# Elements of free modules#

AUTHORS:

EXAMPLES: We create a vector space over $$\QQ$$ and a subspace of this space.

sage: V = QQ^5
sage: W = V.span([V.1, V.2])


Arithmetic operations always return something in the ambient space, since there is a canonical map from $$W$$ to $$V$$ but not from $$V$$ to $$W$$.

sage: parent(W.0 + V.1)
Vector space of dimension 5 over Rational Field
sage: parent(V.1 + W.0)
Vector space of dimension 5 over Rational Field
sage: W.0 + V.1
(0, 2, 0, 0, 0)
sage: W.0 - V.0
(-1, 1, 0, 0, 0)


Next we define modules over $$\ZZ$$ and a finite field.

sage: K = ZZ^5
sage: M = GF(7)^5


Arithmetic between the $$\QQ$$ and $$\ZZ$$ modules is defined, and the result is always over $$\QQ$$, since there is a canonical coercion map to $$\QQ$$.

sage: K.0 + V.1
(1, 1, 0, 0, 0)
sage: parent(K.0 + V.1)
Vector space of dimension 5 over Rational Field


Since there is no canonical coercion map to the finite field from $$\QQ$$ the following arithmetic is not defined:

sage: V.0 + M.0
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Vector space of dimension 5 over Rational Field' and
'Vector space of dimension 5 over Finite Field of size 7'


However, there is a map from $$\ZZ$$ to the finite field, so the following is defined, and the result is in the finite field.

sage: w = K.0 + M.0; w
(2, 0, 0, 0, 0)
sage: parent(w)
Vector space of dimension 5 over Finite Field of size 7
sage: parent(M.0 + K.0)
Vector space of dimension 5 over Finite Field of size 7


Matrix vector multiply:

sage: MS = MatrixSpace(QQ,3)
sage: A = MS([0,1,0,1,0,0,0,0,1])
sage: V = QQ^3
sage: v = V([1,2,3])
sage: v * A
(2, 1, 3)

class sage.modules.free_module_element.FreeModuleElement#

Bases: Vector

An element of a generic free module.

Mod(p)#

EXAMPLES:

sage: V = vector(ZZ, [5, 9, 13, 15])
sage: V.Mod(7)
(5, 2, 6, 1)
sage: parent(V.Mod(7))
Vector space of dimension 4 over Ring of integers modulo 7


Return the additive order of self.

EXAMPLES:

sage: v = vector(Integers(4), [1,2])
4

sage: v = vector([1,2,3])
+Infinity

sage: v = vector(Integers(30), [6, 15]); v
(6, 15)
10
sage: 10*v
(0, 0)

apply_map(phi, R=None, sparse=None)#

Apply the given map phi (an arbitrary Python function or callable object) to this free module element. If R is not given, automatically determine the base ring of the resulting element.

INPUT:
sparse – True or False will control whether the result

is sparse. By default, the result is sparse iff self is sparse.

• phi - arbitrary Python function or callable object

• R - (optional) ring

OUTPUT: a free module element over R

EXAMPLES:

sage: m = vector([1,x,sin(x+1)])                                            # needs sage.symbolic
sage: m.apply_map(lambda x: x^2)                                            # needs sage.symbolic
(1, x^2, sin(x + 1)^2)
sage: m.apply_map(sin)                                                      # needs sage.symbolic
(sin(1), sin(x), sin(sin(x + 1)))

sage: m = vector(ZZ, 9, range(9))
sage: k.<a> = GF(9)                                                         # needs sage.rings.finite_rings
sage: m.apply_map(k)                                                        # needs sage.rings.finite_rings
(0, 1, 2, 0, 1, 2, 0, 1, 2)


In this example, we explicitly specify the codomain.

sage: s = GF(3)
sage: f = lambda x: s(x)
sage: n = m.apply_map(f, k); n                                              # needs sage.rings.finite_rings
(0, 1, 2, 0, 1, 2, 0, 1, 2)
sage: n.parent()                                                            # needs sage.rings.finite_rings
Vector space of dimension 9 over Finite Field in a of size 3^2


If your map sends 0 to a non-zero value, then your resulting vector is not mathematically sparse:

sage: v = vector([0] * 6 + [1], sparse=True); v
(0, 0, 0, 0, 0, 0, 1)
sage: v2 = v.apply_map(lambda x: x+1); v2
(1, 1, 1, 1, 1, 1, 2)


but it’s still represented with a sparse data type:

sage: parent(v2)
Ambient sparse free module of rank 7 over the principal ideal domain Integer Ring


This data type is inefficient for dense vectors, so you may want to specify sparse=False:

sage: v2 = v.apply_map(lambda x: x+1, sparse=False); v2
(1, 1, 1, 1, 1, 1, 2)
sage: parent(v2)
Ambient free module of rank 7 over the principal ideal domain Integer Ring


Or if you have a map that will result in mostly zeroes, you may want to specify sparse=True:

sage: v = vector(srange(10))
sage: v2 = v.apply_map(lambda x: 0 if x else 1, sparse=True); v2
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
sage: parent(v2)
Ambient sparse free module of rank 10 over the principal ideal domain Integer Ring

change_ring(R)#

Change the base ring of this vector.

EXAMPLES:

sage: v = vector(QQ['x,y'], [1..5]); v.change_ring(GF(3))
(1, 2, 0, 1, 2)

column()#

Return a matrix with a single column and the same entries as the vector self.

OUTPUT:

A matrix over the same ring as the vector (or free module element), with a single column. The entries of the column are identical to those of the vector, and in the same order.

EXAMPLES:

sage: v = vector(ZZ, [1,2,3])
sage: w = v.column(); w
[1]
[2]
[3]
sage: w.parent()
Full MatrixSpace of 3 by 1 dense matrices over Integer Ring

sage: x = vector(FiniteField(13), [2,4,8,16])
sage: x.column()
[2]
[4]
[8]
[3]


There is more than one way to get one-column matrix from a vector. The column method is about equally efficient to making a row and then taking a transpose. Notice that supplying a vector to the matrix constructor demonstrates Sage’s preference for rows.

sage: x = vector(RDF, [sin(i*pi/20) for i in range(10)])                    # needs sage.libs.pari sage.symbolic
sage: x.column() == matrix(x).transpose()
True
sage: x.column() == x.row().transpose()
True


Sparse or dense implementations are preserved.

sage: d = vector(RR, [1.0, 2.0, 3.0])
sage: s = vector(CDF, {2: 5.0+6.0*I})                                       # needs sage.symbolic
sage: dm = d.column()
sage: sm = s.column()                                                       # needs sage.symbolic
sage: all([d.is_dense(), dm.is_dense(), s.is_sparse(), sm.is_sparse()])     # needs sage.symbolic
True

conjugate()#

Returns a vector where every entry has been replaced by its complex conjugate.

OUTPUT:

A vector of the same length, over the same ring, but with each entry replaced by the complex conjugate, as implemented by the conjugate() method for elements of the base ring, which is presently always complex conjugation.

EXAMPLES:

sage: v = vector(CDF, [2.3 - 5.4*I, -1.7 + 3.6*I])                          # needs sage.symbolic
sage: w = v.conjugate(); w                                                  # needs sage.symbolic
(2.3 + 5.4*I, -1.7 - 3.6*I)
sage: w.parent()                                                            # needs sage.symbolic
Vector space of dimension 2 over Complex Double Field


Even if conjugation seems nonsensical over a certain ring, this method for vectors cooperates silently.

sage: u = vector(ZZ, range(6))
sage: u.conjugate()
(0, 1, 2, 3, 4, 5)


Sage implements a few specialized subfields of the complex numbers, such as the cyclotomic fields. This example uses such a field containing a primitive 7-th root of unity named a.

sage: # needs sage.rings.number_field
sage: F.<a> = CyclotomicField(7)
sage: v = vector(F, [a^i for i in range(7)])
sage: v
(1, a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1)
sage: v.conjugate()
(1, -a^5 - a^4 - a^3 - a^2 - a - 1, a^5, a^4, a^3, a^2, a)


Sparse vectors are returned as such.

sage: # needs sage.symbolic
sage: v = vector(CC, {1: 5 - 6*I, 3: -7*I}); v
(0.000000000000000, 5.00000000000000 - 6.00000000000000*I, 0.000000000000000, -7.00000000000000*I)
sage: v.is_sparse()
True
sage: vc = v.conjugate(); vc
(0.000000000000000, 5.00000000000000 + 6.00000000000000*I, 0.000000000000000, 7.00000000000000*I)
sage: vc.conjugate()
(0.000000000000000, 5.00000000000000 - 6.00000000000000*I, 0.000000000000000, -7.00000000000000*I)

coordinate_ring()#

Return the ring from which the coefficients of this vector come.

This is different from base_ring(), which returns the ring of scalars.

EXAMPLES:

sage: M = (ZZ^2) * (1/2)
sage: v = M([0,1/2])
sage: v.base_ring()
Integer Ring
sage: v.coordinate_ring()
Rational Field

cross_product(right)#

Return the cross product of self and right, which is only defined for vectors of length 3 or 7.

INPUT:

• right - A vector of the same size as self, either degree three or degree seven.

OUTPUT:

The cross product (vector product) of self and right, a vector of the same size of self and right.

This product is performed under the assumption that the basis vectors are orthonormal. See the method cross_product() of vector fields for more general cases.

EXAMPLES:

sage: v = vector([1,2,3]); w = vector([0,5,-9])
sage: v.cross_product(v)
(0, 0, 0)
sage: u = v.cross_product(w); u
(-33, 9, 5)
sage: u.dot_product(v)
0
sage: u.dot_product(w)
0


The cross product is defined for degree seven vectors as well: see Wikipedia article Cross_product. The 3-D cross product is achieved using the quaternions, whereas the 7-D cross product is achieved using the octonions.

sage: u = vector(QQ, [1, -1/3, 57, -9, 56/4, -4,1])
sage: v = vector(QQ, [37, 55, -99/57, 9, -12, 11/3, 4/98])
sage: u.cross_product(v)
(1394815/2793, -2808401/2793, 39492/49, -48737/399, -9151880/2793, 62513/2793, -326603/171)


The degree seven cross product is anticommutative.

sage: u.cross_product(v) + v.cross_product(u)
(0, 0, 0, 0, 0, 0, 0)


The degree seven cross product is distributive across addition.

sage: v = vector([-12, -8/9, 42, 89, -37, 60/99, 73])
sage: u = vector([31, -42/7, 97, 80, 30/55, -32, 64])
sage: w = vector([-25/4, 40, -89, -91, -72/7, 79, 58])
sage: v.cross_product(u + w) - (v.cross_product(u) + v.cross_product(w))
(0, 0, 0, 0, 0, 0, 0)


The degree seven cross product respects scalar multiplication.

sage: v = vector([2, 17, -11/5, 21, -6, 2/17, 16])
sage: u = vector([-8, 9, -21, -6, -5/3, 12, 99])
sage: (5*v).cross_product(u) - 5*(v.cross_product(u))
(0, 0, 0, 0, 0, 0, 0)
sage: v.cross_product(5*u) - 5*(v.cross_product(u))
(0, 0, 0, 0, 0, 0, 0)
sage: (5*v).cross_product(u) - (v.cross_product(5*u))
(0, 0, 0, 0, 0, 0, 0)


The degree seven cross product respects the scalar triple product.

sage: v = vector([2,6,-7/4,-9/12,-7,12,9])
sage: u = vector([22,-7,-9/11,12,15,15/7,11])
sage: w = vector([-11,17,19,-12/5,44,21/56,-8])
sage: v.dot_product(u.cross_product(w)) - w.dot_product(v.cross_product(u))
0


AUTHOR:

Billy Wonderly (2010-05-11), Added 7-D Cross Product

cross_product_matrix()#

Return the matrix which describes a cross product between self and some other vector.

