{"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":44261,"title":"Multivariate polynomials - sort monomials","description":"In \u003chttps://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array Problem 44260\u003e, multivariate polynomials were defined as a sum of monomial terms using|exponents|, a matrix of integers, and|coefficients|, a vector (follow the above link for an explanation).  It can be useful to order the monomials. But first we need to define the total degree of a monomial as the sum of the exponents. For example, the total degree of |5*x| is 1 and the total degree of |x^3*y^5*z| is 9.\r\n\r\nWrite a function \r\n\r\n  function [coeffs,exponents] = sortMonomials(coeffs,exponents)\r\n\r\nto sort the monomials. Sort them first by descending total degree, and then for a given total degree, by lexicographical order of the exponents (by the first exponent, then the second, and so on, each in descending order). The coefficients should be sorted so they stay with the correct monomial.\r\n\r\nExample: Consider the polynomial |p(x,y,z) = 3*x - 2 + y^2 +4*z^2|, which is represented as:\r\n\r\n  exponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]\r\n\r\nThe sorted version is\r\n\r\n  exponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].\r\n\r\nYou can assume that a given combination of exponents is never repeated.","description_html":"\u003cp\u003eIn \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array\"\u003eProblem 44260\u003c/a\u003e, multivariate polynomials were defined as a sum of monomial terms using|exponents|, a matrix of integers, and|coefficients|, a vector (follow the above link for an explanation).  It can be useful to order the monomials. But first we need to define the total degree of a monomial as the sum of the exponents. For example, the total degree of \u003ctt\u003e5*x\u003c/tt\u003e is 1 and the total degree of \u003ctt\u003ex^3*y^5*z\u003c/tt\u003e is 9.\u003c/p\u003e\u003cp\u003eWrite a function\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003efunction [coeffs,exponents] = sortMonomials(coeffs,exponents)\r\n\u003c/pre\u003e\u003cp\u003eto sort the monomials. Sort them first by descending total degree, and then for a given total degree, by lexicographical order of the exponents (by the first exponent, then the second, and so on, each in descending order). The coefficients should be sorted so they stay with the correct monomial.\u003c/p\u003e\u003cp\u003eExample: Consider the polynomial \u003ctt\u003ep(x,y,z) = 3*x - 2 + y^2 +4*z^2\u003c/tt\u003e, which is represented as:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eexponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]\r\n\u003c/pre\u003e\u003cp\u003eThe sorted version is\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eexponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].\r\n\u003c/pre\u003e\u003cp\u003eYou can assume that a given combination of exponents is never repeated.\u003c/p\u003e","function_template":"function [coeffs,exponents] = sortMonomials(coeffs,exponents)\r\ncoeffs = 0;\r\nexponents = 0;\r\nend","test_suite":"%% Test sortMonomials\r\nfiletext = fileread('sortMonomials.m');\r\nassert(~contains(filetext,'regexp'))\r\n\r\n%%\r\nunsortedCoeffs = [-10 7 -10 -7 6 6 3 1 -7 2]';\r\nunsortedExponents = [5 4 2; 2 5 3; 2 1 5; 1 5 4; 1 4 3; 1 3 3; 1 2 1; 0 4 1; 0 2 1; 0 0 5];\r\n[sortedCoeffs,sortedExponents] = sortMonomials(unsortedCoeffs,unsortedExponents);\r\nsortOrder = [1 2 4 3 5 6 8 10 7 9];\r\nassert(isequal(sortedCoeffs,unsortedCoeffs(sortOrder)))\r\nassert(isequal(sortedExponents,unsortedExponents(sortOrder,:)))\r\n\r\n%%\r\nx = randi(1000); y = randi(1000);\r\n[coeffs,exponents] = sortMonomials(x,y);\r\nassert(isequal([x y],[coeffs exponents]))\r\n\r\n%%\r\nunsortedCoeffs = randi(1000,[4 1]);\r\nough = ['hguot '; 'hguoc '; 'hguolp'; 'hguod '];\r\nunsortedExponents = ough - repmat(randi(100),size(ough));\r\nunsortedExponents = [unsortedExponents -sum(unsortedExponents,2)];\r\n[sortedCoeffs,~] = sortMonomials(unsortedCoeffs,unsortedExponents);\r\n[~,ia] = sort(ough(:,5));\r\nassert(isequal(sortedCoeffs,flipud(unsortedCoeffs(ia))))\r\n\r\n%%\r\nz = [1 3 5+randi(10)];\r\nv1 = perms(z); \r\nv2 = perms(z+[1 