{"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":43553,"title":"Linear Least Squares (L2 fitting problem )","description":"Given a set of real measurements\r\n\r\n     (x(i), y(i))\r\n\r\nfind a line sol(1)x + sol(2) (more specifically furnish the vector with sol=[sol(1),sol(2)])such that it fits the data (it minimises the 2 norm)\r\n\r\n_Example:_ Input:\r\n\r\n  % INPUT\r\n  x=linspace(0,1,50);\r\n  y=4*x-1+ randn(50,1); % perturbed observations\r\n  % SOLUTION:\r\n  sol=[4,-1]\r\n\r\n*HINT :* This problem can be expressed as a convex optimisation problem:\r\n\r\n  min_{sol} sum(sol(1)*x+sol(2)-y)^2   \r\n\r\n*Suggestion:* use the following code to test your function:\r\n\r\n  plot(x,y,'.') % plot the data\r\n  hold on\r\n  plot(x,sol(1)*x+sol(2))\r\n  legend('measurements', 'L2 fit')","description_html":"\u003cp\u003eGiven a set of real measurements\u003c/p\u003e\u003cpre\u003e     (x(i), y(i))\u003c/pre\u003e\u003cp\u003efind a line sol(1)x + sol(2) (more specifically furnish the vector with sol=[sol(1),sol(2)])such that it fits the data (it minimises the 2 norm)\u003c/p\u003e\u003cp\u003e\u003ci\u003eExample:\u003c/i\u003e Input:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e% INPUT\r\nx=linspace(0,1,50);\r\ny=4*x-1+ randn(50,1); % perturbed observations\r\n% SOLUTION:\r\nsol=[4,-1]\r\n\u003c/pre\u003e\u003cp\u003e\u003cb\u003eHINT :\u003c/b\u003e This problem can be expressed as a convex optimisation problem:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003emin_{sol} sum(sol(1)*x+sol(2)-y)^2   \r\n\u003c/pre\u003e\u003cp\u003e\u003cb\u003eSuggestion:\u003c/b\u003e use the following code to test your function:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eplot(x,y,'.') % plot the data\r\nhold on\r\nplot(x,sol(1)*x+sol(2))\r\nlegend('measurements', 'L2 fit')\r\n\u003c/pre\u003e","function_template":"function sol = Lls2(x,y) % Linear least squares with 2 norm\r\n  y = sol(1)*x+sol(2);  \r\nend","test_suite":"%%\r\nx=linspace(0,1,50)';\r\ny=4*x-1+ randn(50,1); % perturbed observations\r\nsol = Lls2(x,y)\r\nassert(abs(sum(Lls2(x,y))-4+1)\u003c2)\r\n\r\n\r\n%%\r\nx=linspace(0,1,50)';\r\ny=7*x+45+ randn(50,1); % perturbed observations\r\nsol = 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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 set of real measurements\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[     (x(i), y(i))]]\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\u003efind a line sol(1)x + sol(2) (more specifically furnish the vector with sol=[sol(1),sol(2)])such that it fits the data (it minimises the 2 norm)\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:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Input:\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[% INPUT\\nx=linspace(0,1,50);\\ny=4*x-1+ randn(50,1); % perturbed observations\\n% SOLUTION:\\nsol=[4,-1]]]\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHINT :\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e This problem can be expressed as a convex optimisation problem:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle 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\r\n\r\n*Suggestion:* use the following code to test your function:\r\n\r\n  plot(x,y,'.') % plot the data\r\n  hold on\r\n  plot(x,sol(1)*x+sol(2))\r\n  legend('measurements', 'L2 fit')","description_html":"\u003cp\u003eGiven a set of real measurements\u003c/p\u003e\u003cpre\u003e     (x(i), y(i))\u003c/pre\u003e\u003cp\u003efind a line sol(1)x + sol(2) (more specifically furnish the vector with sol=[sol(1),sol(2)])such that it fits the data (it minimises the 2 norm)\u003c/p\u003e\u003cp\u003e\u003ci\u003eExample:\u003c/i\u003e Input:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e% INPUT\r\nx=linspace(0,1,50);\r\ny=4*x-1+ randn(50,1); % perturbed observations\r\n% SOLUTION:\r\nsol=[4,-1]\r\n\u003c/pre\u003e\u003cp\u003e\u003cb\u003eHINT :\u003c/b\u003e This problem can be expressed as a convex optimisation problem:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003emin_{sol} sum(sol(1)*x+sol(2)-y)^2   \r\n\u003c/pre\u003e\u003cp\u003e\u003cb\u003eSuggestion:\u003c/b\u003e use the following code to test your function:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eplot(x,y,'.') % plot the data\r\nhold on\r\nplot(x,sol(1)*x+sol(2))\r\nlegend('measurements', 'L2 fit')\r\n\u003c/pre\u003e","function_template":"function sol = Lls2(x,y) % Linear least squares with 2 norm\r\n  y = sol(1)*x+sol(2);  \r\nend","test_suite":"%%\r\nx=linspace(0,1,50)';\r\ny=4*x-1+ randn(50,1); % perturbed observations\r\nsol = Lls2(x,y)\r\nassert(abs(sum(Lls2(x,y))-4+1)\u003c2)\r\n\r\n\r\n%%\r\nx=linspace(0,1,50)';\r\ny=7*x+45+ randn(50,1); % perturbed observations\r\nsol = 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