This operation is sometimes written using the hat operator: see Wikipedia article Hat_operator#Cross_product. It is only defined for vectors of length 3 or 7. For a vector $$v$$ the cross product matrix $$\hat{v}$$ is a matrix which satisfies $$\hat{v} \cdot w = v \times w$$ and also $$w \cdot \hat{v} = w \times v$$ for all vectors $$w$$. The basis vectors are assumed to be orthonormal.

OUTPUT:

The cross product matrix of this vector.

EXAMPLES:

sage: v = vector([1, 2, 3])
sage: vh = v.cross_product_matrix()
sage: vh
[ 0 -3  2]
[ 3  0 -1]
[-2  1  0]
sage: w = random_vector(3, x=1, y=100)
sage: vh*w == v.cross_product(w)
True
sage: w*vh == w.cross_product(v)
True
sage: vh.is_alternating()
True

curl(variables=None)#

Return the curl of this two-dimensional or three-dimensional vector function.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: vector([-y, x, 0]).curl()
(0, 0, 2)
sage: vector([y, -x, x*y*z]).curl()
(x*z, -y*z, -2)
sage: vector([y^2, 0, 0]).curl()
(0, 0, -2*y)
sage: (R^3).random_element().curl().div()
0


For rings where the variable order is not well defined, it must be defined explicitly:

sage: v = vector(SR, [-y, x, 0])                                            # needs sage.symbolic
sage: v.curl()                                                              # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: Unable to determine ordered variable names for Symbolic Ring
sage: v.curl([x, y, z])                                                     # needs sage.symbolic
(0, 0, 2)


Note that callable vectors have well defined variable orderings:

sage: v(x, y, z) = (-y, x, 0)                                               # needs sage.symbolic
sage: v.curl()                                                              # needs sage.symbolic
(x, y, z) |--> (0, 0, 2)


In two dimensions, this returns a scalar value:

sage: R.<x,y> = QQ[]
sage: vector([-y, x]).curl()
2


curl() of vector fields on Euclidean spaces (and more generally pseudo-Riemannian manifolds), in particular for computing the curl in curvilinear coordinates.

degree()#

Return the degree of this vector, which is simply the number of entries.

EXAMPLES:

sage: sage.modules.free_module_element.FreeModuleElement(QQ^389).degree()
389
sage: vector([1,2/3,8]).degree()
3

denominator()#

Return the least common multiple of the denominators of the entries of self.

EXAMPLES:

sage: v = vector([1/2,2/5,3/14])
sage: v.denominator()
70
sage: 2*5*7
70

sage: M = (ZZ^2)*(1/2)
sage: M.basis()[0].denominator()
2

dense_vector()#

Return dense version of self. If self is dense, just return self; otherwise, create and return correspond dense vector.

EXAMPLES:

sage: vector([-1,0,3,0,0,0]).dense_vector().is_dense()
True
sage: vector([-1,0,3,0,0,0],sparse=True).dense_vector().is_dense()
True
sage: vector([-1,0,3,0,0,0],sparse=True).dense_vector()
(-1, 0, 3, 0, 0, 0)

derivative(*args)#

Derivative with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

diff() is an alias of this function.

EXAMPLES:

sage: # needs sage.symbolic
sage: v = vector([1,x,x^2])
sage: v.derivative(x)
(0, 1, 2*x)
sage: type(v.derivative(x)) == type(v)
True
sage: v = vector([1,x,x^2], sparse=True)
sage: v.derivative(x)
(0, 1, 2*x)
sage: type(v.derivative(x)) == type(v)
True
sage: v.derivative(x,x)
(0, 0, 2)

dict(copy=True)#

Return dictionary of nonzero entries of self.

More precisely, this returns a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

INPUT:

• copy – (default: True) if self is internally represented by a dictionary d, then make a copy of d; if False, then this can cause undesired behavior by mutating d

OUTPUT:

• Python dictionary

EXAMPLES:

sage: v = vector([0,0,0,0,1/2,0,3/14])
sage: v.dict()
{4: 1/2, 6: 3/14}
sage: sorted(v.support())
[4, 6]


In some cases, when copy=False, we get back a dangerous reference:

sage: v = vector({0:5, 2:3/7}, sparse=True)
sage: v.dict(copy=False)
{0: 5, 2: 3/7}
sage: v.dict(copy=False)[0] = 18
sage: v
(18, 0, 3/7)

diff(*args)#

Derivative with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

diff() is an alias of this function.

EXAMPLES:

sage: # needs sage.symbolic
sage: v = vector([1,x,x^2])
sage: v.derivative(x)
(0, 1, 2*x)
sage: type(v.derivative(x)) == type(v)
True
sage: v = vector([1,x,x^2], sparse=True)
sage: v.derivative(x)
(0, 1, 2*x)
sage: type(v.derivative(x)) == type(v)
True
sage: v.derivative(x,x)
(0, 0, 2)

div(variables=None)#

Return the divergence of this vector function.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: vector([x, y, z]).div()
3
sage: vector([x*y, y*z, z*x]).div()
x + y + z

sage: R.<x,y,z,w> = QQ[]
sage: vector([x*y, y*z, z*x]).div([x, y, z])
x + y + z
sage: vector([x*y, y*z, z*x]).div([z, x, y])
0
sage: vector([x*y, y*z, z*x]).div([x, y, w])
y + z

sage: vector(SR, [x*y, y*z, z*x]).div()                                     # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: Unable to determine ordered variable names for Symbolic Ring
sage: vector(SR, [x*y, y*z, z*x]).div([x, y, z])                            # needs sage.symbolic
x + y + z


divergence() of vector fields on Euclidean spaces (and more generally pseudo-Riemannian manifolds), in particular for computing the divergence in curvilinear coordinates.

dot_product(right)#

Return the dot product of self and right, which is the sum of the product of the corresponding entries.

INPUT:

• right – a vector of the same degree as self. It does not need to belong to the same parent as self, so long as the necessary products and sums are defined.

OUTPUT:

If self and right are the vectors $$\vec{x}$$ and $$\vec{y}$$, of degree $$n$$, then this method returns

$\sum_{i=1}^{n}x_iy_i$

Note

The inner_product() is a more general version of this method, and the hermitian_inner_product() method may be more appropriate if your vectors have complex entries.

EXAMPLES:

sage: V = FreeModule(ZZ, 3)
sage: v = V([1,2,3])
sage: w = V([4,5,6])
sage: v.dot_product(w)
32

sage: R.<x> = QQ[]
sage: v = vector([x,x^2,3*x]); w = vector([2*x,x,3+x])
sage: v*w
x^3 + 5*x^2 + 9*x
sage: (x*2*x) + (x^2*x) + (3*x*(3+x))
x^3 + 5*x^2 + 9*x
sage: w*v
x^3 + 5*x^2 + 9*x


The vectors may be from different vector spaces, provided the necessary operations make sense. Notice that coercion will generate a result of the same type, even if the order of the arguments is reversed.:

sage: v = vector(ZZ, [1,2,3])
sage: w = vector(FiniteField(3), [0,1,2])
sage: ip = w.dot_product(v); ip
2
sage: ip.parent()
Finite Field of size 3

sage: ip = v.dot_product(w); ip
2
sage: ip.parent()
Finite Field of size 3


The dot product of a vector with itself is the 2-norm, squared.

sage: v = vector(QQ, [3, 4, 7])
sage: v.dot_product(v) - v.norm()^2                                         # needs sage.symbolic
0

element()#

Simply returns self. This is useful, since for many objects, self.element() returns a vector corresponding to self.

EXAMPLES:

sage: v = vector([1/2,2/5,0]); v
(1/2, 2/5, 0)
sage: v.element()
(1/2, 2/5, 0)

get(i)#

Like __getitem__ but without bounds checking: $$i$$ must satisfy 0 <= i < self.degree.

EXAMPLES:

sage: vector(SR, [1/2,2/5,0]).get(0)                                        # needs sage.symbolic
1/2

hamming_weight()#

Return the number of positions i such that self[i] != 0.

EXAMPLES:

sage: vector([-1,0,3,0,0,0,0.01]).hamming_weight()
3

hermitian_inner_product(right)#

Returns the dot product, but with the entries of the first vector conjugated beforehand.

INPUT:

• right - a vector of the same degree as self

OUTPUT:

If self and right are the vectors $$\vec{x}$$ and $$\vec{y}$$ of degree $$n$$ then this routine computes

$\sum_{i=1}^{n}\overline{x}_i{y}_i$

where the bar indicates complex conjugation.

Note

If your vectors do not contain complex entries, then dot_product() will return the same result without the overhead of conjugating elements of self.

If you are not computing a weighted inner product, and your vectors do not have complex entries, then the dot_product() will return the same result.

EXAMPLES:

sage: # needs sage.symbolic
sage: v = vector(CDF, [2+3*I, 5-4*I])
sage: w = vector(CDF, [6-4*I, 2+3*I])
sage: v.hermitian_inner_product(w)
-2.0 - 3.0*I


Sage implements a few specialized fields over the complex numbers, such as cyclotomic fields and quadratic number fields. So long as the base rings have a conjugate method, then the Hermitian inner product will be available.

sage: # needs sage.rings.number_field
sage: a^2
-7
sage: v = vector(Q, [3+a, 5-2*a])
sage: w = vector(Q, [6, 4+3*a])
sage: v.hermitian_inner_product(w)
17*a - 4


The Hermitian inner product should be additive in each argument (we only need to test one), linear in each argument (with conjugation on the first scalar), and anti-commutative.

sage: # needs sage.rings.complex_double sage.symbolic
sage: alpha = CDF(5.0 + 3.0*I)
sage: u = vector(CDF, [2+4*I, -3+5*I, 2-7*I])
sage: v = vector(CDF, [-1+3*I, 5+4*I, 9-2*I])
sage: w = vector(CDF, [8+3*I, -4+7*I, 3-6*I])
sage: (u+v).hermitian_inner_product(w) == u.hermitian_inner_product(w) + v.hermitian_inner_product(w)
True
sage: (alpha*u).hermitian_inner_product(w) == alpha.conjugate()*u.hermitian_inner_product(w)
True
sage: u.hermitian_inner_product(alpha*w) == alpha*u.hermitian_inner_product(w)
True
sage: u.hermitian_inner_product(v) == v.hermitian_inner_product(u).conjugate()
True


For vectors with complex entries, the Hermitian inner product has a more natural relationship with the 2-norm (which is the default for the norm() method). The norm squared equals the Hermitian inner product of the vector with itself.

sage: # needs sage.rings.complex_double sage.symbolic
sage: v = vector(CDF, [-0.66+0.47*I, -0.60+0.91*I, -0.62-0.87*I, 0.53+0.32*I])
sage: abs(v.norm()^2 - v.hermitian_inner_product(v)) < 1.0e-10
True

inner_product(right)#

Returns the inner product of self and right, possibly using an inner product matrix from the parent of self.

