{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":231,"title":"Differential equations I","description":"Given a function handle |f| an initial condition |y0| and a final time |tf|, solve numerically the differential equation\r\n\r\n  dy/dt = f(y)\r\n\r\nfor the function |y(t)| between |t=0| and |t=tf|. Give as a result |res=y(tf)|.\r\n\r\nExample:\r\n\r\n   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\r\n\r\n =\u003e y(tf) = 1/e = 0.367879441171442\r\n\r\nRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].","description_html":"\u003cp\u003eGiven a function handle \u003ctt\u003ef\u003c/tt\u003e an initial condition \u003ctt\u003ey0\u003c/tt\u003e and a final time \u003ctt\u003etf\u003c/tt\u003e, solve numerically the differential equation\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003edy/dt = f(y)\r\n\u003c/pre\u003e\u003cp\u003efor the function \u003ctt\u003ey(t)\u003c/tt\u003e between \u003ctt\u003et=0\u003c/tt\u003e and \u003ctt\u003et=tf\u003c/tt\u003e. Give as a result \u003ctt\u003eres=y(tf)\u003c/tt\u003e.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\u003c/pre\u003e\u003cpre\u003e =\u003e y(tf) = 1/e = 0.367879441171442\u003c/pre\u003e\u003cp\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/p\u003e","function_template":"function res = deqnsolve(f,y0,tf)\r\n  res = 0;\r\nend","test_suite":"%% \r\nf = @(x) -x;\r\ntf =1;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(-1)) \u003c 1e-5)\r\n\r\n%% \r\nf = @sin;\r\ntf =1;\r\ny0 =1/2;\r\nassert(abs(deqnsolve(f,y0,tf)-2*acot(exp(-1)*cot(1/4))) \u003c 1e-5)\r\n\r\n%% \r\nf = @(x) 1/(x+1);\r\ntf =6;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-3) \u003c 1e-5)\r\n\r\n%% a randomized one\r\na = rand*0.9;\r\nf = @(x) x-a*x^2;\r\ntf = rand+1.5;\r\ny0=1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(tf)/(1-a+a*exp(tf))) \u003c 1e-5)","published":true,"deleted":false,"likes_count":3,"comments_count":5,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":190,"test_suite_updated_at":"2012-02-02T15:10:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-02T14:57:14.000Z","updated_at":"2026-03-13T19:43:52.000Z","published_at":"2012-02-02T15:20:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a function handle\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e an initial condition\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and a final time\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, solve numerically the differential equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[dy/dt = f(y)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey(t)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=tf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Give as a result\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eres=y(tf)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   f = @(x) -x;\\n   tf= 1;\\n   y0= 1;\\n\\n =\u003e y(tf) = 1/e = 0.367879441171442]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":179,"title":"Monte-Carlo integration","description":"Write a function that estimates a d-dimensional integral to at least 1% relative precision.\r\n\r\nInputs:\r\n\r\n* d: positive integer. The dimension of the integral.\r\n* fun: function handle. The function accepts a row-vector of length d as an argument and returns a real scalar as a result.\r\n\r\nOutput:\r\n\r\n* I: is the integral over fun from 0 to 1 in each direction.\r\n\r\n     1     1        1            \r\n     /     /        /           \r\n I = |dx_1 |dx_2 ...| dx_d  fun([x_1,x_2,...,x_d])\r\n     /     /        /            \r\n     0     0        0            \r\n\r\nExample:\r\n\r\n  fun = @(x) x(1)*x(2)\r\n  d = 2\r\n\r\nThe result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.25-0.2501)/0.25 which is smaller than 1%.\r\n\r\nThe functions in the test-suite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.","description_html":"\u003cp\u003eWrite a function that estimates a d-dimensional integral to at least 1% relative precision.\u003c/p\u003e\u003cp\u003eInputs:\u003c/p\u003e\u003cul\u003e\u003cli\u003ed: positive integer. The dimension of the integral.\u003c/li\u003e\u003cli\u003efun: function handle. The function accepts a row-vector of length d as an argument and returns a real scalar as a result.