Problem 44259. Product of two multivariate polynomials

Created by Andrew Newell

Solve Later

{"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>MATLAB</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/polynomials.html\"><w:r><w:t>has a few functions</w:t></w:r></w:hyperlink><w:r><w:t> for creating and manipulating single-variable polynomials, but outside of the Symbolic Math Toolbox there is nothing for multivariate polynomials such as</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-x^2</w:t></w:r><w:r><w:t> or</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^2+y^2+z^2-1</w:t></w:r><w:r><w:t>. Generalizing the approach for one variable, we can define an array of coefficients with one dimension for each variable. We will index them in decreasing order, for example if</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) = A + B x^3</w:t></w:r><w:r><w:t>, then the coefficients are</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[c = [B 0 0 A].]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Note this is same order as used by the builtin functions. A couple more examples: if</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) = x - y^2</w:t></w:r><w:r><w:t>, then</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[c = [0 -1; 0 0; 1 0].]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>If</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) = z-2</w:t></w:r><w:r><w:t>, then</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>c</w:t></w:r><w:r><w:t> has dimensions</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>[1 1 2]</w:t></w:r><w:r><w:t> with</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>c(:,:,1) = 1</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>c(:,:,2) = -2</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The challenge is to create a function</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>polyMult</w:t></w:r><w:r><w:t> that takes two arrays of coefficients for polynomials</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>p1</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>p2</w:t></w:r><w:r><w:t> and returns the coefficients 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>p1*p2</w:t></w:r><w:r><w:t>. See the tests for examples.</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>"}]}

MATLAB <a href = "https://www.mathworks.com/help/matlab/polynomials.html">has a few functions</a> for creating and manipulating single-variable polynomials, but outside of the Symbolic Math Toolbox there is nothing for multivariate polynomials such as <tt>y-x^2</tt> or <tt>x^2+y^2+z^2-1</tt>. Generalizing the approach for one variable, we can define an array of coefficients with one dimension for each variable. We will index them in decreasing order, for example if <tt>p(x) = A + B x^3</tt>, then the coefficients are<pre class="language-matlab">c = [B 0 0 A].
</pre>Note this is same order as used by the builtin functions. A couple more examples: if <tt>p(x,y) = x - y^2</tt>, then<pre class="language-matlab">c = [0 -1; 0 0; 1 0].
</pre>If <tt>p(x,y,z) = z-2</tt>, then <tt>c</tt> has dimensions <tt>[1 1 2]</tt> with <tt>c(:,:,1) = 1</tt> and <tt>c(:,:,2) = -2</tt>.The challenge is to create a function <tt>polyMult</tt> that takes two arrays of coefficients for polynomials <tt>p1</tt> and <tt>p2</tt> and returns the coefficients for <tt>p1*p2</tt>. See the tests for examples.

Solve

Solution Stats

49 Solutions
17 Solvers

Last Solution submitted on May 23, 2023

Last 200 Solutions

Problem Comments

1 Comment

1 Comment

Predrag Vukovic on 23 Nov 2020

Can you explain why is the same matrix c = [0 -1; 0 0 ; 1 0] used to represent two different polynomials p(x,y) = x-y^2 (as in problem definition above) and p(x,y) = y - x^2 as in test example 3 on the solution page?

Solution Comments

Show comments

Problem Recent Solvers17

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 44259. Product of two multivariate polynomials

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers17

Suggested Problems

More from this Author9

Problem Tags

Community Treasure Hunt

Problem 44259. Product of two multivariate polynomials

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers17

Suggested Problems

More from this Author9

Problem Tags

Community Treasure Hunt

Players