INPUT:

• right - a vector of the same degree as self

OUTPUT:

If the parent vector space does not have an inner product matrix defined, then this is the usual dot product (dot_product()). If self and right are considered as single column matrices, $$\vec{x}$$ and $$\vec{y}$$, and $$A$$ is the inner product matrix, then this method computes

$\left(\vec{x}\right)^tA\vec{y}$

where $$t$$ indicates the transpose.

Note

If your vectors have complex entries, the hermitian_inner_product() may be more appropriate for your purposes.

EXAMPLES:

sage: v = vector(QQ, [1,2,3])
sage: w = vector(QQ, [-1,2,-3])
sage: v.inner_product(w)
-6
sage: v.inner_product(w) == v.dot_product(w)
True


The vector space or free module that is the parent to self can have an inner product matrix defined, which will be used by this method. This matrix will be passed through to subspaces.

sage: ipm = matrix(ZZ,[[2,0,-1], [0,2,0], [-1,0,6]])
sage: M = FreeModule(ZZ, 3, inner_product_matrix=ipm)
sage: v = M([1,0,0])
sage: v.inner_product(v)
2
sage: K = M.span_of_basis([[0/2,-1/2,-1/2], [0,1/2,-1/2], [2,0,0]])
sage: (K.0).inner_product(K.0)
2
sage: w = M([1,3,-1])
sage: v = M([2,-4,5])
sage: w.row()*ipm*v.column() == w.inner_product(v)
True


Note that the inner product matrix comes from the parent of self. So if a vector is not an element of the correct parent, the result could be a source of confusion.

sage: V = VectorSpace(QQ, 2, inner_product_matrix=[[1,2],[2,1]])
sage: v = V([12, -10])
sage: w = vector(QQ, [10,12])
sage: v.inner_product(w)
88
sage: w.inner_product(v)
0
sage: w = V(w)
sage: w.inner_product(v)
88


Note

The use of an inner product matrix makes no restrictions on the nature of the matrix. In particular, in this context it need not be Hermitian and positive-definite (as it is in the example above).

integral(*args, **kwds)#

Returns a symbolic integral of the vector, component-wise.

integrate() is an alias of the function.

EXAMPLES:

sage: # needs sage.symbolic
sage: t = var('t')
sage: r = vector([t,t^2,sin(t)])
sage: r.integral(t)
(1/2*t^2, 1/3*t^3, -cos(t))
sage: integrate(r, t)
(1/2*t^2, 1/3*t^3, -cos(t))
sage: r.integrate(t, 0, 1)
(1/2, 1/3, -cos(1) + 1)

integrate(*args, **kwds)#

Returns a symbolic integral of the vector, component-wise.

integrate() is an alias of the function.

EXAMPLES:

sage: # needs sage.symbolic
sage: t = var('t')
sage: r = vector([t,t^2,sin(t)])
sage: r.integral(t)
(1/2*t^2, 1/3*t^3, -cos(t))
sage: integrate(r, t)
(1/2*t^2, 1/3*t^3, -cos(t))
sage: r.integrate(t, 0, 1)
(1/2, 1/3, -cos(1) + 1)

is_dense()#

Return True if this is a dense vector, which is just a statement about the data structure, not the number of nonzero entries.

EXAMPLES:

sage: vector([1/2, 2/5, 0]).is_dense()
True
sage: vector([1/2, 2/5, 0], sparse=True).is_dense()
False

is_sparse()#

Return True if this is a sparse vector, which is just a statement about the data structure, not the number of nonzero entries.

EXAMPLES:

sage: vector([1/2, 2/5, 0]).is_sparse()
False
sage: vector([1/2, 2/5, 0], sparse=True).is_sparse()
True

is_vector()#

Return True, since this is a vector.

EXAMPLES:

sage: vector([1/2, 2/5, 0]).is_vector()
True

items()#

Return an iterator over self.

EXAMPLES:

sage: v = vector([1,2/3,pi])                                                # needs sage.symbolic
sage: v.items()                                                             # needs sage.symbolic
<...generator object at ...>
sage: list(v.items())                                                       # needs sage.symbolic
[(0, 1), (1, 2/3), (2, pi)]

iteritems()#

Return an iterator over self.

EXAMPLES:

sage: v = vector([1,2/3,pi])                                                # needs sage.symbolic
sage: v.items()                                                             # needs sage.symbolic
<...generator object at ...>
sage: list(v.items())                                                       # needs sage.symbolic
[(0, 1), (1, 2/3), (2, pi)]

lift()#

Lift self to the cover ring.

OUTPUT:

Return a lift of self to the covering ring of the base ring $$R$$, which is by definition the ring returned by calling cover_ring() on $$R$$, or just $$R$$ itself if the cover_ring() method is not defined.

EXAMPLES:

sage: V = vector(Integers(7), [5, 9, 13, 15]) ; V
(5, 2, 6, 1)
sage: V.lift()
(5, 2, 6, 1)
sage: parent(V.lift())
Ambient free module of rank 4 over the principal ideal domain Integer Ring


If the base ring does not have a cover method, return a copy of the vector:

sage: W = vector(QQ, [1, 2, 3])
sage: W1 = W.lift()
sage: W is W1
False
sage: parent(W1)
Vector space of dimension 3 over Rational Field

lift_centered()#

Lift to a congruent, centered vector.

INPUT:

• self A vector with coefficients in $$Integers(n)$$.

OUTPUT:

• The unique integer vector $$v$$ such that foreach $$i$$, $$Mod(v[i],n) = Mod(self[i],n)$$ and $$-n/2 < v[i] \leq n/2$$.

EXAMPLES:

sage: V = vector(Integers(7), [5, 9, 13, 15]) ; V
(5, 2, 6, 1)
sage: V.lift_centered()
(-2, 2, -1, 1)
sage: parent(V.lift_centered())
Ambient free module of rank 4 over the principal ideal domain Integer Ring

list(copy=True)#

Return list of elements of self.

INPUT:

• copy – bool, whether returned list is a copy that is safe to change, is ignored.

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: v = vector([x,y,z], sparse=True)
sage: type(v)
<class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
sage: a = v.list(); a
[x, y, z]
sage: a[0] = x*y; v
(x, y, z)


The optional argument copy is ignored:

sage: a = v.list(copy=False); a
[x, y, z]
sage: a[0] = x*y; v
(x, y, z)

list_from_positions(positions)#

Return list of elements chosen from this vector using the given positions of this vector.

INPUT:

• positions – iterable of ints

EXAMPLES:

sage: v = vector([1, 2/3, pi])                                              # needs sage.symbolic
sage: v.list_from_positions([0,0,0,2,1])                                    # needs sage.symbolic
[1, 1, 1, pi, 2/3]

monic()#

Return this vector divided through by the first nonzero entry of this vector.

EXAMPLES:

sage: v = vector(QQ, [0, 4/3, 5, 1, 2])
sage: v.monic()
(0, 1, 15/4, 3/4, 3/2)
sage: v = vector(QQ, [])
sage: v.monic()
()

monomial_coefficients(copy=True)#

Return dictionary of nonzero entries of self.

More precisely, this returns a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

INPUT:

• copy – (default: True) if self is internally represented by a dictionary d, then make a copy of d; if False, then this can cause undesired behavior by mutating d

OUTPUT:

• Python dictionary

EXAMPLES:

sage: v = vector([0,0,0,0,1/2,0,3/14])
sage: v.dict()
{4: 1/2, 6: 3/14}
sage: sorted(v.support())
[4, 6]


In some cases, when copy=False, we get back a dangerous reference:

sage: v = vector({0:5, 2:3/7}, sparse=True)
sage: v.dict(copy=False)
{0: 5, 2: 3/7}
sage: v.dict(copy=False)[0] = 18
sage: v
(18, 0, 3/7)

nintegral(*args, **kwds)#

Returns a numeric integral of the vector, component-wise, and the result of the nintegral command on each component of the input.

nintegrate() is an alias of the function.

EXAMPLES:

sage: # needs sage.symbolic
sage: t = var('t')
sage: r = vector([t,t^2,sin(t)])
sage: vec  # abs tol 1e-15
(0.5, 0.3333333333333334, 0.4596976941318602)
sage: type(vec)
<class 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
[(0.5, 5.55111512312578...e-15, 21, 0),
(0.3333333333333..., 3.70074341541719...e-15, 21, 0),
(0.45969769413186..., 5.10366964392284...e-15, 21, 0)]

sage: # needs sage.symbolic
sage: r = vector([t,0,1], sparse=True)
sage: r.nintegral(t, 0, 1)
((0.5, 0.0, 1.0),
{0: (0.5, 5.55111512312578...e-15, 21, 0),
2: (1.0, 1.11022302462515...e-14, 21, 0)})

nintegrate(*args, **kwds)#

Returns a numeric integral of the vector, component-wise, and the result of the nintegral command on each component of the input.

nintegrate() is an alias of the function.

EXAMPLES:

sage: # needs sage.symbolic
sage: t = var('t')
sage: r = vector([t,t^2,sin(t)])
sage: vec  # abs tol 1e-15
(0.5, 0.3333333333333334, 0.4596976941318602)
sage: type(vec)
<class 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
[(0.5, 5.55111512312578...e-15, 21, 0),
(0.3333333333333..., 3.70074341541719...e-15, 21, 0),
(0.45969769413186..., 5.10366964392284...e-15, 21, 0)]

sage: # needs sage.symbolic
sage: r = vector([t,0,1], sparse=True)
sage: r.nintegral(t, 0, 1)
((0.5, 0.0, 1.0),
{0: (0.5, 5.55111512312578...e-15, 21, 0),
2: (1.0, 1.11022302462515...e-14, 21, 0)})

nonzero_positions()#

Return the sorted list of integers i such that self[i] != 0.

EXAMPLES:

sage: vector([-1,0,3,0,0,0,0.01]).nonzero_positions()
[0, 2, 6]

norm(p='__two__')#

Return the $$p$$-norm of self.