\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eOutput:\u003c/p\u003e\u003cul\u003e\u003cli\u003eI: is the integral over fun from 0 to 1 in each direction.\u003c/li\u003e\u003c/ul\u003e\u003cpre\u003e     1     1        1            \r\n     /     /        /           \r\n I = |dx_1 |dx_2 ...| dx_d  fun([x_1,x_2,...,x_d])\r\n     /     /        /            \r\n     0     0        0            \u003c/pre\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003efun = @(x) x(1)*x(2)\r\nd = 2\r\n\u003c/pre\u003e\u003cp\u003eThe result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.25-0.2501)/0.25 which is smaller than 1%.\u003c/p\u003e\u003cp\u003eThe functions in the test-suite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.\u003c/p\u003e","function_template":"function I = dquad(d,fun)\r\n  I=fun(0.5*ones(1,d));\r\nend","test_suite":"%% 1d integral: integrate x^2 from 0 to 1\r\nfun = @(x) x^2;\r\nassert(abs((dquad(1,fun) - 1/3)*3)\u003c0.01)\r\n\r\n%% 2d integral from the example\r\nfun = @(x) x(1)*x(2);\r\nassert(abs((dquad(2,fun) - 0.25)*4)\u003c0.01)\r\n\r\n%% constant in most dimensions\r\nfun = @(x) 1+sin(x(1));\r\nassert(abs((dquad(50,fun) -  1.45969769)/1.45969769)\u003c0.01)\r\n\r\n%% volume of d-dimensional 2^d box with a spherical hole, d between 5 and 10\r\nd = randi([5 10],1)\r\nr = rand*0.8\r\nfun = @(x) 2^d*(norm(x)\u003er);\r\ndball = exp(d/2*log(pi)+d*log(r)-gammaln(d/2+1));\r\nassert(abs((dquad(d,fun) - 2^d+dball)/(2^d-dball))\u003c0.01)\r\n","published":true,"deleted":false,"likes_count":7,"comments_count":8,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":153,"test_suite_updated_at":"2012-01-31T21:50:54.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-01-30T15:04:16.000Z","updated_at":"2026-04-03T20:58:12.000Z","published_at":"2012-02-01T19:02:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that estimates a d-dimensional integral to at least 1% relative precision.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInputs:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ed: positive integer. The dimension of the integral.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efun: function handle. The function accepts a row-vector of length d as an argument and returns a real scalar as a result.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI: is the integral over fun from 0 to 1 in each direction.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[     1     1        1            \\n     /     /        /           \\n I = |dx_1 |dx_2 ...| dx_d  fun([x_1,x_2,...,x_d])\\n     /     /        /            \\n     0     0        0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[fun = @(x) x(1)*x(2)\\nd = 2]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.25-0.2501)/0.25 which is smaller than 1%.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe functions in the test-suite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1197,"title":"Numerical Integration","description":"Input\r\n\r\n* |x0|, a real number greater than 0\r\n\r\nOutput\r\n\r\n* |I|, a numerical estimate of the integral\r\n\r\n      x0\r\n      /\r\n  I = |  cosh(x) / sqrt(cosh(x0) - cosh(x)) dx\r\n      /\r\n      0\r\n\r\nExample:\r\n\r\n   x0=1.0  --\u003e  I = 2.6405789412796\r\n\r\n\r\nRemarks:\r\n\r\n* Aim at a relative precision better than 1e-10\r\n* The problem arises studying the frictionless movement of a mass point on a hanging wire, which follows the curve cosh(x).","description_html":"\u003cp\u003eInput\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003ex0\u003c/tt\u003e, a real number greater than 0\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eOutput\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003eI\u003c/tt\u003e, a numerical estimate of the integral\u003c/li\u003e\u003c/ul\u003e\u003cpre\u003e      x0\r\n      /\r\n  I = |  cosh(x) / sqrt(cosh(x0) - cosh(x)) dx\r\n      /\r\n      0\u003c/pre\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e   x0=1.0  --\u003e  I = 2.6405789412796\u003c/pre\u003e\u003cp\u003eRemarks:\u003c/p\u003e\u003cul\u003e\u003cli\u003eAim at a relative precision better than 1e-10\u003c/li\u003e\u003cli\u003eThe problem arises studying the frictionless movement of a mass point on a hanging wire, which follows the curve cosh(x).