This is a compiled implementation of sparse matrices over
for
small.
TODO: - move vectors into a pyrex vector class - add _add_ and _mul_ methods.
EXAMPLES:
sage: a = matrix(Integers(37),3,3,range(9),sparse=True); a
[0 1 2]
[3 4 5]
[6 7 8]
sage: type(a)
<type 'sage.matrix.matrix_modn_sparse.Matrix_modn_sparse'>
sage: parent(a)
Full MatrixSpace of 3 by 3 sparse matrices over Ring of integers modulo 37
sage: a^2
[15 18 21]
[ 5 17 29]
[32 16 0]
sage: a+a
[ 0 2 4]
[ 6 8 10]
[12 14 16]
sage: b = a.new_matrix(2,3,range(6)); b
[0 1 2]
[3 4 5]
sage: a*b
...
TypeError: unsupported operand parent(s) for '*': 'Full MatrixSpace of 3 by 3 sparse matrices over Ring of integers modulo 37' and 'Full MatrixSpace of 2 by 3 sparse matrices over Ring of integers modulo 37'
sage: b*a
[15 18 21]
[ 5 17 29]
sage: a == loads(dumps(a))
True
sage: b == loads(dumps(b))
True
sage: a.echelonize(); a
[ 1 0 36]
[ 0 1 2]
[ 0 0 0]
sage: b.echelonize(); b
[ 1 0 36]
[ 0 1 2]
sage: a.pivots()
[0, 1]
sage: b.pivots()
[0, 1]
sage: a.rank()
2
sage: b.rank()
2
sage: a[2,2] = 5
sage: a.rank()
3
Unsafe version of the dict method, mainly for internal use. This may return the dict of elements, but as an unsafe reference to the underlying dict of the object. It might be dangerous if you change entries of the returned dict.
Replace self by its reduction to reduced row echelon form.
ALGORITHM: We use Gauss elimination, in a slightly intelligent way, in that we clear each column using a row with the minimum number of nonzero entries.
TODO: Implement switching to a dense method when the matrix gets dense.
Multiply self by the sparse matrix _right, and return the result as a dense matrix.
sage: a = matrix(GF(10007), 2, [1,2,3,4], sparse=True) sage: b = matrix(GF(10007), 2, 3, [1..6], sparse=True) sage: a * b [ 9 12 15] [19 26 33] sage: c = a._matrix_times_matrix_dense(b); c [ 9 12 15] [19 26 33] sage: type(c) <type ‘sage.matrix.matrix_modn_dense.Matrix_modn_dense’>
sage: a = matrix(GF(2), 20, 20, sparse=True) sage: a*a == a._matrix_times_matrix_dense(a) True sage: type(a._matrix_times_matrix_dense(a)) <type ‘sage.matrix.matrix_mod2_dense.Matrix_mod2_dense’>
Returns the list of pairs (i,j) such that self[i,j] != 0.
It is safe to change the resulting list (unless you give the option copy=False).
TESTS:
sage: M = Matrix( GF(2), [[1,1,1,1,0,0,0,0,0,0]], sparse=True )
sage: loads(dumps(M))
[1 1 1 1 0 0 0 0 0 0]
sage: loads(dumps(M)) == M
True
If self is a matrix , then this function returns a
vector or matrix
such that
. If
is a vector then
is a vector and if
is a matrix, then
is a matrix.
Note
In Sage one can also write A B for A.solve_right(B), i.e., Sage implements the “the MATLAB/Octave backslash operator”.
INPUT:
OUTPUT: a matrix or vector
EXAMPLES:
sage: A = matrix(GF(127), 3, [1,2,3,-1,2,5,2,3,1], sparse=True)
sage: b = vector(GF(127),[1,2,3])
sage: x = A \ b; x
(73, 76, 10)
sage: A * x
(1, 2, 3)
Return the density of self, i.e., the ratio of the number of nonzero entries of self to the total size of self.
EXAMPLES:
sage: A = matrix(QQ,3,3,[0,1,2,3,0,0,6,7,8],sparse=True)
sage: A.density()
2/3
Notice that the density parameter does not ensure the density of a matrix; it is only an upper bound.
sage: A = random_matrix(GF(127),200,200,density=0.3, sparse=True)
sage: A.density()
257/1000
Return lift of this matrix to a sparse matrix over the integers.
Subdivisions are preserved when lifting:
sage: a.subdivide([], [1,1]); a
[1||2 3]
[4||5 6]
sage: a.lift()
[1||2 3]
[4||5 6]
Return the matrix constructed from self using columns with indices in the columns list.
EXAMPLES:
sage: M = MatrixSpace(GF(127),3,3,sparse=True)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: A.matrix_from_columns([2,1])
[2 1]
[5 4]
[8 7]
Return the matrix constructed from self using rows with indices in the rows list.
INPUT:
EXAMPLE:
sage: M = MatrixSpace(GF(127),3,3,sparse=True)
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: A.matrix_from_rows([2,1])
[6 7 8]
[3 4 5]
Compute the rank of self.
INPUT:
EXAMPLE:
sage: A = random_matrix(GF(127),200,200,density=0.01,sparse=True)
sage: r1 = A.rank(gauss=False)
sage: r2 = A.rank(gauss=True)
sage: r3 = A.rank(gauss='native')
sage: r1 == r2 == r3
True
sage: r1
155
ALGORITHM: Uses LinBox or native implementation.
REFERENCES:
Note
For very sparse matrices Gaussian elimination is faster because it barly has anything to do. If the fill in needs to be considered, ‘Symbolic Reordering’ is usually much faster.
Return the transpose of self.
EXAMPLE:
sage: A = matrix(GF(127),3,3,[0,1,0,2,0,0,3,0,0],sparse=True)
sage: A
[0 1 0]
[2 0 0]
[3 0 0]
sage: A.transpose()
[0 2 3]
[1 0 0]
[0 0 0]
Write a PNG image to ‘filename’ which visualizes self by putting black pixels in those positions which have nonzero entries.
White pixels are put at positions with zero entries. If ‘maxsize’ is given, then the maximal dimension in either x or y direction is set to ‘maxsize’ depending on which is bigger. If the image is scaled, the darkness of the pixel reflects how many of the represented entries are nonzero. So if e.g. one image pixel actually represents a 2x2 submatrix, the dot is darker the more of the four values are nonzero.
INPUT: