{"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":258,"title":"linear least squares fitting","description":"Inputs:\r\n\r\n* |f|: cell-array of function handles\r\n* |x|: column vector of |x| values\r\n* |y|: column vector of |y| values, same length as |x|\r\n\r\nOutput:\r\n\r\n* |a|: column vector of coefficients, same length as |f|\r\n\r\nIn a correct answer the coefficients |a| take values such that the function\r\n\r\n   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)\r\n\r\nminimizes the sum of the squared deviations between |fit(x)| and |y|, i.e.\r\n    sum((fit(x)-y).^2)\r\nis minimal. \r\n\r\nRemarks:\r\n\r\n* The functions will all be vectorized, so e.g. |f{1}(x)| will return results for the whole vector x\r\n* The absolute errors of |a| must be smaller than 1e-6 to pass the tests","description_html":"\u003cp\u003eInputs:\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003ef\u003c/tt\u003e: cell-array of function handles\u003c/li\u003e\u003cli\u003e\u003ctt\u003ex\u003c/tt\u003e: column vector of \u003ctt\u003ex\u003c/tt\u003e values\u003c/li\u003e\u003cli\u003e\u003ctt\u003ey\u003c/tt\u003e: column vector of \u003ctt\u003ey\u003c/tt\u003e values, same length as \u003ctt\u003ex\u003c/tt\u003e\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eOutput:\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003ea\u003c/tt\u003e: column vector of coefficients, same length as \u003ctt\u003ef\u003c/tt\u003e\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eIn a correct answer the coefficients \u003ctt\u003ea\u003c/tt\u003e take values such that the function\u003c/p\u003e\u003cpre\u003e   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)\u003c/pre\u003e\u003cp\u003eminimizes the sum of the squared deviations between \u003ctt\u003efit(x)\u003c/tt\u003e and \u003ctt\u003ey\u003c/tt\u003e, i.e.\r\n    sum((fit(x)-y).^2)\r\nis minimal.\u003c/p\u003e\u003cp\u003eRemarks:\u003c/p\u003e\u003cul\u003e\u003cli\u003eThe functions will all be vectorized, so e.g. \u003ctt\u003ef{1}(x)\u003c/tt\u003e will return results for the whole vector x\u003c/li\u003e\u003cli\u003eThe absolute errors of \u003ctt\u003ea\u003c/tt\u003e must be smaller than 1e-6 to pass the tests\u003c/li\u003e\u003c/ul\u003e","function_template":"function a = fit_coefficients(f,x,y)\r\n  a = zeros(length(f),1);\r\nend","test_suite":"%%% first test: fit to a constant\r\nx = [1,2,3,4]';\r\ny = rand(4,1);\r\nf{1} = @(x) ones(size(x));\r\naref=mean(y);\r\nassert(norm(fit_coefficients(f,x,y)-aref)\u003c1e-6)\r\n\r\n%%% second test: fit to a straight line (linear regression)\r\nx = [1,2,3,4,5]' + randn(5,1);\r\ny = [1,2,3,4,5]' + randn(5,1);\r\nf{1} = @(x) ones(size(x));\r\nf{2} = @(x) x;\r\naref(2) = sum((x-mean(x)).*(y-mean(y)))/sum((x-mean(x)).^2);\r\naref(1) = mean(y)-aref(2)*mean(x);\r\nassert(norm(fit_coefficients(f,x,y)-aref')\u003c1e-6)\r\n\r\n%%% third test: polynomial fit\r\nx = [1:15]' + randn(15,1);\r\ny = -10+0.2*x-0.5*x.^2+0.4*x.^3+0.001*log(abs(x)) + 0.2*randn(15,1);\r\nf{1} = @(x) ones(size(x));\r\nf{2} = @(x) x;\r\nf{3} = @(x) x.^2;\r\nf{4} = @(x) x.^3;\r\naref = fliplr(polyfit(x,y,3));\r\nassert(norm(fit_coefficients(f,x,y)-aref')\u003c1e-6)\r\n\r\n%%% fourth test: non-polynomial fit (yes, we are that crazy)\r\nx = [0:0.1:2*pi]';\r\ny = 0.123 + 0.456*sin(x).*exp(0.1*x);\r\nf{1} = @(x) ones(size(x));\r\nf{2} = @(x) sin(x).*exp(0.1*x);\r\naref=[0.123 0.456]';\r\nassert(norm(fit_coefficients(f,x,y)-aref)\u003c1e-6)","published":true,"deleted":false,"likes_count":5,"comments_count":6,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":163,"test_suite_updated_at":"2013-01-10T10:23:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-04T19:59:00.000Z","updated_at":"2026-03-29T20:40:56.000Z","published_at":"2013-01-09T22:29:23.000Z","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 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w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e values, same length as\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\u003ex\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\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: column vector of coefficients, same length as\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\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\u003eIn a correct answer the coefficients\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\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e take values such that the function\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[   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)]]\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\u003eminimizes the sum of the squared deviations 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\u003efit(x)\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\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, i.e. sum((fit(x)-y).