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<head><title>LLL(..., ChangeMatrix => ...) -- also find change of basis matrix</title>
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<div><h1>LLL(..., ChangeMatrix => ...) -- also find change of basis matrix</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>(B,U) = LLL(A,ChangeMatrix=>true)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, over the integers, of size d by n</span></li>
</ul>
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</li>
<li><div class="single">Consequences:<ul><li>The routine returns a pair of matrices, rather than just one</li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>B</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the LLL matrix (also having d rows)</span></li>
<li><span><tt>U</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the n by n invertible transform matrix</span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div>Constructs the change of basis matrix U from the basis A to the basis B. This is an invertible matrix U such that<i>AU = B</i>.<table class="examples"><tr><td><pre>i1 : A = matrix randomMutableMatrix(10,10,.5,100000)
o1 = | 237 38588 -40039 0 0 0 -20475 0 -76935 0
| 0 0 0 -79732 94821 0 30541 0 0 0
| 46152 -93180 -41278 0 -21080 -97290 -99965 55546 -70145 -56762
| -76501 0 0 99877 0 0 0 0 -66526 0
| -55912 78522 -52131 0 -47791 0 0 -11760 0 0
| 0 0 1881 -96808 9018 -3842 0 0 0 0
| 0 -50664 0 0 0 9706 -61055 71458 0 85044
| 0 25600 0 0 0 -61364 0 0 0 0
| 0 -13808 52032 0 -94055 -3696 0 0 80482 5067
| 24721 0 0 -94916 0 0 0 81413 -6404 69831
------------------------------------------------------------------------
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10 10
o1 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i2 : (B,U) = LLL(A, ChangeMatrix=>true)
o2 = (| -19024 -19024 57072 -1991 19261 -19024 1152 -1451 -18484 313
| 30541 30541 3198 33739 -30541 30541 33739 0 -3198 21485
| 34493 19475 2767 -4018 11659 -77815 8393 -22150 -40401 12768
| 0 0 0 0 -76501 0 -66526 0 0 -33175
| -26391 -14631 7862 -70660 -29521 -14631 33602 14631 58900 -3456
| -1881 1961 10819 10819 1881 -1881 7057 1881 -10819 -71875
| -10391 -6511 13707 3316 10391 3195 3316 -64250 7087 9948
| -25600 35764 15436 -10164 25600 -25600 -10164 25600 10164 -30492
| -38224 -29461 6787 20595 38224 -33157 -2987 33157 -20595 14621
| 0 -11582 23164 23164 24721 -11582 16760 11582 58249 -38232
------------------------------------------------------------------------
|, | 0 0 0 0 1 0 0 0 0 0 |)
| | -1 -1 3 2 1 -1 2 1 -2 6 |
| | -1 -1 3 3 1 -1 1 1 -3 5 |
| | 0 0 0 0 0 0 0 0 0 1 |
| | 0 0 1 1 0 0 1 0 -1 3 |
| | 0 -1 1 1 0 0 1 0 -1 3 |
| | 1 1 -3 -2 -1 1 -2 0 3 -6 |
| | 0 -1 2 2 0 -1 2 1 -1 6 |
| | 0 0 0 0 0 0 1 0 0 2 |
| | 0 1 -2 -2 0 1 -2 -1 2 -6 |
o2 : Sequence</pre>
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<tr><td><pre>i3 : B == A*U
o3 = true</pre>
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<h2>Further information</h2>
<ul><li><span>Default value: <a href="../../Macaulay2Doc/html/_false.html" title="">false</a></span></li>
<li><span>Function: <span><a href="___L__L__L.html" title="compute an LLL basis">LLL</a> -- compute an LLL basis</span></span></li>
<li><span>Option name: <span><a href="../../Macaulay2Doc/html/___Change__Matrix.html" title="name for an optional argument">ChangeMatrix</a> -- name for an optional argument</span></span></li>
</ul>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="lattice reduction (Lenstra-Lenstra-Lovasz bases)">LLLBases</a> -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</span></li>
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