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<div><h1>remainder -- matrix 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) = remainder(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>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><p>This operation is the same as <a href="___Matrix_sp_pc_sp__Groebner__Basis.html" title="calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis">Matrix % GroebnerBasis</a>.</p>
<p>The equation <tt>g*q+r == f</tt> will hold, where <tt>q</tt> is the map provided by <a href="_quotient.html" title="quotient or division">quotient</a>. The source of <tt>f</tt> should be a free module.</p>
<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 = | y 8x |
| 2x 2x+9y |
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 : remainder(f,g)
o4 = 0
2 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 = | y+1 8x |
| 2x 2x+9y+1 |
2 2
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : remainder(f,g)
o6 = | 1 0 |
| 0 1 |
2 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="_remainder_sq.html" title="matrix quotient and remainder (opposite)">remainder'</a> -- matrix quotient and remainder (opposite)</span></li>
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<div class="waystouse"><h2>Ways to use <tt>remainder</tt> :</h2>
<ul><li>remainder(Matrix,GroebnerBasis)</li>
<li>remainder(Matrix,Matrix)</li>
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