INPUT:

• p - default: 2 – p can be a real number greater than 1, infinity (oo or Infinity), or a symbolic expression.

• $$p=1$$: the taxicab (Manhattan) norm

• $$p=2$$: the usual Euclidean norm (the default)

• $$p=\infty$$: the maximum entry (in absolute value)

Note

EXAMPLES:

sage: v = vector([1,2,-3])
sage: v.norm(5)                                                             # needs sage.symbolic
276^(1/5)


The default is the usual Euclidean norm.

sage: v.norm()                                                              # needs sage.symbolic
sqrt(14)
sage: v.norm(2)                                                             # needs sage.symbolic
sqrt(14)


The infinity norm is the maximum size (in absolute value) of the entries.

sage: v.norm(Infinity)
3
sage: v.norm(oo)
3


Real or symbolic values may be used for p.

sage: v=vector(RDF,[1,2,3])
sage: v.norm(5)
3.077384885394063

sage: # needs sage.symbolic
sage: v.norm(pi/2)    # abs tol 1e-15
4.216595864704748
sage: _ = var('a b c d p'); v = vector([a, b, c, d])
sage: v.norm(p)
(abs(a)^p + abs(b)^p + abs(c)^p + abs(d)^p)^(1/p)


Notice that the result may be a symbolic expression, owing to the necessity of taking a square root (in the default case). These results can be converted to numerical values if needed.

sage: v = vector(ZZ, [3,4])
sage: nrm = v.norm(); nrm
5
sage: nrm.parent()
Rational Field

sage: # needs sage.symbolic
sage: v = vector(QQ, [3, 5])
sage: nrm = v.norm(); nrm
sqrt(34)
sage: nrm.parent()
Symbolic Ring
sage: numeric = N(nrm); numeric
5.83095189484...
sage: numeric.parent()
Real Field with 53 bits of precision

normalized(p='__two__')#

Return the input vector divided by the p-norm.

INPUT:

• “p” - default: 2 - p value for the norm

EXAMPLES:

sage: v = vector(QQ, [4, 1, 3, 2])
sage: v.normalized()                                                        # needs sage.symbolic
(2/15*sqrt(30), 1/30*sqrt(30), 1/10*sqrt(30), 1/15*sqrt(30))
sage: sum(v.normalized(1))
1


Note that normalizing the vector may change the base ring:

sage: v.base_ring() == v.normalized().base_ring()                           # needs sage.symbolic
False
sage: u = vector(RDF, [-3, 4, 6, 9])
sage: u.base_ring() == u.normalized().base_ring()
True

numerical_approx(prec=None, digits=None, algorithm=None)#

Return a numerical approximation of self with prec bits (or decimal digits) of precision, by approximating all entries.

INPUT:

• prec – precision in bits

• digits – precision in decimal digits (only used if prec is not given)

• algorithm – which algorithm to use to compute the approximation of the entries (the accepted algorithms depend on the object)

If neither prec nor digits is given, the default precision is 53 bits (roughly 16 digits).

EXAMPLES:

sage: v = vector(RealField(212), [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: numerical_approx(v)
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: v.n(prec=75)
(1.000000000000000000000, 2.000000000000000000000, 3.000000000000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 75 bits of precision
sage: numerical_approx(v, digits=3)
(1.00, 2.00, 3.00)
sage: _.parent()
Vector space of dimension 3 over Real Field with 14 bits of precision


Both functional and object-oriented usage is possible.

sage: u = vector(QQ, [1/2, 1/3, 1/4])
sage: u.n()
(0.500000000000000, 0.333333333333333, 0.250000000000000)
sage: u.numerical_approx()
(0.500000000000000, 0.333333333333333, 0.250000000000000)
sage: n(u)
(0.500000000000000, 0.333333333333333, 0.250000000000000)
sage: N(u)
(0.500000000000000, 0.333333333333333, 0.250000000000000)
sage: numerical_approx(u)
(0.500000000000000, 0.333333333333333, 0.250000000000000)


Precision (bits) and digits (decimal) may be specified. When both are given, prec wins.

sage: u = vector(QQ, [1/2, 1/3, 1/4])
sage: n(u, prec=15)
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)
(0.50000, 0.33333, 0.25000)
sage: n(u, prec=30, digits=100)
(0.50000000, 0.33333333, 0.25000000)


These are some legacy doctests that were part of various specialized versions of the numerical approximation routine that were removed as part of github issue #12195.

sage: v = vector(ZZ, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: v.n(prec=75)
(1.000000000000000000000, 2.000000000000000000000, 3.000000000000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 75 bits of precision

sage: v = vector(RDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: v = vector(CDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Complex Field with 53 bits of precision

sage: v = vector(Integers(8), [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: v.n(prec=75)
(1.000000000000000000000, 2.000000000000000000000, 3.000000000000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 75 bits of precision

sage: v = vector(QQ, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: v.n(prec=75)
(1.000000000000000000000, 2.000000000000000000000, 3.000000000000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 75 bits of precision

sage: v = vector(GF(2), [1,2,3])
sage: v.n()
(1.00000000000000, 0.000000000000000, 1.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 53 bits of precision
sage: v.n(prec=75)
(1.000000000000000000000, 0.0000000000000000000000, 1.000000000000000000000)
sage: _.parent()
Vector space of dimension 3 over Real Field with 75 bits of precision

numpy(dtype='object')#

Convert self to a numpy array.

INPUT:

• dtype – the numpy dtype of the returned array

EXAMPLES:

sage: # needs numpy
sage: v = vector([1,2,3])
sage: v.numpy()
array([1, 2, 3], dtype=object)
sage: v.numpy() * v.numpy()
array([1, 4, 9], dtype=object)

sage: vector(QQ, [1, 2, 5/6]).numpy()                                       # needs numpy
array([1, 2, 5/6], dtype=object)


By default, the object dtype is used. Alternatively, the desired dtype can be passed in as a parameter:

sage: # needs numpy
sage: v = vector(QQ, [1, 2, 5/6])
sage: v.numpy()
array([1, 2, 5/6], dtype=object)
sage: v.numpy(dtype=float)
array([1.        , 2.        , 0.83333333])
sage: v.numpy(dtype=int)
array([1, 2, 0])
sage: import numpy
sage: v.numpy(dtype=numpy.uint8)
array([1, 2, 0], dtype=uint8)


Passing a dtype of None will let numpy choose a native type, which can be more efficient but may have unintended consequences:

sage: # needs numpy
sage: v.numpy(dtype=None)
array([1.        , 2.        , 0.83333333])

sage: w = vector(ZZ, [0, 1, 2^63 -1]); w
(0, 1, 9223372036854775807)
sage: wn = w.numpy(dtype=None); wn                                          # needs numpy
array([                  0,                   1, 9223372036854775807]...)
sage: wn.dtype                                                              # needs numpy
dtype('int64')
sage: w.dot_product(w)
85070591730234615847396907784232501250
sage: wn.dot(wn)        # overflow                                          # needs numpy
2


Numpy can give rather obscure errors; we wrap these to give a bit of context:

sage: vector([1, 1/2, QQ['x'].0]).numpy(dtype=float)                        # needs numpy
Traceback (most recent call last):
...
ValueError: Could not convert vector over Univariate Polynomial Ring in x
over Rational Field to numpy array of type <... 'float'>:
setting an array element with a sequence.

outer_product(right)#

Returns a matrix, the outer product of two vectors self and right.

INPUT:

• right - a vector (or free module element) of any size, whose elements are compatible (with regard to multiplication) with the elements of self.

OUTPUT:

The outer product of two vectors $$x$$ and $$y$$ (respectively self and right) can be described several ways. If we interpret $$x$$ as a $$m\times 1$$ matrix and interpret $$y$$ as a $$1\times n$$ matrix, then the outer product is the $$m\times n$$ matrix from the usual matrix product $$xy$$. Notice how this is the “opposite” in some ways from an inner product (which would require $$m=n$$).

If we just consider vectors, use each entry of $$x$$ to create a scalar multiples of the vector $$y$$ and use these vectors as the rows of a matrix. Or use each entry of $$y$$ to create a scalar multiples of $$x$$ and use these vectors as the columns of a matrix.

EXAMPLES:

sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
sage: v = vector(ZZ, [60, 180, 600])
sage: u.outer_product(v)
[ 30  90 300]
[ 20  60 200]
[ 15  45 150]
[ 12  36 120]
sage: M = v.outer_product(u); M
[ 30  20  15  12]
[ 90  60  45  36]
[300 200 150 120]
sage: M.parent()
Full MatrixSpace of 3 by 4 dense matrices over Rational Field


The more general sage.matrix.matrix2.tensor_product() is an operation on a pair of matrices. If we construct a pair of vectors as a column vector and a row vector, then an outer product and a tensor product are identical. Thus $$tensor_product$$ is a synonym for this method.

sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
sage: v = vector(ZZ, [60, 180, 600])
sage: u.tensor_product(v) == (u.column()).tensor_product(v.row())
True


The result is always a dense matrix, no matter if the two vectors are, or are not, dense.

sage: d = vector(ZZ,[4,5], sparse=False)
sage: s = vector(ZZ, [1,2,3], sparse=True)
sage: dd = d.outer_product(d)
sage: ds = d.outer_product(s)
sage: sd = s.outer_product(d)
sage: ss = s.outer_product(s)
sage: all([dd.is_dense(), ds.is_dense(), sd.is_dense(), dd.is_dense()])
True


Vectors with no entries do the right thing.

sage: v = vector(ZZ, [])
sage: z = v.outer_product(v)
sage: z.parent()
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring


There is a fair amount of latitude in the value of the right vector, and the matrix that results can have entries from a new ring large enough to contain the result. If you know better, you can sometimes bring the result down to a less general ring.

sage: R.<t> = ZZ[]
sage: v = vector(R, [12, 24*t])
sage: w = vector(QQ, [1/2, 1/3, 1/4])
sage: op = v.outer_product(w); op
[   6    4    3]
[12*t  8*t  6*t]
sage: op.base_ring()
Univariate Polynomial Ring in t over Rational Field
sage: m = op.change_ring(R); m
[   6    4    3]
[12*t  8*t  6*t]
sage: m.base_ring()
Univariate Polynomial Ring in t over Integer Ring


But some inputs are not compatible, even if vectors.

sage: w = vector(GF(5), [1,2])
sage: v = vector(GF(7), [1,2,3,4])
sage: z = w.outer_product(v)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 1 dense matrices over Finite Field of size 5' and
'Full MatrixSpace of 1 by 4 dense matrices over Finite Field of size 7'


And some inputs don’t make any sense at all.

sage: w = vector(QQ, [5,10])
sage: z = w.outer_product(6)
Traceback (most recent call last):
...
TypeError: right operand in an outer product must be a vector,
not an element of Integer Ring

pairwise_product(right)#

Return the pairwise product of self and right, which is a vector of the products of the corresponding entries.

