Problem 44260. Multivariate polynomials - convert monomial form to array

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/44259-product-of-two-multidimensional-polynomials\"><w:r><w:t>Problem 44259</w:t></w:r></w:hyperlink><w:r><w:t> I asked you to multiply two multidimensional polynomials that were represented by an array that is a generalization of the way MATLAB handles one-variable polynomials. However, that representation has at least two problems:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>Defining a polynomial is an indexing headache, with a high probability of errors.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>Polynomials often have a small number of terms, so if they are higher order there will be a lot of wasted storage.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Here, we will represent a polynomial as a sum of monomials. For example, 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) = 2*x^5*y + 3*x*y^5</w:t></w:r><w:r><w:t> is the sum of two monomials in</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</w:t></w:r><w:r><w:t> and</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>y</w:t></w:r><w:r><w:t>. We will represent this by</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>exponents</w:t></w:r><w:r><w:t>, a matrix of integers with each row representing the exponents of one monomial (including zeros); and a column vector</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>coefficients</w:t></w:r><w:r><w:t> for the coefficient of each monomial. For</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)</w:t></w:r><w:r><w:t>, these are</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[exponents = [5 1; 1 5];\ncoefficients = [2; 3];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Let's hedge our bets, though, and create a function that converts this form to the array form. Your task is to create 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 c = coeffArray(exponents,coefficients)]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>that inputs the exponents and coefficients and returns an array as defined in Problem 44259.</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/44259-product-of-two-multidimensional-polynomials">Problem 44259</a> I asked you to multiply two multidimensional polynomials that were represented by an array that is a generalization of the way MATLAB handles one-variable polynomials. However, that representation has at least two problems:<ol><li>Defining a polynomial is an indexing headache, with a high probability of errors.</li><li>Polynomials often have a small number of terms, so if they are higher order there will be a lot of wasted storage.</li></ol>Here, we will represent a polynomial as a sum of monomials. For example, the polynomial <tt>p(x,y) = 2*x^5*y + 3*x*y^5</tt> is the sum of two monomials in <tt>x</tt> and <tt>y</tt>. We will represent this by <tt>exponents</tt>, a matrix of integers with each row representing the exponents of one monomial (including zeros); and a column vector <tt>coefficients</tt> for the coefficient of each monomial. For <tt>p(x,y)</tt>, these are<pre class="language-matlab">exponents = [5 1; 1 5];
coefficients = [2; 3];
</pre>Let's hedge our bets, though, and create a function that converts this form to the array form. Your task is to create a function<pre class="language-matlab">function c = coeffArray(exponents,coefficients)
</pre>that inputs the exponents and coefficients and returns an array as defined in Problem 44259.

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Last Solution submitted on Dec 29, 2020

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Tim on 19 Jul 2017

What is the matrix form for, say, A*x^2+B*x*y+C*y^2+D*x+E*y+F?

Andrew Newell on 20 Jul 2017

Tim, it is

exponents = [2 0; 1 1; 0 2; 1 0; 0 1; 0 0];
coefficients = [A B C D E F]';

Tim on 21 Jul 2017

What I meant to ask was, what is the correct output from coeffArray for that case?

Peng Liu on 13 Sep 2017

@Tim The correct output from coeffArray for A*x^2+B*x*y+C*y^2+D*x+E*y+F should be [0 0 A;0 B D;C E F]

Rafael S.T. Vieira on 21 Aug 2020

Shouldn't the single variable case be a row vector? "a matrix of integers with each row representing the exponents of one monomial" The test suite is representing it as a column. I guess you meant a matrix of integers with each row representing an exponent of one variable or with each row of size n representing the exponents of n variables.

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Problem 44260. Multivariate polynomials - convert monomial form to array

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