^2) is minimal.\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\u003eThe functions will all be vectorized, so e.g.\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{1}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will return results for the whole vector x\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 absolute errors of\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\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e must be smaller than 1e-6 to pass the tests\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":258,"title":"linear least squares fitting","description":"Inputs:\r\n\r\n* |f|: cell-array of function handles\r\n* |x|: column vector of |x| values\r\n* |y|: column vector of |y| values, same length as |x|\r\n\r\nOutput:\r\n\r\n* |a|: column vector of coefficients, same length as |f|\r\n\r\nIn a correct answer the coefficients |a| take values such that the function\r\n\r\n   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)\r\n\r\nminimizes the sum of the squared deviations between |fit(x)| and |y|, i.e.\r\n    sum((fit(x)-y).^2)\r\nis minimal. \r\n\r\nRemarks:\r\n\r\n* The functions will all be vectorized, so e.g. |f{1}(x)| will return results for the whole vector x\r\n* The absolute errors of |a| must be smaller than 1e-6 to pass the tests","description_html":"\u003cp\u003eInputs:\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003ef\u003c/tt\u003e: cell-array of function handles\u003c/li\u003e\u003cli\u003e\u003ctt\u003ex\u003c/tt\u003e: column vector of \u003ctt\u003ex\u003c/tt\u003e values\u003c/li\u003e\u003cli\u003e\u003ctt\u003ey\u003c/tt\u003e: column vector of \u003ctt\u003ey\u003c/tt\u003e values, same length as \u003ctt\u003ex\u003c/tt\u003e\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eOutput:\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003ctt\u003ea\u003c/tt\u003e: column vector of coefficients, same length as \u003ctt\u003ef\u003c/tt\u003e\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eIn a correct answer the coefficients \u003ctt\u003ea\u003c/tt\u003e take values such that the function\u003c/p\u003e\u003cpre\u003e   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)\u003c/pre\u003e\u003cp\u003eminimizes the sum of the squared deviations between \u003ctt\u003efit(x)\u003c/tt\u003e and \u003ctt\u003ey\u003c/tt\u003e, i.e.\r\n    sum((fit(x)-y).^2)\r\nis minimal.\u003c/p\u003e\u003cp\u003eRemarks:\u003c/p\u003e\u003cul\u003e\u003cli\u003eThe functions will all be vectorized, so e.g. \u003ctt\u003ef{1}(x)\u003c/tt\u003e will return results for the whole vector x\u003c/li\u003e\u003cli\u003eThe absolute errors of \u003ctt\u003ea\u003c/tt\u003e must be smaller than 1e-6 to pass the tests\u003c/li\u003e\u003c/ul\u003e","function_template":"function a = fit_coefficients(f,x,y)\r\n  a = zeros(length(f),1);\r\nend","test_suite":"%%% first test: fit to a constant\r\nx = [1,2,3,4]';\r\ny = rand(4,1);\r\nf{1} = @(x) ones(size(x));\r\naref=mean(y);\r\nassert(norm(fit_coefficients(f,x,y)-aref)\u003c1e-6)\r\n\r\n%%% second test: fit to a straight line (linear regression)\r\nx = [1,2,3,4,5]' + randn(5,1);\r\ny = [1,2,3,4,5]' + randn(5,1);\r\nf{1} = @(x) ones(size(x));\r\nf{2} = @(x) x;\r\naref(2) = sum((x-mean(x)).*(y-mean(y)))/sum((x-mean(x)).^2);\r\naref(1) = mean(y)-aref(2)*mean(x);\r\nassert(norm(fit_coefficients(f,x,y)-aref')\u003c1e-6)\r\n\r\n%%% third test: polynomial fit\r\nx = [1:15]' + randn(15,1);\r\ny = -10+0.2*x-0.5*x.^2+0.4*x.^3+0.001*log(abs(x)) + 0.2*randn(15,1);\r\nf{1} = @(x) ones(size(x));\r\nf{2} = @(x) x;\r\nf{3} = @(x) x.^2;\r\nf{4} = @(x) x.^3;\r\naref = fliplr(polyfit(x,y,3));\r\nassert(norm(fit_coefficients(f,x,y)-aref')\u003c1e-6)\r\n\r\n%%% fourth test: non-polynomial fit (yes, we are that crazy)\r\nx = [0:0.1:2*pi]';\r\ny = 0.123 + 0.456*sin(x).*exp(0.1*x);\r\nf{1} = @(x) ones(size(x));\r\nf{2} = @(x) sin(x).*exp(0.1*x);\r\naref=[0.123 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as\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\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\u003eIn a correct answer the coefficients\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\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e take values such that the function\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[   fit = @(x) a(1)*f{1}(x) + a(2)*f{2}(x) + a(3)*f{3}(x) +...+ a(end)*f{end}(x)]]\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\u003eminimizes the sum of the squared deviations 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\u003efit(x)\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\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, i.e. sum((fit(x)-y).^2) is minimal.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle 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