INPUT:

• right - vector of the same degree as self. It need not be in the same vector space as self, as long as the coefficients can be multiplied.

EXAMPLES:

sage: V = FreeModule(ZZ, 3)
sage: v = V([1,2,3])
sage: w = V([4,5,6])
sage: v.pairwise_product(w)
(4, 10, 18)
sage: sum(v.pairwise_product(w)) == v.dot_product(w)
True

sage: W = VectorSpace(GF(3), 3)
sage: w = W([0,1,2])
sage: w.pairwise_product(v)
(0, 2, 0)
sage: w.pairwise_product(v).parent()
Vector space of dimension 3 over Finite Field of size 3


Implicit coercion is well defined (regardless of order), so we get 2 even if we do the dot product in the other order.

sage: v.pairwise_product(w).parent()
Vector space of dimension 3 over Finite Field of size 3

plot(plot_type=None, start=None, **kwds)#

INPUT:

• plot_type - (default: ‘arrow’ if v has 3 or fewer components,

otherwise ‘step’) type of plot. Options are:

• ‘arrow’ to draw an arrow

• ‘point’ to draw a point at the coordinates specified by the vector

• ‘step’ to draw a step function representing the coordinates of the vector.

Both ‘arrow’ and ‘point’ raise exceptions if the vector has more than 3 dimensions.

• start - (default: origin in correct dimension) may be a tuple, list, or vector.

EXAMPLES:

The following both plot the given vector:

sage: v = vector(RDF, (1,2))
sage: A = plot(v)                                                           # needs sage.plot
sage: B = v.plot()                                                          # needs sage.plot
sage: A + B  # should just show one vector                                  # needs sage.plot
Graphics object consisting of 2 graphics primitives


Examples of the plot types:

sage: # needs sage.plot
sage: A = plot(v, plot_type='arrow')
sage: B = plot(v, plot_type='point', color='green', size=20)
sage: C = plot(v, plot_type='step') # calls v.plot_step()
sage: A+B+C
Graphics object consisting of 3 graphics primitives


You can use the optional arguments for plot_step():

sage: eps = 0.1
sage: plot(v, plot_type='step', eps=eps, xmax=5, hue=0)                     # needs sage.plot
Graphics object consisting of 1 graphics primitive


Three-dimensional examples:

sage: v = vector(RDF, (1,2,1))
sage: plot(v) # defaults to an arrow plot                                   # needs sage.plot
Graphics3d Object

sage: plot(v, plot_type='arrow')                                            # needs sage.plot
Graphics3d Object

sage: from sage.plot.plot3d.shapes2 import frame3d                          # needs sage.plot
sage: plot(v, plot_type='point')+frame3d((0,0,0), v.list())                 # needs sage.plot
Graphics3d Object

sage: plot(v, plot_type='step') # calls v.plot_step()                       # needs sage.plot
Graphics object consisting of 1 graphics primitive

sage: plot(v, plot_type='step', eps=eps, xmax=5, hue=0)                     # needs sage.plot
Graphics object consisting of 1 graphics primitive


With greater than three coordinates, it defaults to a step plot:

sage: v = vector(RDF, (1,2,3,4))
sage: plot(v)                                                               # needs sage.plot
Graphics object consisting of 1 graphics primitive


One dimensional vectors are plotted along the horizontal axis of the coordinate plane:

sage: plot(vector([1]))                                                     # needs sage.plot
Graphics object consisting of 1 graphics primitive


An optional start argument may also be specified by a tuple, list, or vector:

sage: u = vector([1,2]); v = vector([2,5])
sage: plot(u, start=v)                                                      # needs sage.plot
Graphics object consisting of 1 graphics primitive

plot_step(xmin=0, xmax=1, eps=None, res=None, connect=True, **kwds)#

INPUT:

• xmin - (default: 0) start x position to start plotting

• xmax - (default: 1) stop x position to stop plotting

• eps - (default: determined by xmax) we view this vector as defining a function at the points xmin, xmin + eps, xmin + 2*eps, …,

• res - (default: all points) total number of points to include in the graph

• connect - (default: True) if True draws a line; otherwise draw a list of points.

EXAMPLES:

sage: eps = 0.1
sage: v = vector(RDF, [sin(n*eps) for n in range(100)])
sage: v.plot_step(eps=eps, xmax=5, hue=0)                                   # needs sage.plot
Graphics object consisting of 1 graphics primitive

row()#

Return a matrix with a single row and the same entries as the vector self.

OUTPUT:

A matrix over the same ring as the vector (or free module element), with a single row. The entries of the row are identical to those of the vector, and in the same order.

EXAMPLES:

sage: v = vector(ZZ, [1,2,3])
sage: w = v.row(); w
[1 2 3]
sage: w.parent()
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring

sage: x = vector(FiniteField(13), [2,4,8,16])
sage: x.row()
[2 4 8 3]


There is more than one way to get one-row matrix from a vector, but the row method is more efficient than making a column and then taking a transpose. Notice that supplying a vector to the matrix constructor demonstrates Sage’s preference for rows.

sage: x = vector(RDF, [sin(i*pi/20) for i in range(10)])                    # needs sage.symbolic
sage: x.row() == matrix(x)
True
sage: x.row() == x.column().transpose()
True


Sparse or dense implementations are preserved.

sage: d = vector(RR, [1.0, 2.0, 3.0])
sage: s = vector(CDF, {2: 5.0+6.0*I})                                       # needs sage.symbolic
sage: dm = d.row()
sage: sm = s.row()                                                          # needs sage.symbolic
sage: all([d.is_dense(), dm.is_dense(), s.is_sparse(), sm.is_sparse()])     # needs sage.symbolic
True

set(i, value)#

Like __setitem__ but without type or bounds checking: $$i$$ must satisfy 0 <= i < self.degree and value must be an element of the coordinate ring.

EXAMPLES:

sage: v = vector(SR, [1/2,2/5,0]); v                                        # needs sage.symbolic
(1/2, 2/5, 0)
sage: v.set(2, pi); v                                                       # needs sage.symbolic
(1/2, 2/5, pi)

sparse_vector()#

Return sparse version of self. If self is sparse, just return self; otherwise, create and return correspond sparse vector.

EXAMPLES:

sage: vector([-1,0,3,0,0,0]).sparse_vector().is_sparse()
True
sage: vector([-1,0,3,0,0,0]).sparse_vector().is_sparse()
True
sage: vector([-1,0,3,0,0,0]).sparse_vector()
(-1, 0, 3, 0, 0, 0)

subs(in_dict=None, **kwds)#

EXAMPLES:

sage: # needs sage.symbolic
sage: var('a,b,d,e')
(a, b, d, e)
sage: v = vector([a, b, d, e])
sage: v.substitute(a=1)
(1, b, d, e)
sage: v.subs(a=b, b=d)
(b, d, d, e)

support()#

Return the integers i such that self[i] != 0. This is the same as the nonzero_positions function.

EXAMPLES:

sage: vector([-1,0,3,0,0,0,0.01]).support()
[0, 2, 6]

tensor_product(right)#

Returns a matrix, the outer product of two vectors self and right.

INPUT:

• right - a vector (or free module element) of any size, whose elements are compatible (with regard to multiplication) with the elements of self.

OUTPUT:

The outer product of two vectors $$x$$ and $$y$$ (respectively self and right) can be described several ways. If we interpret $$x$$ as a $$m\times 1$$ matrix and interpret $$y$$ as a $$1\times n$$ matrix, then the outer product is the $$m\times n$$ matrix from the usual matrix product $$xy$$. Notice how this is the “opposite” in some ways from an inner product (which would require $$m=n$$).

If we just consider vectors, use each entry of $$x$$ to create a scalar multiples of the vector $$y$$ and use these vectors as the rows of a matrix. Or use each entry of $$y$$ to create a scalar multiples of $$x$$ and use these vectors as the columns of a matrix.

EXAMPLES:

sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
sage: v = vector(ZZ, [60, 180, 600])
sage: u.outer_product(v)
[ 30  90 300]
[ 20  60 200]
[ 15  45 150]
[ 12  36 120]
sage: M = v.outer_product(u); M
[ 30  20  15  12]
[ 90  60  45  36]
[300 200 150 120]
sage: M.parent()
Full MatrixSpace of 3 by 4 dense matrices over Rational Field


The more general sage.matrix.matrix2.tensor_product() is an operation on a pair of matrices. If we construct a pair of vectors as a column vector and a row vector, then an outer product and a tensor product are identical. Thus $$tensor_product$$ is a synonym for this method.

sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
sage: v = vector(ZZ, [60, 180, 600])
sage: u.tensor_product(v) == (u.column()).tensor_product(v.row())
True


The result is always a dense matrix, no matter if the two vectors are, or are not, dense.

sage: d = vector(ZZ,[4,5], sparse=False)
sage: s = vector(ZZ, [1,2,3], sparse=True)
sage: dd = d.outer_product(d)
sage: ds = d.outer_product(s)
sage: sd = s.outer_product(d)
sage: ss = s.outer_product(s)
sage: all([dd.is_dense(), ds.is_dense(), sd.is_dense(), dd.is_dense()])
True


Vectors with no entries do the right thing.

sage: v = vector(ZZ, [])
sage: z = v.outer_product(v)
sage: z.parent()
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring


There is a fair amount of latitude in the value of the right vector, and the matrix that results can have entries from a new ring large enough to contain the result. If you know better, you can sometimes bring the result down to a less general ring.

sage: R.<t> = ZZ[]
sage: v = vector(R, [12, 24*t])
sage: w = vector(QQ, [1/2, 1/3, 1/4])
sage: op = v.outer_product(w); op
[   6    4    3]
[12*t  8*t  6*t]
sage: op.base_ring()
Univariate Polynomial Ring in t over Rational Field
sage: m = op.change_ring(R); m
[   6    4    3]
[12*t  8*t  6*t]
sage: m.base_ring()
Univariate Polynomial Ring in t over Integer Ring


But some inputs are not compatible, even if vectors.

sage: w = vector(GF(5), [1,2])
sage: v = vector(GF(7), [1,2,3,4])
sage: z = w.outer_product(v)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 1 dense matrices over Finite Field of size 5' and
'Full MatrixSpace of 1 by 4 dense matrices over Finite Field of size 7'


And some inputs don’t make any sense at all.

sage: w = vector(QQ, [5,10])
sage: z = w.outer_product(6)
Traceback (most recent call last):
...
TypeError: right operand in an outer product must be a vector,
not an element of Integer Ring

class sage.modules.free_module_element.FreeModuleElement_generic_dense#

A generic dense element of a free module.

function(*args)#

Return a vector over a callable symbolic expression ring.

