in the Javascript doesn't mess up the search for\n        \/\/ the MATLAB code.\n        t1='e85e1a810aa74c19b4d62655f2a1c423 ' + '##### ' + 'SOURCE BEGIN' + ' #####';\n        t2='##### ' + 'SOURCE END' + ' #####' + ' e85e1a810aa74c19b4d62655f2a1c423';\n    \n        b=document.getElementsByTagName('body')[0];\n        i1=b.innerHTML.indexOf(t1)+t1.length;\n        i2=b.innerHTML.indexOf(t2);\n \n        code_string = b.innerHTML.substring(i1, i2);\n        code_string = code_string.replace(\/REPLACE_WITH_DASH_DASH\/g,'--');\n\n        \/\/ Use \/x3C\/g instead of the less-than character to avoid errors \n        \/\/ in the XML parser.\n        \/\/ Use '\\x26#60;' instead of '<' so that the XML parser\n        \/\/ doesn't go ahead and substitute the less-than character. \n        code_string = code_string.replace(\/\\x3C\/g, '\\x26#60;');\n\n        author = 'Loren Shure';\n        copyright = 'Copyright 2011 The MathWorks, Inc.';\n\n        w = window.open();\n        d = w.document;\n        d.write('\n\n\n\n\n\n<pre>\\n');\r\n        d.write(code_string);\r\n\r\n        \/\/ Add author and copyright lines at the bottom if specified.\r\n        if ((author.length > 0) || (copyright.length > 0)) {\r\n            d.writeln('');\r\n            d.writeln('%%');\r\n            if (author.length > 0) {\r\n                d.writeln('% _' + author + '_');\r\n            }\r\n            if (copyright.length > 0) {\r\n                d.writeln('% _' + copyright + '_');\r\n            }\r\n        }\r\n\r\n        d.write('<\/pre>\n\n\n\n\n\n\n\\n');\n      \n      d.title = title + ' (MATLAB code)';\n      d.close();\n      }\n\/\/ ]]><\/script><\/p>\n<p style=\"text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;\"><a><span style=\"font-size: x-small; font-style: italic;\">Get<br \/>\nthe MATLAB code<br \/>\n<noscript>(requires JavaScript)<\/noscript><\/span><\/a><\/p>\n<p>Published with MATLAB\u00ae 7.12<\/p>\n<\/div>\n<p><!--\ne85e1a810aa74c19b4d62655f2a1c423 ##### SOURCE BEGIN #####\n%% Update on the Chebfun Project\n% A while ago,\n% <https:\/\/blogs.mathworks.com\/loren\/2010\/01\/12\/the-chebfun-project\/ I % wrote> about the <http:\/\/www2.maths.ox.ac.uk\/chebfun\/ Chebfun Project>.\n% At the time, version 3 had recently come out.  I recently visited Nick\n% Trefethen and Nick Hale, part of the\n% <http:\/\/www2.maths.ox.ac.uk\/chebfun\/team.html Chebfun team>, and they\n% showed me the latest and greatest version 4.\n%\n% As I said in my earlier\n% post, the package is designed to have syntax very much like regular\n% vector notation in MATLAB, though the entities represent functions.\n%\n% Here's a description from the Chebfun website:\n%\n% <html>\n\n\n<blockquote>Chebfun is a collection of algorithms and a software\n% system in object-oriented MATLAB which extends familiar powerful methods\n% of numerical computation involving numbers to continuous or\n% piecewise-continuous functions. It also implements continuous analogues\n% of linear algebra notions like the QR decomposition and the SVD, and\n% solves ordinary differential equations. The mathematical basis of the\n% system combines tools of Chebyshev expansions, fast Fourier transform,\n% barycentric interpolation, recursive zerofinding, and automatic\n% differentiation.<\/blockquote>\n\n\n<\/html>\n%% Nonlinear Operators\n% One recent addition includes being able to use nonlinear operators more\n% naturally.  So, to solve a nonlinear system, guess what?!  You get to use\n% the MATLAB operator\n% <https:\/\/www.mathworks.com\/help\/releases\/R2011a\/techdoc\/\/ref\/mldivide.html % |\\|>! See <http:\/\/www2.maths.ox.ac.uk\/chebfun\/guide\/html\/guide7.shtml#49 % Chapter 7> of the Chebfun Guide for a nice example.\n%% Graphical User Interface\n% Nick and Nick ran the GUI for Chebfun (and have since).  You can use\n% it to formulate your problems.  In addition, you can use it to access the\n% demos and examples.  Find a problem similar to on your want to solve,\n% select it in the GUI where it will populate all the required fields, and\n% you are ready to run the code or tinker.\n%% Rich Set of Examples\n% <http:\/\/www2.maths.ox.ac.uk\/chebfun\/examples This page> lists an overview\n% of the categories of examples solved using Chebfun.\n% <http:\/\/www2.maths.ox.ac.uk\/chebfun\/examples\/listing.shtml Here is a>\n% full list of examples>.  You can get the code, or the example in PDF or\n% HTML form.\n%%\n% I'll mention just a few of my favorites to show you the breadth of\n% possible problems that Chebfun can solve.\n%\n% * Orr-Somerfield eigenvalues (hydrodynamic stability)\n% * Carrier equation (ode: boundary value problem)\n% * Double well Schrodinger eigenstates (quantum mechanics)\n% * Coupled system of reaction-diffusion equations\n%\n% And that's just the tip of the iceberg.\n\n%% What Basis Functions Do You Use When Required?\n% Many people use splines as basis functions for approximations, solving\n% differential equations, and so on.  Some people use wavelets for similar\n% purposes.  The package discussed here relies on Chebyshev functions.  I\n% hope from the examples you can see the broad applicability of the Chebfun\n% project. What do you use?  And why?  Let me know\n% <https:\/\/blogs.mathworks.com\/loren\/?p=279#respond here>.\n##### SOURCE END ##### e85e1a810aa74c19b4d62655f2a1c423\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\nA while ago, I wrote about the Chebfun Project. At the time, version 3 had recently come out. I recently visited Nick Trefethen and Nick Hale, part of the Chebfun team, and they showed me the latest... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/loren\/2011\/06\/21\/update-on-the-chebfun-project\/\">read more >><\/a><\/p>\n","protected":false},"author":39,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[25,31,13],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/posts\/279"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/users\/39"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/comments?post=279"}],"version-history":[{"count":2,"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/posts\/279\/revisions"}],"predecessor-version":[{"id":1847,"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/posts\/279\/revisions\/1847"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/media?parent=279"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/categories?post=279"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/loren\/wp-json\/wp\/v2\/tags?post=279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}