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<div><h1>quotientRemainder -- matrix quotient and remainder</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 target 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 division by <tt>g</tt></span></li>
<li><span><tt>r</tt>, the remainder of <tt>f</tt> upon division by <tt>g</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The equation <tt>g*q+r == f</tt> will hold. The source of <tt>f</tt> should be a free module.<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,R^{2:-1})
o2 = | 4x+7y 4x+3y |
| 3x+6y 6x+7y |
2 2
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : g = vars R ++ vars R
o3 = | x y 0 0 |
| 0 0 x y |
2 4
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : (q,r) = quotientRemainder(f,g)
o4 = ({1} | 4 4 |, 0)
{1} | 7 3 |
{1} | 3 6 |
{1} | 6 7 |
o4 : Sequence</pre>
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<tr><td><pre>i5 : g*q+r == f
o5 = true</pre>
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<tr><td><pre>i6 : f = f + map(target f, source f, id_(R^2))
o6 = | 4x+7y+1 4x+3y |
| 3x+6y 6x+7y+1 |
2 2
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : (q,r) = quotientRemainder(f,g)
o7 = ({1} | 4 4 |, | 1 0 |)
{1} | 7 3 | | 0 1 |
{1} | 3 6 |
{1} | 6 7 |
o7 : Sequence</pre>
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<tr><td><pre>i8 : g*q+r == f
o8 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_quotient__Remainder_sq.html" title="matrix quotient and remainder (opposite)">quotientRemainder'</a> -- matrix quotient and remainder (opposite)</span></li>
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<div class="waystouse"><h2>Ways to use <tt>quotientRemainder</tt> :</h2>
<ul><li>quotientRemainder(Matrix,GroebnerBasis)</li>
<li>quotientRemainder(Matrix,Matrix)</li>
<li><span><tt>quotientRemainder(InexactNumber,RingElement)</tt> (missing documentation<!-- tag: (quotientRemainder,InexactNumber,RingElement) -->)</span></li>
<li><span><tt>quotientRemainder(RingElement,InexactNumber)</tt> (missing documentation<!-- tag: (quotientRemainder,RingElement,InexactNumber) -->)</span></li>
<li><span>quotientRemainder(Number,RingElement), see <span><a href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html" title="quotient and remainder">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></span></li>
<li><span>quotientRemainder(RingElement,Number), see <span><a href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html" title="quotient and remainder">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></span></li>
<li><span><a href="_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html" title="quotient and remainder">quotientRemainder(RingElement,RingElement)</a> -- quotient and remainder</span></li>
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