# Sparse Matrices over a general ring#

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: M = MatrixSpace(QQ['x'], 2, 3, sparse=True); M
Full MatrixSpace of 2 by 3 sparse matrices over
Univariate Polynomial Ring in x over Rational Field
sage: a = M(range(6)); a
[0 1 2]
[3 4 5]
sage: b = M([x^n for n in range(6)]); b
[  1   x x^2]
[x^3 x^4 x^5]
sage: a * b.transpose()
[            2*x^2 + x           2*x^5 + x^4]
[      5*x^2 + 4*x + 3 5*x^5 + 4*x^4 + 3*x^3]
sage: pari(a)*pari(b.transpose())                                                   # needs sage.libs.pari
[2*x^2 + x, 2*x^5 + x^4; 5*x^2 + 4*x + 3, 5*x^5 + 4*x^4 + 3*x^3]
sage: c = copy(b); c
[  1   x x^2]
[x^3 x^4 x^5]
sage: c[0,0] = 5; c
[  5   x x^2]
[x^3 x^4 x^5]
sage: b[0,0]
1
sage: c.dict()
{(0, 0): 5, (0, 1): x, (0, 2): x^2, (1, 0): x^3, (1, 1): x^4, (1, 2): x^5}
sage: c.list()
[5, x, x^2, x^3, x^4, x^5]
sage: c.rows()
[(5, x, x^2), (x^3, x^4, x^5)]
sage: TestSuite(c).run()
sage: d = c.change_ring(CC['x']); d
[5.00000000000000                x              x^2]
[             x^3              x^4              x^5]
sage: latex(c)
\left(\begin{array}{rrr}
5 & x & x^{2} \\
x^{3} & x^{4} & x^{5}
\end{array}\right)
sage: c.sparse_rows()
[(5, x, x^2), (x^3, x^4, x^5)]
sage: d = c.dense_matrix(); d
[  5   x x^2]
[x^3 x^4 x^5]
sage: parent(d)
Full MatrixSpace of 2 by 3 dense matrices
over Univariate Polynomial Ring in x over Rational Field
sage: c.sparse_matrix() is c
True
sage: c.is_sparse()
True

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> M = MatrixSpace(QQ['x'], Integer(2), Integer(3), sparse=True); M
Full MatrixSpace of 2 by 3 sparse matrices over
Univariate Polynomial Ring in x over Rational Field
>>> a = M(range(Integer(6))); a
[0 1 2]
[3 4 5]
>>> b = M([x**n for n in range(Integer(6))]); b
[  1   x x^2]
[x^3 x^4 x^5]
>>> a * b.transpose()
[            2*x^2 + x           2*x^5 + x^4]
[      5*x^2 + 4*x + 3 5*x^5 + 4*x^4 + 3*x^3]
>>> pari(a)*pari(b.transpose())                                                   # needs sage.libs.pari
[2*x^2 + x, 2*x^5 + x^4; 5*x^2 + 4*x + 3, 5*x^5 + 4*x^4 + 3*x^3]
>>> c = copy(b); c
[  1   x x^2]
[x^3 x^4 x^5]
>>> c[Integer(0),Integer(0)] = Integer(5); c
[  5   x x^2]
[x^3 x^4 x^5]
>>> b[Integer(0),Integer(0)]
1
>>> c.dict()
{(0, 0): 5, (0, 1): x, (0, 2): x^2, (1, 0): x^3, (1, 1): x^4, (1, 2): x^5}
>>> c.list()
[5, x, x^2, x^3, x^4, x^5]
>>> c.rows()
[(5, x, x^2), (x^3, x^4, x^5)]
>>> TestSuite(c).run()
>>> d = c.change_ring(CC['x']); d
[5.00000000000000                x              x^2]
[             x^3              x^4              x^5]
>>> latex(c)
\left(\begin{array}{rrr}
5 & x & x^{2} \\
x^{3} & x^{4} & x^{5}
\end{array}\right)
>>> c.sparse_rows()
[(5, x, x^2), (x^3, x^4, x^5)]
>>> d = c.dense_matrix(); d
[  5   x x^2]
[x^3 x^4 x^5]
>>> parent(d)
Full MatrixSpace of 2 by 3 dense matrices
over Univariate Polynomial Ring in x over Rational Field
>>> c.sparse_matrix() is c
True
>>> c.is_sparse()
True

class sage.matrix.matrix_generic_sparse.Matrix_generic_sparse[source]#

Bases: Matrix_sparse

Generic sparse matrix.