EXAMPLES:

sage: # needs sage.symbolic
sage: x, y = var('x,y')
sage: v = vector([x, y, x*sin(y)])
sage: w = v.function([x,y]); w
(x, y) |--> (x, y, x*sin(y))
sage: w.coordinate_ring()
Callable function ring with arguments (x, y)
sage: w(1,2)
(1, 2, sin(2))
sage: w(2,1)
(2, 1, 2*sin(1))
sage: w(y=1,x=2)
(2, 1, 2*sin(1))

sage: # needs sage.symbolic
sage: x,y = var('x,y')
sage: v = vector([x, y, x*sin(y)])
sage: w = v.function([x]); w
x |--> (x, y, x*sin(y))
sage: w.coordinate_ring()
Callable function ring with argument x
sage: w(4)
(4, y, 4*sin(y))

list(copy=True)#

Return list of elements of self.

INPUT:

• copy – bool, return list of underlying entries

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: v = vector([x,y,z])
sage: type(v)
<class 'sage.modules.free_module_element.FreeModuleElement_generic_dense'>
sage: a = v.list(); a
[x, y, z]
sage: a[0] = x*y; v
(x, y, z)
sage: a = v.list(copy=False); a
[x, y, z]
sage: a[0] = x*y; v
(x*y, y, z)

class sage.modules.free_module_element.FreeModuleElement_generic_sparse#

A generic sparse free module element is a dictionary with keys ints i and entries in the base ring.

denominator()#

Return the least common multiple of the denominators of the entries of self.

EXAMPLES:

sage: v = vector([1/2,2/5,3/14], sparse=True)
sage: v.denominator()
70

dict(copy=True)#

Return dictionary of nonzero entries of self.

More precisely, this returns a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

INPUT:

• copy – (default: True) if self is internally represented by a dictionary d, then make a copy of d; if False, then this can cause undesired behavior by mutating d

OUTPUT:

• Python dictionary

EXAMPLES:

sage: v = vector([0,0,0,0,1/2,0,3/14], sparse=True)
sage: v.dict()
{4: 1/2, 6: 3/14}
sage: sorted(v.support())
[4, 6]

hamming_weight()#

Returns the number of positions i such that self[i] != 0.

EXAMPLES:

sage: v = vector({1: 1, 3: -2})
sage: w = vector({1: 4, 3: 2})
sage: v+w
(0, 5, 0, 0)
sage: (v+w).hamming_weight()
1

items()#

Return an iterator over the entries of self.

EXAMPLES:

sage: v = vector([1,2/3,pi], sparse=True)                                   # needs sage.symbolic
sage: next(v.items())                                                       # needs sage.symbolic
(0, 1)
sage: list(v.items())                                                       # needs sage.symbolic
[(0, 1), (1, 2/3), (2, pi)]

iteritems()#

Return an iterator over the entries of self.

EXAMPLES:

sage: v = vector([1,2/3,pi], sparse=True)                                   # needs sage.symbolic
sage: next(v.items())                                                       # needs sage.symbolic
(0, 1)
sage: list(v.items())                                                       # needs sage.symbolic
[(0, 1), (1, 2/3), (2, pi)]

list(copy=True)#

Return list of elements of self.

INPUT:

• copy – ignored for sparse vectors

EXAMPLES:

sage: R.<x> = QQ[]
sage: M = FreeModule(R, 3, sparse=True) * (1/x)
sage: v = M([-x^2, 3/x, 0])
sage: type(v)
<class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
sage: a = v.list()
sage: a
[-x^2, 3/x, 0]
sage: [parent(c) for c in a]
[Fraction Field of Univariate Polynomial Ring in x over Rational Field,
Fraction Field of Univariate Polynomial Ring in x over Rational Field,
Fraction Field of Univariate Polynomial Ring in x over Rational Field]

monomial_coefficients(copy=True)#

Return dictionary of nonzero entries of self.

More precisely, this returns a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

INPUT:

• copy – (default: True) if self is internally represented by a dictionary d, then make a copy of d; if False, then this can cause undesired behavior by mutating d

OUTPUT:

• Python dictionary

EXAMPLES:

sage: v = vector([0,0,0,0,1/2,0,3/14], sparse=True)
sage: v.dict()
{4: 1/2, 6: 3/14}
sage: sorted(v.support())
[4, 6]

nonzero_positions()#

Returns the list of numbers i such that self[i] != 0.

EXAMPLES:

sage: v = vector({1: 1, 3: -2})
sage: w = vector({1: 4, 3: 2})
sage: v+w
(0, 5, 0, 0)
sage: (v+w).nonzero_positions()
[1]

numerical_approx(prec=None, digits=None, algorithm=None)#

Return a numerical approximation of self with prec bits (or decimal digits) of precision, by approximating all entries.

INPUT:

• prec – precision in bits

• digits – precision in decimal digits (only used if prec is not given)

• algorithm – which algorithm to use to compute the approximation of the entries (the accepted algorithms depend on the object)

If neither prec nor digits is given, the default precision is 53 bits (roughly 16 digits).

EXAMPLES:

sage: v = vector(RealField(200), [1,2,3], sparse=True)
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Sparse vector space of dimension 3 over Real Field with 53 bits of precision
sage: v.n(prec=75)
(1.000000000000000000000, 2.000000000000000000000, 3.000000000000000000000)
sage: _.parent()
Sparse vector space of dimension 3 over Real Field with 75 bits of precision

sage.modules.free_module_element.free_module_element(arg0, arg1=None, arg2=None, sparse=None, immutable=False)#

Return a vector or free module element with specified entries.

CALL FORMATS:

This constructor can be called in several different ways. In each case, sparse=True or sparse=False as well as immutable=True or immutable=False can be supplied as an option. free_module_element() is an alias for vector().

1. vector(object)

2. vector(ring, object)

3. vector(object, ring)

4. vector(ring, degree, object)

5. vector(ring, degree)

INPUT:

• object – a list, dictionary, or other iterable containing the entries of the vector, including any object that is palatable to the Sequence constructor

• ring – a base ring (or field) for the vector space or free module, which contains all of the elements

• degree – an integer specifying the number of entries in the vector or free module element

• sparse – boolean, whether the result should be a sparse vector

• immutable – boolean (default: False); whether the result should be an immutable vector

In call format 4, an error is raised if the degree does not match the length of object so this call can provide some safeguards. Note however that using this format when object is a dictionary is unlikely to work properly.

OUTPUT:

An element of the ambient vector space or free module with the given base ring and implied or specified dimension or rank, containing the specified entries and with correct degree.

In call format 5, no entries are specified, so the element is populated with all zeros.

If the sparse option is not supplied, the output will generally have a dense representation. The exception is if object is a dictionary, then the representation will be sparse.

EXAMPLES:

sage: v = vector([1,2,3]); v
(1, 2, 3)
sage: v.parent()
Ambient free module of rank 3 over the principal ideal domain Integer Ring
sage: v = vector([1,2,3/5]); v
(1, 2, 3/5)
sage: v.parent()
Vector space of dimension 3 over Rational Field


All entries must canonically coerce to some common ring:

sage: v = vector([17, GF(11)(5), 19/3]); v
Traceback (most recent call last):
...
TypeError: unable to find a common ring for all elements

sage: v = vector([17, GF(11)(5), 19]); v
(6, 5, 8)
sage: v.parent()
Vector space of dimension 3 over Finite Field of size 11
sage: v = vector([17, GF(11)(5), 19], QQ); v
(17, 5, 19)
sage: v.parent()
Vector space of dimension 3 over Rational Field
sage: v = vector((1,2,3), QQ); v
(1, 2, 3)
sage: v.parent()
Vector space of dimension 3 over Rational Field
sage: v = vector(QQ, (1,2,3)); v
(1, 2, 3)
sage: v.parent()
Vector space of dimension 3 over Rational Field
sage: v = vector(vector([1,2,3])); v
(1, 2, 3)
sage: v.parent()
Ambient free module of rank 3 over the principal ideal domain Integer Ring


You can also use free_module_element, which is the same as vector.

sage: free_module_element([1/3, -4/5])
(1/3, -4/5)


We make a vector mod 3 out of a vector over $$\ZZ$$.

sage: vector(vector([1,2,3]), GF(3))
(1, 2, 0)


The degree of a vector may be specified:

sage: vector(QQ, 4, [1,1/2,1/3,1/4])
(1, 1/2, 1/3, 1/4)


But it is an error if the degree and size of the list of entries are mismatched:

sage: vector(QQ, 5, [1,1/2,1/3,1/4])
Traceback (most recent call last):
...
ValueError: incompatible degrees in vector constructor


Providing no entries populates the vector with zeros, but of course, you must specify the degree since it is not implied. Here we use a finite field as the base ring.

sage: w = vector(FiniteField(7), 4); w
(0, 0, 0, 0)
sage: w.parent()
Vector space of dimension 4 over Finite Field of size 7


The fastest method to construct a zero vector is to call the zero_vector() method directly on a free module or vector space, since vector(…) must do a small amount of type checking. Almost as fast as the zero_vector() method is the zero_vector() constructor, which defaults to the integers.

sage: vector(ZZ, 5)          # works fine
(0, 0, 0, 0, 0)
sage: (ZZ^5).zero_vector()   # very tiny bit faster
(0, 0, 0, 0, 0)
sage: zero_vector(ZZ, 5)     # similar speed to vector(...)
(0, 0, 0, 0, 0)
sage: z = zero_vector(5); z
(0, 0, 0, 0, 0)
sage: z.parent()
Ambient free module of rank 5 over
the principal ideal domain Integer Ring


Here we illustrate the creation of sparse vectors by using a dictionary:

sage: vector({1:1.1, 3:3.14})
(0.000000000000000, 1.10000000000000, 0.000000000000000, 3.14000000000000)


With no degree given, a dictionary of entries implicitly declares a degree by the largest index (key) present. So you can provide a terminal element (perhaps a zero?) to set the degree. But it is probably safer to just include a degree in your construction.

sage: v = vector(QQ, {0:1/2, 4:-6, 7:0}); v
(1/2, 0, 0, 0, -6, 0, 0, 0)
sage: v.degree()
8
sage: v.is_sparse()
True
sage: w = vector(QQ, 8, {0:1/2, 4:-6})
sage: w == v
True


It is an error to specify a negative degree.

sage: vector(RR, -4, [1.0, 2.0, 3.0, 4.0])
Traceback (most recent call last):
...
ValueError: cannot specify the degree of a vector as a negative integer (-4)


It is an error to create a zero vector but not provide a ring as the first argument.

sage: vector('junk', 20)
Traceback (most recent call last):
...
TypeError: first argument must be base ring of zero vector, not junk


And it is an error to specify an index in a dictionary that is greater than or equal to a requested degree.

sage: vector(ZZ, 10, {3:4, 7:-2, 10:637})
Traceback (most recent call last):
...
ValueError: dictionary of entries has a key (index) exceeding the requested degree


