Ticket #1937: trac1937_LLL_doc.patch

File trac1937_LLL_doc.patch, 3.3 KB (added by malb, 14 years ago)
  • sage/matrix/matrix_integer_dense.pyx

    # HG changeset patch
    # User Martin Albrecht <malb@informatik.uni-bremen.de>
    # Date 1201345112 0
    # Node ID e1d15edce9a1a181024f7bb023e66667a915536e
    # Parent  91998f13b187fc782b1f2ec71ecf8005a9c5f825
    documentation fixes for LLL documentation (fixes #1937)
    
    diff -r 91998f13b187 -r e1d15edce9a1 sage/matrix/matrix_integer_dense.pyx
    a b cdef class Matrix_integer_dense(matrix_d 
    15481548    def LLL(self, delta=None, eta=None, algorithm=None, fp=None, prec=0, early_red = False, use_givens = False):
    15491549        r"""
    15501550        Returns LLL reduced or approximated LLL reduced lattice R for
    1551         self.
     1551        this matrix interpreted as a lattice.
    15521552
    1553         A lattice is (delta, eta)-LLL-reduce if the two following
    1554         conditions hold:
     1553        A lattice $(b_1, b_2, ..., b_d)$ is $(\delta, \eta)$-LLL-reduced
     1554        if the two following conditions hold:
    15551555       
    15561556        (a) For any $i>j$, we have $|mu_{i, j}| <= \eta$,
    15571557        (b) For any $i<d$, we have
    15581558        $\delta |b_i^*|^2 <= |b_{i + 1}^* + mu_{i + 1, i} b_{i + 1}^* |^2$,
    15591559
    1560         The lattice is returned as a matrix. Also the rank (and the
    1561         determinant) of self are cached if those are computed during
    1562         the reduction.
     1560        where $mu_{i,j} = <b_i, b_j^*>/<b_j^*,b_j^*>$ and $b_i^*$ is
     1561        the $i$-th vector of the Gram-Schmidt orthogonalisation of
     1562        $(b_1, b_2, ..., b_d)$.
    15631563
    15641564        The default reduction parameters are $\delta=3/4$ and
    15651565        $eta=0.501$. The parameters $\delta$ and $\eta$ must satisfy:
    1566         $0.25 < \delta <= 1.0$ and $0.5 <= \eta < sqrt(\delta)$.
     1566        $0.25 < \delta <= 1.0$ and $0.5 <= \eta <
     1567        sqrt(\delta)$. Polynomial time complexity is only guaranteed
     1568        for $\delta < 1$.
    15671569
    1568         If we can compute the determinant of self using this method,
    1569         we also cache it. Note that in general this only happens when
     1570        The lattice is returned as a matrix. Also the rank (and the
     1571        determinant) of self are cached if those are computed during
     1572        the reduction. Note that in general this only happens when
    15701573        self.rank() == self.ncols() and the exact algorithm is used.
    15711574
    15721575        INPUT:
    cdef class Matrix_integer_dense(matrix_d 
    15851588            use_givens -- use Givens orthogonalization (default: False)
    15861589                          only applicable to approximate reductions and NTL.
    15871590                          This is more stable but slower.
    1588                          
    15891591
    15901592        Also, if the verbose level is >= 2, some more verbose output
    15911593        is printed during the calculation if NTL is used.
    cdef class Matrix_integer_dense(matrix_d 
    16251627        sage: add([Q[i]*M[i] for i in range(n)])
    16261628        -1
    16271629           
    1628         ALGORITHM: Uses NTL or fpLLL.
     1630        ALGORITHM: Uses the NTL library by Victor Shoup or fpLLL
     1631        library by Damien Stehle depending on the chosen algorithm.
    16291632
    1630         REFERENCES:
    1631             ntl.mat_ZZ or sage.libs.fplll.fplll for details on the
    1632             used algorithms.
     1633        REFERENCES: \code{ntl.mat_ZZ} or \code{sage.libs.fplll.fplll}
     1634            for details on the used algorithms.
    16331635        """
    16341636
    16351637        import sage.libs.ntl.all