Problem 44261. Multivariate polynomials - sort monomials

Created by Andrew Newell

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{"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":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array\"><w:r><w:t>Problem 44260</w:t></w:r></w:hyperlink><w:r><w:t>, 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</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>5*x</w:t></w:r><w:r><w:t> is 1 and the total degree of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>x^3*y^5*z</w:t></w:r><w:r><w:t> is 9.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Write a function</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[function [coeffs,exponents] = sortMonomials(coeffs,exponents)]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>to 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.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Example: Consider the polynomial</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>p(x,y,z) = 3*x - 2 + y^2 +4*z^2</w:t></w:r><w:r><w:t>, which is represented as:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[exponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The sorted version is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[exponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>You can assume that a given combination of exponents is never repeated.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

In <a href = "https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array">Problem 44260</a>, 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 <tt>5*x</tt> is 1 and the total degree of <tt>x^3*y^5*z</tt> is 9.Write a function<pre class="language-matlab">function [coeffs,exponents] = sortMonomials(coeffs,exponents)
</pre>to 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.Example: Consider the polynomial <tt>p(x,y,z) = 3*x - 2 + y^2 +4*z^2</tt>, which is represented as:<pre class="language-matlab">exponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]
</pre>The sorted version is<pre class="language-matlab">exponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].
</pre>You can assume that a given combination of exponents is never repeated.

In <https://www.mathworks.com/matlabcentral/cody/problems/44260-multidimensional-polynomials-convert-monomial-form-to-array Problem 44260>, 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.

Write a function

function [coeffs,exponents] = sortMonomials(coeffs,exponents)

to 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.

Example: Consider the polynomial |p(x,y,z) = 3*x - 2 + y^2 +4*z^2|, which is represented as:

exponents = [1 0 0; 0 0 0; 0 2 0; 0 0 2], coefficients = [3; -2; 1; 4]

The sorted version is

exponents = [0 2 0; 0 0 2; 1 0 0; 0 0 0], coefficients = [1; 3; 1; 4].

You can assume that a given combination of exponents is never repeated.

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Last Solution submitted on Jul 26, 2023

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Peng Liu on 12 Sep 2017

The sorted coefficients in your example should be [1; 4; 3; -2].

William on 23 Jan 2019

The test suite is also comparing against an incorrect result in the last problem.

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