A 1-dimensional numpy array of type float or complex may be passed to vector. Unless an explicit ring is given, the result will be a vector in the appropriate dimensional vector space over the real double field or the complex double field. The data in the array must be contiguous, so column-wise slices of numpy matrices will raise an exception.

sage: # needs numpy
sage: import numpy
sage: x = numpy.random.randn(10)
sage: y = vector(x)
sage: parent(y)
Vector space of dimension 10 over Real Double Field
sage: parent(vector(RDF, x))
Vector space of dimension 10 over Real Double Field
sage: parent(vector(CDF, x))
Vector space of dimension 10 over Complex Double Field
sage: parent(vector(RR, x))
Vector space of dimension 10 over Real Field with 53 bits of precision
sage: v = numpy.random.randn(10) * complex(0,1)
sage: w = vector(v)
sage: parent(w)
Vector space of dimension 10 over Complex Double Field


Multi-dimensional arrays are not supported:

sage: # needs numpy
sage: import numpy as np
sage: a = np.array([[1, 2, 3], [4, 5, 6]], np.float64)
sage: vector(a)
Traceback (most recent call last):
...
TypeError: cannot convert 2-dimensional array to a vector


If any of the arguments to vector have Python type int, real, or complex, they will first be coerced to the appropriate Sage objects. This fixes github issue #3847.

sage: v = vector([int(0)]); v
(0)
sage: v[0].parent()
Integer Ring
sage: v = vector(range(10)); v
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
sage: v[3].parent()
Integer Ring
sage: v = vector([float(23.4), int(2), complex(2+7*I), 1]); v                   # needs sage.symbolic
(23.4, 2.0, 2.0 + 7.0*I, 1.0)
sage: v[1].parent()                                                             # needs sage.symbolic
Complex Double Field


If the argument is a vector, it doesn’t change the base ring. This fixes github issue #6643:

sage: # needs sage.rings.number_field
sage: u = vector(K, (1/2, sqrt3/2))
sage: vector(u).base_ring()
Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?
sage: v = vector(K, (0, 1))
sage: vector(v).base_ring()
Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?


Constructing a vector from a numpy array behaves as expected:

sage: # needs numpy
sage: import numpy
sage: a = numpy.array([1,2,3])
sage: v = vector(a); v
(1, 2, 3)
sage: parent(v)
Ambient free module of rank 3 over the principal ideal domain Integer Ring


Complex numbers can be converted naturally to a sequence of length 2. And then to a vector.

sage: c = CDF(2 + 3*I)                                                          # needs sage.rings.complex_double sage.symbolic
sage: v = vector(c); v                                                          # needs sage.rings.complex_double sage.symbolic
(2.0, 3.0)


A generator, or other iterable, may also be supplied as input. Anything that can be converted to a Sequence is a possible input.

sage: type(i^2 for i in range(3))
<... 'generator'>
sage: v = vector(i^2 for i in range(3)); v
(0, 1, 4)


An empty list, without a ring given, will default to the integers.

sage: x = vector([]); x
()
sage: x.parent()
Ambient free module of rank 0 over the principal ideal domain Integer Ring


The immutable switch allows to create an immutable vector.

sage: v = vector(QQ, {0:1/2, 4:-6, 7:0}, immutable=True); v
(1/2, 0, 0, 0, -6, 0, 0, 0)
sage: v.is_immutable()
True


The immutable switch works regardless of the type of valid input to the constructor.

sage: v = vector(ZZ, 4, immutable=True)
sage: v.is_immutable()
True
sage: w = vector(ZZ, [1,2,3])
sage: v = vector(w, ZZ, immutable=True)
sage: v.is_immutable()
True
sage: v = vector(QQ, w, immutable=True)
sage: v.is_immutable()
True

sage: # needs numpy sage.symbolic
sage: import numpy as np
sage: w = np.array([1, 2, pi], float)
sage: v = vector(w, immutable=True)
sage: v.is_immutable()
True
sage: w = np.array([i, 2, 3], complex)
sage: v = vector(w, immutable=True)
sage: v.is_immutable()
True

sage.modules.free_module_element.is_FreeModuleElement(x)#

EXAMPLES:

sage: sage.modules.free_module_element.is_FreeModuleElement(0)
False
sage: sage.modules.free_module_element.is_FreeModuleElement(vector([1,2,3]))
True

sage.modules.free_module_element.make_FreeModuleElement_generic_dense(parent, entries, degree)#

EXAMPLES:

sage: sage.modules.free_module_element.make_FreeModuleElement_generic_dense(QQ^3, [1,2,-3/7], 3)
(1, 2, -3/7)

sage.modules.free_module_element.make_FreeModuleElement_generic_dense_v1(parent, entries, degree, is_mutable)#

EXAMPLES:

sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_dense_v1(QQ^3, [1,2,-3/7], 3, True); v
(1, 2, -3/7)
sage: v[0] = 10; v
(10, 2, -3/7)
sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_dense_v1(QQ^3, [1,2,-3/7], 3, False); v
(1, 2, -3/7)
sage: v[0] = 10
Traceback (most recent call last):
...

sage.modules.free_module_element.make_FreeModuleElement_generic_sparse(parent, entries, degree)#

EXAMPLES:

sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_sparse(QQ^3, {2:5/2}, 3); v
(0, 0, 5/2)

sage.modules.free_module_element.make_FreeModuleElement_generic_sparse_v1(parent, entries, degree, is_mutable)#

EXAMPLES:

sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_sparse_v1(QQ^3, {2:5/2}, 3, False); v
(0, 0, 5/2)
sage: v.is_mutable()
False

sage.modules.free_module_element.prepare(v, R, degree=None)#

Converts an object describing elements of a vector into a list of entries in a common ring.

INPUT:

• v - a dictionary with non-negative integers as keys, or a list or other object that can be converted by the Sequence constructor

• R - a ring containing all the entries, possibly given as None

• degree - a requested size for the list when the input is a dictionary, otherwise ignored

OUTPUT:

A pair.

The first item is a list of the values specified in the object v. If the object is a dictionary , entries are placed in the list according to the indices that were their keys in the dictionary, and the remainder of the entries are zero. The value of degree is assumed to be larger than any index provided in the dictionary and will be used as the number of entries in the returned list.

The second item returned is a ring that contains all of the entries in the list. If R is given, the entries are coerced in. Otherwise a common ring is found. For more details, see the Sequence object. When v has no elements and R is None, the ring returned is the integers.

EXAMPLES:

sage: from sage.modules.free_module_element import prepare
sage: prepare([1, 2/3, 5], None)
([1, 2/3, 5], Rational Field)

sage: prepare([1, 2/3, 5], RR)
([1.00000000000000, 0.666666666666667, 5.00000000000000],
Real Field with 53 bits of precision)

sage: prepare({1: 4, 3: -2}, ZZ, 6)
([0, 4, 0, -2, 0, 0], Integer Ring)

sage: prepare({3: 1, 5: 3}, QQ, 6)
([0, 0, 0, 1, 0, 3], Rational Field)

sage: prepare([1, 2/3, '10', 5], RR)
([1.00000000000000, 0.666666666666667, 10.0000000000000, 5.00000000000000],
Real Field with 53 bits of precision)

sage: prepare({}, QQ, 0)
([], Rational Field)

sage: prepare([1, 2/3, '10', 5], None)
Traceback (most recent call last):
...
TypeError: unable to find a common ring for all elements


Some objects can be converted to sequences even if they are not always thought of as sequences.

sage: c = CDF(2 + 3*I)                                                          # needs sage.symbolic
sage: prepare(c, None)                                                          # needs sage.symbolic
([2.0, 3.0], Real Double Field)


This checks a bug listed at github issue #10595. Without good evidence for a ring, the default is the integers.

sage: prepare([], None)
([], Integer Ring)

sage.modules.free_module_element.random_vector(ring, degree=None, *args, **kwds)#

Returns a vector (or module element) with random entries.

INPUT:

• ring – default: ZZ - the base ring for the entries

• degree – a non-negative integer for the number of entries in the vector

• sparse – default: False - whether to use a sparse implementation

• args, kwds - additional arguments and keywords are passed to the random_element() method of the ring

OUTPUT:

A vector, or free module element, with degree elements from ring, chosen randomly from the ring according to the ring’s random_element() method.

Note

See below for examples of how random elements are generated by some common base rings.

EXAMPLES:

First, module elements over the integers. The default distribution is tightly clustered around -1, 0, 1. Uniform distributions can be specified by giving bounds, though the upper bound is never met. See sage.rings.integer_ring.IntegerRing_class.random_element() for several other variants.

sage: random_vector(10).parent()
Ambient free module of rank 10 over the principal ideal domain Integer Ring
sage: random_vector(20).parent()
Ambient free module of rank 20 over the principal ideal domain Integer Ring

sage: v = random_vector(ZZ, 20, x=4)
sage: all(i in range(4) for i in v)
True

sage: v = random_vector(ZZ, 20, x=-20, y=100)
sage: all(i in range(-20, 100) for i in v)
True


If the ring is not specified, the default is the integers, and parameters for the random distribution may be passed without using keywords. This is a random vector with 20 entries uniformly distributed between -20 and 100.

sage: random_vector(20, -20, 100).parent()
Ambient free module of rank 20 over the principal ideal domain Integer Ring


Now over the rationals. Note that bounds on the numerator and denominator may be specified. See sage.rings.rational_field.RationalField.random_element() for documentation.

sage: random_vector(QQ, 10).parent()
Vector space of dimension 10 over Rational Field

sage: v = random_vector(QQ, 10, num_bound=15, den_bound=5)
sage: v.parent()
Vector space of dimension 10 over Rational Field
sage: all(q.numerator() <= 15 and q.denominator() <= 5 for q in v)
True


Inexact rings may be used as well. The reals have uniform distributions, with the range $$(-1,1)$$ as the default. More at: sage.rings.real_mpfr.RealField_class.random_element()

sage: v = random_vector(RR, 5)
sage: v.parent()
Vector space of dimension 5 over Real Field with 53 bits of precision
sage: all(-1 <= r <= 1 for r in v)
True

sage: v = random_vector(RR, 5, min=8, max=14)
sage: v.parent()
Vector space of dimension 5 over Real Field with 53 bits of precision
sage: all(8 <= r <= 14 for r in v)
True


Any ring with a random_element() method may be used.

sage: F = FiniteField(23)
sage: hasattr(F, 'random_element')
True
sage: v = random_vector(F, 10)
sage: v.parent()
Vector space of dimension 10 over Finite Field of size 23


The default implementation is a dense representation, equivalent to setting sparse=False.

sage: v = random_vector(10)
sage: v.is_sparse()
False

sage: w = random_vector(ZZ, 20, sparse=True)
sage: w.is_sparse()
True


The elements are chosen using the ring’s random_element method:

sage: from sage.misc.randstate import current_randstate
sage: seed = current_randstate().seed()
sage: set_random_seed(seed)
sage: v1 = random_vector(ZZ, 20, distribution="1/n")
sage: v2 = random_vector(ZZ, 15, x=-1000, y=1000)
sage: v3 = random_vector(QQ, 10)
sage: v4 = random_vector(FiniteField(17), 10)
sage: v5 = random_vector(RR, 10)
sage: set_random_seed(seed)
sage: w1 = vector(ZZ.random_element(distribution="1/n") for _ in range(20))
sage: w2 = vector(ZZ.random_element(x=-1000, y=1000) for _ in range(15))
sage: w3 = vector(QQ.random_element() for _ in range(10))
sage: [v1, v2, v3] == [w1, w2, w3]
True
sage: w4 = vector(FiniteField(17).random_element() for _ in range(10))
sage: v4 == w4
True
sage: w5 = vector(RR.random_element() for _ in range(10))
sage: v5 == w5
True


Inputs get checked before constructing the vector.

sage: random_vector('junk')
Traceback (most recent call last):
...
TypeError: degree of a random vector must be an integer, not None

sage: random_vector('stuff', 5)
Traceback (most recent call last):
...
TypeError: elements of a vector, or module element, must come from a ring, not stuff

sage: random_vector(ZZ, -9)
Traceback (most recent call last):
...
ValueError: degree of a random vector must be non-negative, not -9

sage.modules.free_module_element.vector(arg0, arg1=None, arg2=None, sparse=None, immutable=False)#

Return a vector or free module element with specified entries.