\u003c/li\u003e\u003c/ul\u003e","function_template":"function I = coshint(x0)\r\n  I = x0;\r\nend","test_suite":"%%\r\nx0 = 1;\r\nI_correct = 2.6405789412796;\r\nfprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\nassert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\n\r\n%%\r\nx0 = 2;\r\nI_correct = 3.9464053536380;\r\nfprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\nassert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\n\r\n%%\r\nx0 = 13;\r\nI_correct = 9.4065231838369e+02;\r\nfprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\nassert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\n\r\n%% randomized test for small values of x0 where \r\n % cosh(x) ~ 1 + x^2/2 + ...\r\n % and up to x0=1e-5 Integrating (analytically) the approximation is\r\n % accurate enough\r\nfor l=1:5\r\n   x0 = 1e-6 * (1+rand);\r\n   I_correct = pi*(4+x0^2)/4/sqrt(2);\r\n   fprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\n   assert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":9,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":106,"test_suite_updated_at":"2013-01-11T15:08:44.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-01-11T14:11:17.000Z","updated_at":"2026-03-13T22:59:58.000Z","published_at":"2013-01-11T15:08:44.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInput\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a real number greater than 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eI\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a numerical estimate of the integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[      x0\\n      /\\n  I = |  cosh(x) / sqrt(cosh(x0) - cosh(x)) dx\\n      /\\n      0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   x0=1.0  --\u003e  I = 2.6405789412796]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRemarks:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAim at a relative precision better than 1e-10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe problem arises studying the frictionless movement of a mass point on a hanging wire, which follows the curve cosh(x).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":231,"title":"Differential equations I","description":"Given a function handle |f| an initial condition |y0| and a final time |tf|, solve numerically the differential equation\r\n\r\n  dy/dt = f(y)\r\n\r\nfor the function |y(t)| between |t=0| and |t=tf|. Give as a result |res=y(tf)|.\r\n\r\nExample:\r\n\r\n   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\r\n\r\n =\u003e y(tf) = 1/e = 0.367879441171442\r\n\r\nRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].","description_html":"\u003cp\u003eGiven a function handle \u003ctt\u003ef\u003c/tt\u003e an initial condition \u003ctt\u003ey0\u003c/tt\u003e and a final time \u003ctt\u003etf\u003c/tt\u003e, solve numerically the differential equation\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003edy/dt = f(y)\r\n\u003c/pre\u003e\u003cp\u003efor the function \u003ctt\u003ey(t)\u003c/tt\u003e between \u003ctt\u003et=0\u003c/tt\u003e and \u003ctt\u003et=tf\u003c/tt\u003e. Give as a result \u003ctt\u003eres=y(tf)\u003c/tt\u003e.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\u003c/pre\u003e\u003cpre\u003e =\u003e y(tf) = 1/e = 0.367879441171442\u003c/pre\u003e\u003cp\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/p\u003e","function_template":"function res = deqnsolve(f,y0,tf)\r\n  res = 0;\r\nend","test_suite":"%% \r\nf = @(x) -x;\r\ntf =1;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(-1)) \u003c 1e-5)\r\n\r\n%% \r\nf = @sin;\r\ntf =1;\r\ny0 =1/2;\r\nassert(abs(deqnsolve(f,y0,tf)-2*acot(exp(-1)*cot(1/4))) \u003c 1e-5)\r\n\r\n%% \r\nf = @(x) 1/(x+1);\r\ntf =6;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-3) \u003c 1e-5)\r\n\r\n%% a randomized one\r\na = rand*0.9;\r\nf = @(x) x-a*x^2;\r\ntf = rand+1.5;\r\ny0=1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(tf)/(1-a+a*exp(tf))) \u003c 1e-5)","published":true,"deleted":false,"likes_count":3,"comments_count":5,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":190,"test_suite_updated_at":"2012-02-02T15:10:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-02T14:57:14.000Z","updated_at":"2026-03-13T19:43:52.000Z","published_at":"2012-02-02T15:20:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a