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<head><title>quotient(Matrix,GroebnerBasis) -- matrix quotient</title>
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<div><h1>quotient(Matrix,GroebnerBasis) -- matrix quotient</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) = quotient(f,g)</tt></div>
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<li><span>Function: <a href="_quotient.html" title="quotient or division">quotient</a></span></li>
<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>, <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>, <span>a <a href="___Matrix.html">matrix</a></span>, the quotient of <tt>f</tt> upon division by <tt>g</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_quotient_lp..._cm_sp__Basis__Element__Limit_sp_eq_gt_sp..._rp.html">BasisElementLimit => ...</a>, </span></li>
<li><span><a href="_quotient_lp..._cm_sp__Degree__Limit_sp_eq_gt_sp..._rp.html">DegreeLimit => ...</a>, </span></li>
<li><span><a href="_quotient_lp..._cm_sp__Minimal__Generators_sp_eq_gt_sp..._rp.html">MinimalGenerators => ...</a>, -- Decides whether quotient computes and outputs a trimmed set of generators; default is true</span></li>
<li><span><a href="_quotient_lp..._cm_sp__Pair__Limit_sp_eq_gt_sp..._rp.html">PairLimit => ...</a>, </span></li>
<li><span><a href="_quotient_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- Possible strategies are: Iterate, Linear, and Quotient</span></li>
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<div class="single"><h2>Description</h2>
<div>The equation <tt>g*q+r == f</tt> will hold, where <tt>r</tt> is the map provided by <a href="_remainder.html" title="matrix remainder">remainder</a>. 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 = | 7x+2y 3x |
| 4x 6x |
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 : quotient(f,g)
o4 = {1} | 7 3 |
{1} | 2 0 |
{1} | 4 6 |
{1} | 0 0 |
4 2
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : f = f + map(target f, source f, id_(R^2))
o5 = | 7x+2y+1 3x |
| 4x 6x+1 |
2 2
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : quotient(f,g)
o6 = {1} | 7 3 |
{1} | 2 0 |
{1} | 4 6 |
{1} | 0 0 |
4 2
o6 : Matrix R <--- R</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>
<li><span><a href="_quotient_sq.html" title="matrix quotient (opposite)">quotient'</a> -- matrix quotient (opposite)</span></li>
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