0 0]);\r\nv = [v2; v1];\r\nunsortedCoeffs = randi(1000,[size(v,1) 1]);\r\nunsortedExponents = v(randperm(size(v,1)),:);\r\n[sortedCoeffs,sortedExponents] = sortMonomials(unsortedCoeffs,unsortedExponents);\r\nassert(isequal(sortedExponents,v))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":1011,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2017-07-13T18:24:47.000Z","updated_at":"2017-07-15T05:42:59.000Z","published_at":"2017-07-13T18:25:15.000Z","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\u003eIn\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 44260\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, multivariate polynomials were defined as a sum of monomial terms using|exponents|, a matrix of integers, and|coefficients|, a vector (follow the above link for an explanation). It can be useful to order the monomials. But first we need to define the total degree of a monomial as the sum of the exponents. For example, the total degree 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\u003e5*x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is 1 and the total degree 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\u003ex^3*y^5*z\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is 9.\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\u003eWrite a 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[function [coeffs,exponents] = sortMonomials(coeffs,exponents)]]\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\u003eto sort the monomials. Sort them first by descending total degree, and then for a given total degree, by lexicographical order of the exponents (by the first exponent, then the second, and so on, each in descending order). The coefficients should be sorted so they stay with the correct monomial.\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: Consider the polynomial\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\u003ep(x,y,z) = 3*x - 2 + y^2 +4*z^2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which is represented as:\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[exponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]]]\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 sorted version is\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[exponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].]]\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\u003eYou can assume that a given combination of exponents is never repeated.\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":44261,"title":"Multivariate polynomials - sort monomials","description":"In \u003chttps://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array Problem 44260\u003e, multivariate polynomials were defined as a sum of monomial terms using|exponents|, a matrix of integers, and|coefficients|, a vector (follow the above link for an explanation).  It can be useful to order the monomials. But first we need to define the total degree of a monomial as the sum of the exponents. For example, the total degree of |5*x| is 1 and the total degree of |x^3*y^5*z| is 9.\r\n\r\nWrite a function \r\n\r\n  function [coeffs,exponents] = sortMonomials(coeffs,exponents)\r\n\r\nto sort the monomials. Sort them first by descending total degree, and then for a given total degree, by lexicographical order of the exponents (by the first exponent, then the second, and so on, each in descending order). The coefficients should be sorted so they stay with the correct monomial.\r\n\r\nExample: Consider the polynomial |p(x,y,z) = 3*x - 2 + y^2 +4*z^2|, which is represented as:\r\n\r\n  exponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]\r\n\r\nThe sorted version is\r\n\r\n  exponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].\r\n\r\nYou can assume that a given combination of exponents is never repeated.","description_html":"\u003cp\u003eIn \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array\"\u003eProblem 44260\u003c/a\u003e, multivariate polynomials were defined as a sum of monomial terms using|exponents|, a matrix of integers, and|coefficients|, a vector (follow the above link for an explanation).  