function handle\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e an initial condition\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and a final time\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, solve numerically the differential equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[dy/dt = f(y)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey(t)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=tf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Give as a result\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eres=y(tf)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   f = @(x) -x;\\n   tf= 1;\\n   y0= 1;\\n\\n =\u003e y(tf) = 1/e = 0.367879441171442]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":179,"title":"Monte-Carlo integration","description":"Write a function that estimates a d-dimensional integral to at least 1% relative precision.\r\n\r\nInputs:\r\n\r\n* d: positive integer. The dimension of the integral.\r\n* fun: function handle. The function accepts a row-vector of length d as an argument and returns a real scalar as a result.\r\n\r\nOutput:\r\n\r\n* I: is the integral over fun from 0 to 1 in each direction.\r\n\r\n     1     1        1            \r\n     /     /        /           \r\n I = |dx_1 |dx_2 ...| dx_d  fun([x_1,x_2,...,x_d])\r\n     /     /        /            \r\n     0     0        0            \r\n\r\nExample:\r\n\r\n  fun = @(x) x(1)*x(2)\r\n  d = 2\r\n\r\nThe result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.25-0.2501)/0.25 which is smaller than 1%.\r\n\r\nThe functions in the test-suite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.","description_html":"\u003cp\u003eWrite a function that estimates a d-dimensional integral to at least 1% relative precision.\u003c/p\u003e\u003cp\u003eInputs:\u003c/p\u003e\u003cul\u003e\u003cli\u003ed: positive integer. The dimension of the integral.\u003c/li\u003e\u003cli\u003efun: function handle. The function accepts a row-vector of length d as an argument and returns a real scalar as a result.\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eOutput:\u003c/p\u003e\u003cul\u003e\u003cli\u003eI: is the integral over fun from 0 to 1 in each direction.\u003c/li\u003e\u003c/ul\u003e\u003cpre\u003e     1     1        1            \r\n     /     /        /           \r\n I = |dx_1 |dx_2 ...| dx_d  fun([x_1,x_2,...,x_d])\r\n     /     /        /            \r\n     0     0        0            \u003c/pre\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003efun = @(x) x(1)*x(2)\r\nd = 2\r\n\u003c/pre\u003e\u003cp\u003eThe result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.25-0.2501)/0.25 which is smaller than 1%.\u003c/p\u003e\u003cp\u003eThe functions in the test-suite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.\u003c/p\u003e","function_template":"function I = dquad(d,fun)\r\n  I=fun(0.5*ones(1,d));\r\nend","test_suite":"%% 1d integral: integrate x^2 from 0 to 1\r\nfun = @(x) x^2;\r\nassert(abs((dquad(1,fun) - 1/3)*3)\u003c0.01)\r\n\r\n%% 2d integral from the example\r\nfun = @(x) x(1)*x(2);\r\nassert(abs((dquad(2,fun) - 0.25)*4)\u003c0.01)\r\n\r\n%% constant in most dimensions\r\nfun = @(x) 1+sin(x(1));\r\nassert(abs((dquad(50,fun) -  1.45969769)/1.45969769)\u003c0.01)\r\n\r\n%% volume of d-dimensional 2^d box with a spherical hole, d between 5 and 10\r\nd = randi([5 10],1)\r\nr = rand*0.8\r\nfun = @(x) 2^d*(norm(x)\u003er);\r\ndball = exp(d/2*log(pi)+d*log(r)-gammaln(d/2+1));\r\nassert(abs((dquad(d,fun) - 2^d+dball)/(2^d-dball))\u003c0.01)\r\n","published":true,"deleted":false,"likes_count":7,"comments_count":8,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":153,"test_suite_updated_at":"2012-01-31T21:50:54.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-01-30T15:04:16.000Z","updated_at":"2026-04-03T20:58:12.000Z","published_at":"2012-02-01T19:02:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that estimates a d-dimensional integral to at least 1% relative precision.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInputs:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ed: positive integer. The dimension of the integral.