Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 2 1 0] [-1 1 3] We compute the extended GCD of a list of integers using LLL, this example is from the Magma handbook: sage: Q = [ 67015143, 248934363018, 109210, 25590011055, 74631449, \ 10230248, 709487, 68965012139, 972065, 864972271 ] sage: n = len(Q) sage: S = 100 sage: X = Matrix(ZZ, n, n + 1) sage: for i in xrange(n): ... X[i,i + 1] = 1 sage: for i in xrange(n): ... X[i,0] = S*Q[i] sage: L = X.LLL() sage: M = L.row(n-1).list()[1:] sage: M [-3, -1, 13, -1, -4, 2, 3, 4, 5, -1] sage: add([Q[i]*M[i] for i in range(n)]) -1 TESTS: sage: matrix(ZZ, 0, 0).LLL() [] sage: matrix(ZZ, 3, 0).LLL() [] sage: matrix(ZZ, 0, 3).LLL() [] sage: M = matrix(ZZ, [[1,2,3],[31,41,51],[101,201,301]]) sage: A = M.LLL() sage: A [ 0 0 0] [-1 0 1] [ 1 1 1] sage: B = M.LLL(algorithm=’NTL:LLL’) sage: C = M.LLL(algorithm=’NTL:LLL’, fp=None) sage: D = M.LLL(algorithm=’NTL:LLL’, fp=’fp’) sage: F = M.LLL(algorithm=’NTL:LLL’, fp=’xd’) sage: G = M.LLL(algorithm=’NTL:LLL’, fp=’rr’) sage: A == B == C == D == F == G True sage: H = M.LLL(algorithm=’NTL:LLL’, fp=’qd’) Traceback (most recent call last): ... TypeError: algorithm NTL:LLL_QD not supported ALGORITHM: Uses the NTL library by Victor Shoup or fpLLL library by Damien Stehle depending on the chosen algorithm. REFERENCES: •ntl.mat_ZZ or sage.libs.fplll.fplll for details on the used algorithms. LLL_gram() LLL reduction of the lattice whose gram matrix is self. INPUT: •M - gram matrix of a definite quadratic form 316 Chapter 17. Dense matrices over the integer ring
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 OUTPUT: •U - unimodular transformation matrix such that U.transpose() * M * U is LLL-reduced. ALGORITHM: Use PARI EXAMPLES: sage: M = Matrix(ZZ, 2, 2, [5,3,3,2]) ; M [5 3] [3 2] sage: U = M.LLL_gram(); U [-1 1] [ 1 -2] sage: U.transpose() * M * U [1 0] [0 1] Semidefinite and indefinite forms no longer raise a ValueError: sage: Matrix(ZZ,2,2,[2,6,6,3]).LLL_gram() [-3 -1] [ 1 0] sage: Matrix(ZZ,2,2,[1,0,0,-1]).LLL_gram() [ 0 -1] [ 1 0] antitranspose() Returns the antitranspose of self, without changing self. EXAMPLES: sage: A = matrix(2,3,range(6)) sage: type(A) sage: A.antitranspose() [5 2] [4 1] [3 0] sage: A [0 1 2] [3 4 5] sage: A.subdivide(1,2); A [0 1|2] [---+-] [3 4|5] sage: A.antitranspose() [5|2] [-+-] [4|1] [3|0] augment(right, subdivide=False) Returns a new matrix formed by appending the matrix (or vector) right on the right side of self. INPUT: •right - a matrix, vector or free module element, whose dimensions are compatible with self. •subdivide - default: False - request the resulting matrix to have a new subdivision, separating self from right. 317
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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