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    <title>LLL -- compute an LLL basis</title>
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o1 = | 3   11  31   69   131   223   351   521   739   1011   |
     | 7   36  113  262  507   872   1381  2058  2927  4012   |
     | 13  77  249  583  1133  1953  3097  4619  6573  9013   |
     | 21  134 439  1032 2009  3466  5499  8204  11677 16014  |
     | 31  207 683  1609 3135  5411  8587  12813 18239 25015  |
     | 43  296 981  2314 4511  7788  12361 18446 26259 36016  |
     | 57  401 1333 3147 6137  10597 16821 25103 35737 49017  |
     | 73  522 1739 4108 8013  13838 21967 32784 46673 64018  |
     | 91  659 2199 5197 10139 17511 27799 41489 59067 81019  |
     | 111 812 2713 6414 12515 21616 34317 51218 72919 100020 |

10       10
o1 : Matrix ZZ   &lt;-- ZZ</code></pre>
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              <pre><code class="language-macaulay2">i2 : m = syz m1

o2 = | -3 -1632481 -2819632 -4476373  -6680969  -9511685  -13046786 |
     | 8  4353284  7519023  11937004  17815934  25364520  34791469  |
     | -7 -3809126 -6579152 -10444892 -15588965 -22193990 -30442586 |
     | 2  1088324  1879762  2984262   4454001   6341156   8697904   |
     | 0  -1       0        0         0         0         0         |
     | 0  0        -1       0         0         0         0         |
     | 0  0        0        -1        0         0         0         |
     | 0  0        0        0         -1        0         0         |
     | 0  0        0        0         0         -1        0         |
     | 0  0        0        0         0         0         -1        |

10       7
o2 : Matrix ZZ   &lt;-- ZZ</code></pre>
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              <pre><code class="language-macaulay2">i3 : LLL m

o3 = | 1  0  1  0  1  1  1  |
     | -1 1  0  1  0  -1 -2 |
     | -1 -1 -1 -2 -2 0  1  |
     | 0  -1 -1 0  1  -1 1  |
     | 2  1  -1 1  -1 1  -2 |
     | -1 -1 2  1  1  0  0  |
     | 0  2  0  0  -1 0  2  |
     | 0  -1 1  -2 1  -1 -1 |
     | 0  0  -1 1  1  2  0  |
     | 0  0  0  0  -1 -1 0  |

10       7
o3 : Matrix ZZ   &lt;-- ZZ</code></pre>
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It is also possible to get the change of basis matrix from the original basis to the LLL basis.  For example,        <table class="examples">
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              <pre><code class="language-macaulay2">i4 : (n,c) = LLL(m, Strategy => NTL, ChangeMatrix=>true)

o4 = (| 1  0  1  0  1  1  1  |, | 148443 361542 392022 200620 -47785 309365
      | -1 1  0  1  0  -1 -2 |  | -2     -1     1      -1     1      -1    
      | -1 -1 -1 -2 -2 0  1  |  | 1      1      -2     -1     -1     0     
      | 0  -1 -1 0  1  -1 1  |  | 0      -2     0      0      1      0     
      | 2  1  -1 1  -1 1  -2 |  | 0      1      -1     2      -1     1     
      | -1 -1 2  1  1  0  0  |  | 0      0      1      -1     -1     -2    
      | 0  2  0  0  -1 0  2  |  | 0      0      0      0      1      1     
      | 0  -1 1  -2 1  -1 -1 |
      | 0  0  -1 1  1  2  0  |
      | 0  0  0  0  -1 -1 0  |
     ------------------------------------------------------------------------
     -331062 |)
     2       |
     0       |
     -2      |
     1       |
     0       |
     0       |

o4 : Sequence</code></pre>
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              <pre><code class="language-macaulay2">i5 : m * c == n

o5 = true</code></pre>
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      <div>
        <h2>Caveat</h2>
If the strategy given is not an NTL strategy, then the columns of the matrix m must be linearly independent.In any case, the matrix must be defined over the ring ZZ.      </div>
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        <h2>See also</h2>
        <ul>
          <li><span><a title="is a basis an LLL basis?" href="_is__L__L__L.html">isLLL</a> -- is a basis an LLL basis?</span></li>
          <li><span><a title="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</a> -- compute the gcd of integers, and small multipliers</span></li>
          <li><span><span class="tt">kernelLLL</span> (missing documentation)
</span></li>
          <li><span><span class="tt">hermite</span> (missing documentation)
</span></li>
        </ul>
      </div>
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        <div class="waystouse">
          <h2>Ways to use <span class="tt">LLL</span>:</h2>
          <ul>
            <li><kbd>LLL(Matrix)</kbd></li>
            <li><kbd>LLL(MutableMatrix)</kbd></li>
          </ul>
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        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="compute an LLL basis" href="___L__L__L.html">LLL</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">LLLBases.m2:851:0</span>.</p>
        </div>
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Edit File: _LL__L.html