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efun: function handle. The function accepts a row-vector of length d as an argument and returns a real scalar as a result.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI: is the integral over fun from 0 to 1 in each direction.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[     1     1        1            \\n     /     /        /           \\n I = |dx_1 |dx_2 ...| dx_d  fun([x_1,x_2,...,x_d])\\n     /     /        /            \\n     0     0        0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[fun = @(x) x(1)*x(2)\\nd = 2]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe result should be 0.25. An output I=0.2501 would be acceptable, because the relative deviation would be abs(0.25-0.2501)/0.25 which is smaller than 1%.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe functions in the test-suite are all positive and generally 'well behaved', i.e. not fluctuating too much. Some of the tests hav a relatively large d.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1197,"title":"Numerical Integration","description":"Input\r\n\r\n* |x0|, a real number greater than 0\r\n\r\nOutput\r\n\r\n* |I|, a numerical estimate of the integral\r\n\r\n      x0\r\n      /\r\n  I = |  cosh(x) / sqrt(cosh(x0) - cosh(x)) dx\r\n      /\r\n      0\r\n\r\nExample:\r\n\r\n   x0=1.0  --\u003e  I = 2.6405789412796\r\n\r\n\r\nRemarks:\r\n\r\n* Aim at a relative precision better than 1e-10\r\n* The problem arises studying the frictionless movement of a mass point on a hanging wire, which follows the curve cosh(x).","description_html":"\u003cp\u003eInput\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003ex0\u003c/tt\u003e, a real number greater than 0\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eOutput\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003eI\u003c/tt\u003e, a numerical estimate of the integral\u003c/li\u003e\u003c/ul\u003e\u003cpre\u003e      x0\r\n      /\r\n  I = |  cosh(x) / sqrt(cosh(x0) - cosh(x)) dx\r\n      /\r\n      0\u003c/pre\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e   x0=1.0  --\u003e  I = 2.6405789412796\u003c/pre\u003e\u003cp\u003eRemarks:\u003c/p\u003e\u003cul\u003e\u003cli\u003eAim at a relative precision better than 1e-10\u003c/li\u003e\u003cli\u003eThe problem arises studying the frictionless movement of a mass point on a hanging wire, which follows the curve cosh(x).\u003c/li\u003e\u003c/ul\u003e","function_template":"function I = coshint(x0)\r\n  I = x0;\r\nend","test_suite":"%%\r\nx0 = 1;\r\nI_correct = 2.6405789412796;\r\nfprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\nassert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\n\r\n%%\r\nx0 = 2;\r\nI_correct = 3.9464053536380;\r\nfprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\nassert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\n\r\n%%\r\nx0 = 13;\r\nI_correct = 9.4065231838369e+02;\r\nfprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\nassert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\n\r\n%% randomized test for small values of x0 where \r\n % cosh(x) ~ 1 + x^2/2 + ...\r\n % and up to x0=1e-5 Integrating (analytically) the approximation is\r\n % accurate enough\r\nfor l=1:5\r\n   x0 = 1e-6 * (1+rand);\r\n   I_correct = pi*(4+x0^2)/4/sqrt(2);\r\n   fprintf('Relative difference to reference solution: %e\\n',norm(coshint(x0)-I_correct)/I_correct)\r\n   assert(norm(coshint(x0)-I_correct)/I_correct \u003c= 1e-10)\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":9,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":106,"test_suite_updated_at":"2013-01-11T15:08:44.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-01-11T14:11:17.000Z","updated_at":"2026-03-13T22:59:58.000Z","published_at":"2013-01-11T15:08:44.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInput\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a real number greater than 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eI\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a numerical estimate of the integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[      x0\\n      /\\n  I = |  cosh(x) / sqrt(cosh(x0) - cosh(x)) dx\\n      /\\n      0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   x0=1.0  --\u003e  I = 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