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<head><title>quotientRemainder' -- matrix quotient and remainder (opposite)</title>
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<div><h1>quotientRemainder' -- matrix quotient and remainder (opposite)</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>(q,r) = quotientRemainder'(f,g)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
<li><span><tt>g</tt>, <span>a <a href="___Groebner__Basis.html">Groebner basis</a></span> or <span>a <a href="___Matrix.html">matrix</a></span>, with the same source as <tt>f</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>q</tt>, the quotient of <tt>f</tt> upon (opposite) division by <tt>g</tt></span></li>
<li><span><tt>r</tt>, the remainder of <tt>f</tt> upon (opposite) division by <tt>g</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The equation <tt>q*g+r == f</tt> will hold. The sources and targets of the maps should be free modules. This function is obtained from <tt>quotientRemainder</tt> by transposing the inputs and outputs.<table class="examples"><tr><td><pre>i1 : R = ZZ[x,y]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : f = random(R^{2:1},R^2)
o2 = {-1} | 8y 6x+6y |
{-1} | 6x+4y 0 |
2 2
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : g = transpose (vars R ++ vars R)
o3 = {-1} | x 0 |
{-1} | y 0 |
{-1} | 0 x |
{-1} | 0 y |
4 2
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : (q,r) = quotientRemainder'(f,g)
o4 = ({-1} | 0 8 6 6 |, 0)
{-1} | 6 4 0 0 |
o4 : Sequence</pre>
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<tr><td><pre>i5 : q*g+r == f
o5 = true</pre>
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<tr><td><pre>i6 : f = f + map(target f, source f, id_(R^2))
o6 = {-1} | 8y+1 6x+6y |
{-1} | 6x+4y 1 |
2 2
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : (q,r) = quotientRemainder'(f,g)
o7 = ({-1} | 0 8 6 6 |, {-1} | 1 0 |)
{-1} | 6 4 0 0 | {-1} | 0 1 |
o7 : Sequence</pre>
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<tr><td><pre>i8 : q*g+r == f
o8 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_quotient__Remainder.html" title="matrix quotient and remainder">quotientRemainder</a> -- matrix quotient and remainder</span></li>
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<div class="waystouse"><h2>Ways to use <tt>quotientRemainder'</tt> :</h2>
<ul><li>quotientRemainder'(Matrix,Matrix)</li>
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<div class="waystouse"><div class="single"><h2>Code</h2>
<pre>function 'quotientRemainder'': source code not available</pre>
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