The Matrix_generic_sparse class derives from Matrix_sparse, and defines functionality for sparse matrices over any base ring. A generic sparse matrix is represented using a dictionary whose keys are pairs of integers $$(i,j)$$ and values in the base ring. The values of the dictionary must never be zero.

EXAMPLES:

sage: R.<a,b> = PolynomialRing(ZZ,'a,b')
sage: M = MatrixSpace(R,5,5,sparse=True)
sage: M({(0,0):5*a+2*b, (3,4): -a})
[5*a + 2*b         0         0         0         0]
[        0         0         0         0         0]
[        0         0         0         0         0]
[        0         0         0         0        -a]
[        0         0         0         0         0]
sage: M(3)
[3 0 0 0 0]
[0 3 0 0 0]
[0 0 3 0 0]
[0 0 0 3 0]
[0 0 0 0 3]
sage: V = FreeModule(ZZ, 5,sparse=True)
sage: m = M([V({0:3}), V({2:2, 4:-1}), V(0), V(0), V({1:2})])
sage: m
[ 3  0  0  0  0]
[ 0  0  2  0 -1]
[ 0  0  0  0  0]
[ 0  0  0  0  0]
[ 0  2  0  0  0]

>>> from sage.all import *
>>> R = PolynomialRing(ZZ,'a,b', names=('a', 'b',)); (a, b,) = R._first_ngens(2)
>>> M = MatrixSpace(R,Integer(5),Integer(5),sparse=True)
>>> M({(Integer(0),Integer(0)):Integer(5)*a+Integer(2)*b, (Integer(3),Integer(4)): -a})
[5*a + 2*b         0         0         0         0]
[        0         0         0         0         0]
[        0         0         0         0         0]
[        0         0         0         0        -a]
[        0         0         0         0         0]
>>> M(Integer(3))
[3 0 0 0 0]
[0 3 0 0 0]
[0 0 3 0 0]
[0 0 0 3 0]
[0 0 0 0 3]
>>> V = FreeModule(ZZ, Integer(5),sparse=True)
>>> m = M([V({Integer(0):Integer(3)}), V({Integer(2):Integer(2), Integer(4):-Integer(1)}), V(Integer(0)), V(Integer(0)), V({Integer(1):Integer(2)})])
>>> m
[ 3  0  0  0  0]
[ 0  0  2  0 -1]
[ 0  0  0  0  0]
[ 0  0  0  0  0]
[ 0  2  0  0  0]


Note

The datastructure can potentially be optimized. Firstly, as noticed in Issue #17663, we lose time in using 2-tuples to store indices. Secondly, there is no fast way to access non-zero elements in a given row/column.

sage.matrix.matrix_generic_sparse.Matrix_sparse_from_rows(X)[source]#

INPUT:

• X – nonempty list of SparseVector rows

OUTPUT: Sparse_matrix with those rows.

EXAMPLES:

sage: V = VectorSpace(QQ,20,sparse=True)
sage: v = V(0)
sage: v[9] = 4
sage: from sage.matrix.matrix_generic_sparse import Matrix_sparse_from_rows
sage: Matrix_sparse_from_rows([v])
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
sage: Matrix_sparse_from_rows([v, v, v, V(0)])
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

>>> from sage.all import *
>>> V = VectorSpace(QQ,Integer(20),sparse=True)
>>> v = V(Integer(0))
>>> v[Integer(9)] = Integer(4)
>>> from sage.matrix.matrix_generic_sparse import Matrix_sparse_from_rows
>>> Matrix_sparse_from_rows([v])
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
>>> Matrix_sparse_from_rows([v, v, v, V(Integer(0))])
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]