It can be useful to order the monomials. But first we need to define the total degree of a monomial as the sum of the exponents. For example, the total degree of \u003ctt\u003e5*x\u003c/tt\u003e is 1 and the total degree of \u003ctt\u003ex^3*y^5*z\u003c/tt\u003e is 9.\u003c/p\u003e\u003cp\u003eWrite a function\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003efunction [coeffs,exponents] = sortMonomials(coeffs,exponents)\r\n\u003c/pre\u003e\u003cp\u003eto sort the monomials. Sort them first by descending total degree, and then for a given total degree, by lexicographical order of the exponents (by the first exponent, then the second, and so on, each in descending order). The coefficients should be sorted so they stay with the correct monomial.\u003c/p\u003e\u003cp\u003eExample: Consider the polynomial \u003ctt\u003ep(x,y,z) = 3*x - 2 + y^2 +4*z^2\u003c/tt\u003e, which is represented as:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eexponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]\r\n\u003c/pre\u003e\u003cp\u003eThe sorted version is\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eexponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].\r\n\u003c/pre\u003e\u003cp\u003eYou can assume that a given combination of exponents is never repeated.\u003c/p\u003e","function_template":"function [coeffs,exponents] = sortMonomials(coeffs,exponents)\r\ncoeffs = 0;\r\nexponents = 0;\r\nend","test_suite":"%% Test sortMonomials\r\nfiletext = fileread('sortMonomials.m');\r\nassert(~contains(filetext,'regexp'))\r\n\r\n%%\r\nunsortedCoeffs = [-10 7 -10 -7 6 6 3 1 -7 2]';\r\nunsortedExponents = [5 4 2; 2 5 3; 2 1 5; 1 5 4; 1 4 3; 1 3 3; 1 2 1; 0 4 1; 0 2 1; 0 0 5];\r\n[sortedCoeffs,sortedExponents] = sortMonomials(unsortedCoeffs,unsortedExponents);\r\nsortOrder = [1 2 4 3 5 6 8 10 7 9];\r\nassert(isequal(sortedCoeffs,unsortedCoeffs(sortOrder)))\r\nassert(isequal(sortedExponents,unsortedExponents(sortOrder,:)))\r\n\r\n%%\r\nx = randi(1000); y = randi(1000);\r\n[coeffs,exponents] = sortMonomials(x,y);\r\nassert(isequal([x y],[coeffs exponents]))\r\n\r\n%%\r\nunsortedCoeffs = randi(1000,[4 1]);\r\nough = ['hguot '; 'hguoc '; 'hguolp'; 'hguod '];\r\nunsortedExponents = ough - repmat(randi(100),size(ough));\r\nunsortedExponents = [unsortedExponents -sum(unsortedExponents,2)];\r\n[sortedCoeffs,~] = sortMonomials(unsortedCoeffs,unsortedExponents);\r\n[~,ia] = sort(ough(:,5));\r\nassert(isequal(sortedCoeffs,flipud(unsortedCoeffs(ia))))\r\n\r\n%%\r\nz = [1 3 5+randi(10)];\r\nv1 = perms(z); \r\nv2 = perms(z+[1 0 0]);\r\nv = [v2; v1];\r\nunsortedCoeffs = randi(1000,[size(v,1) 1]);\r\nunsortedExponents = v(randperm(size(v,1)),:);\r\n[sortedCoeffs,sortedExponents] = sortMonomials(unsortedCoeffs,unsortedExponents);\r\nassert(isequal(sortedExponents,v))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":1011,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2017-07-13T18:24:47.000Z","updated_at":"2017-07-15T05:42:59.000Z","published_at":"2017-07-13T18:25:15.000Z","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\u003eIn\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 44260\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, multivariate polynomials were defined as a sum of monomial terms using|exponents|, a matrix of integers, and|coefficients|, a vector (follow the above link for an explanation). It can be useful to order the monomials. But first we need to define the total degree of a monomial as the sum of the exponents. For example, the total degree 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\u003e5*x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is 1 and the total degree 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\u003ex^3*y^5*z\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is 9.\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\u003eWrite a 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[function [coeffs,exponents] = sortMonomials(coeffs,exponents)]]\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\u003eto sort the monomials. Sort them first by descending total degree, and then for a given total degree, by lexicographical order of the exponents (by the first exponent, then the second, and so on, each in descending order). 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