CALL FORMATS:

This constructor can be called in several different ways. In each case, sparse=True or sparse=False as well as immutable=True or immutable=False can be supplied as an option. free_module_element() is an alias for vector().

1. vector(object)

2. vector(ring, object)

3. vector(object, ring)

4. vector(ring, degree, object)

5. vector(ring, degree)

INPUT:

• object – a list, dictionary, or other iterable containing the entries of the vector, including any object that is palatable to the Sequence constructor

• ring – a base ring (or field) for the vector space or free module, which contains all of the elements

• degree – an integer specifying the number of entries in the vector or free module element

• sparse – boolean, whether the result should be a sparse vector

• immutable – boolean (default: False); whether the result should be an immutable vector

In call format 4, an error is raised if the degree does not match the length of object so this call can provide some safeguards. Note however that using this format when object is a dictionary is unlikely to work properly.

OUTPUT:

An element of the ambient vector space or free module with the given base ring and implied or specified dimension or rank, containing the specified entries and with correct degree.

In call format 5, no entries are specified, so the element is populated with all zeros.

If the sparse option is not supplied, the output will generally have a dense representation. The exception is if object is a dictionary, then the representation will be sparse.

EXAMPLES:

sage: v = vector([1,2,3]); v
(1, 2, 3)
sage: v.parent()
Ambient free module of rank 3 over the principal ideal domain Integer Ring
sage: v = vector([1,2,3/5]); v
(1, 2, 3/5)
sage: v.parent()
Vector space of dimension 3 over Rational Field


All entries must canonically coerce to some common ring:

sage: v = vector([17, GF(11)(5), 19/3]); v
Traceback (most recent call last):
...
TypeError: unable to find a common ring for all elements

sage: v = vector([17, GF(11)(5), 19]); v
(6, 5, 8)
sage: v.parent()
Vector space of dimension 3 over Finite Field of size 11
sage: v = vector([17, GF(11)(5), 19], QQ); v
(17, 5, 19)
sage: v.parent()
Vector space of dimension 3 over Rational Field
sage: v = vector((1,2,3), QQ); v
(1, 2, 3)
sage: v.parent()
Vector space of dimension 3 over Rational Field
sage: v = vector(QQ, (1,2,3)); v
(1, 2, 3)
sage: v.parent()
Vector space of dimension 3 over Rational Field
sage: v = vector(vector([1,2,3])); v
(1, 2, 3)
sage: v.parent()
Ambient free module of rank 3 over the principal ideal domain Integer Ring


You can also use free_module_element, which is the same as vector.

sage: free_module_element([1/3, -4/5])
(1/3, -4/5)


We make a vector mod 3 out of a vector over $$\ZZ$$.

sage: vector(vector([1,2,3]), GF(3))
(1, 2, 0)


The degree of a vector may be specified:

sage: vector(QQ, 4, [1,1/2,1/3,1/4])
(1, 1/2, 1/3, 1/4)


But it is an error if the degree and size of the list of entries are mismatched:

sage: vector(QQ, 5, [1,1/2,1/3,1/4])
Traceback (most recent call last):
...
ValueError: incompatible degrees in vector constructor


Providing no entries populates the vector with zeros, but of course, you must specify the degree since it is not implied. Here we use a finite field as the base ring.

sage: w = vector(FiniteField(7), 4); w
(0, 0, 0, 0)
sage: w.parent()
Vector space of dimension 4 over Finite Field of size 7


The fastest method to construct a zero vector is to call the zero_vector() method directly on a free module or vector space, since vector(…) must do a small amount of type checking. Almost as fast as the zero_vector() method is the zero_vector() constructor, which defaults to the integers.

sage: vector(ZZ, 5)          # works fine
(0, 0, 0, 0, 0)
sage: (ZZ^5).zero_vector()   # very tiny bit faster
(0, 0, 0, 0, 0)
sage: zero_vector(ZZ, 5)     # similar speed to vector(...)
(0, 0, 0, 0, 0)
sage: z = zero_vector(5); z
(0, 0, 0, 0, 0)
sage: z.parent()
Ambient free module of rank 5 over
the principal ideal domain Integer Ring


Here we illustrate the creation of sparse vectors by using a dictionary:

sage: vector({1:1.1, 3:3.14})
(0.000000000000000, 1.10000000000000, 0.000000000000000, 3.14000000000000)


With no degree given, a dictionary of entries implicitly declares a degree by the largest index (key) present. So you can provide a terminal element (perhaps a zero?) to set the degree. But it is probably safer to just include a degree in your construction.

sage: v = vector(QQ, {0:1/2, 4:-6, 7:0}); v
(1/2, 0, 0, 0, -6, 0, 0, 0)
sage: v.degree()
8
sage: v.is_sparse()
True
sage: w = vector(QQ, 8, {0:1/2, 4:-6})
sage: w == v
True


It is an error to specify a negative degree.

sage: vector(RR, -4, [1.0, 2.0, 3.0, 4.0])
Traceback (most recent call last):
...
ValueError: cannot specify the degree of a vector as a negative integer (-4)


It is an error to create a zero vector but not provide a ring as the first argument.

sage: vector('junk', 20)
Traceback (most recent call last):
...
TypeError: first argument must be base ring of zero vector, not junk


And it is an error to specify an index in a dictionary that is greater than or equal to a requested degree.

sage: vector(ZZ, 10, {3:4, 7:-2, 10:637})
Traceback (most recent call last):
...
ValueError: dictionary of entries has a key (index) exceeding the requested degree


A 1-dimensional numpy array of type float or complex may be passed to vector. Unless an explicit ring is given, the result will be a vector in the appropriate dimensional vector space over the real double field or the complex double field. The data in the array must be contiguous, so column-wise slices of numpy matrices will raise an exception.

sage: # needs numpy
sage: import numpy
sage: x = numpy.random.randn(10)
sage: y = vector(x)
sage: parent(y)
Vector space of dimension 10 over Real Double Field
sage: parent(vector(RDF, x))
Vector space of dimension 10 over Real Double Field
sage: parent(vector(CDF, x))
Vector space of dimension 10 over Complex Double Field
sage: parent(vector(RR, x))
Vector space of dimension 10 over Real Field with 53 bits of precision
sage: v = numpy.random.randn(10) * complex(0,1)
sage: w = vector(v)
sage: parent(w)
Vector space of dimension 10 over Complex Double Field


Multi-dimensional arrays are not supported:

sage: # needs numpy
sage: import numpy as np
sage: a = np.array([[1, 2, 3], [4, 5, 6]], np.float64)
sage: vector(a)
Traceback (most recent call last):
...
TypeError: cannot convert 2-dimensional array to a vector


If any of the arguments to vector have Python type int, real, or complex, they will first be coerced to the appropriate Sage objects. This fixes github issue #3847.

sage: v = vector([int(0)]); v
(0)
sage: v[0].parent()
Integer Ring
sage: v = vector(range(10)); v
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
sage: v[3].parent()
Integer Ring
sage: v = vector([float(23.4), int(2), complex(2+7*I), 1]); v                   # needs sage.symbolic
(23.4, 2.0, 2.0 + 7.0*I, 1.0)
sage: v[1].parent()                                                             # needs sage.symbolic
Complex Double Field


If the argument is a vector, it doesn’t change the base ring. This fixes github issue #6643:

sage: # needs sage.rings.number_field
sage: u = vector(K, (1/2, sqrt3/2))
sage: vector(u).base_ring()
Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?
sage: v = vector(K, (0, 1))
sage: vector(v).base_ring()
Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?


Constructing a vector from a numpy array behaves as expected:

sage: # needs numpy
sage: import numpy
sage: a = numpy.array([1,2,3])
sage: v = vector(a); v
(1, 2, 3)
sage: parent(v)
Ambient free module of rank 3 over the principal ideal domain Integer Ring


Complex numbers can be converted naturally to a sequence of length 2. And then to a vector.

sage: c = CDF(2 + 3*I)                                                          # needs sage.rings.complex_double sage.symbolic
sage: v = vector(c); v                                                          # needs sage.rings.complex_double sage.symbolic
(2.0, 3.0)


A generator, or other iterable, may also be supplied as input. Anything that can be converted to a Sequence is a possible input.

sage: type(i^2 for i in range(3))
<... 'generator'>
sage: v = vector(i^2 for i in range(3)); v
(0, 1, 4)


An empty list, without a ring given, will default to the integers.

sage: x = vector([]); x
()
sage: x.parent()
Ambient free module of rank 0 over the principal ideal domain Integer Ring


The immutable switch allows to create an immutable vector.

sage: v = vector(QQ, {0:1/2, 4:-6, 7:0}, immutable=True); v
(1/2, 0, 0, 0, -6, 0, 0, 0)
sage: v.is_immutable()
True


The immutable switch works regardless of the type of valid input to the constructor.

sage: v = vector(ZZ, 4, immutable=True)
sage: v.is_immutable()
True
sage: w = vector(ZZ, [1,2,3])
sage: v = vector(w, ZZ, immutable=True)
sage: v.is_immutable()
True
sage: v = vector(QQ, w, immutable=True)
sage: v.is_immutable()
True

sage: # needs numpy sage.symbolic
sage: import numpy as np
sage: w = np.array([1, 2, pi], float)
sage: v = vector(w, immutable=True)
sage: v.is_immutable()
True
sage: w = np.array([i, 2, 3], complex)
sage: v = vector(w, immutable=True)
sage: v.is_immutable()
True

sage.modules.free_module_element.zero_vector(arg0, arg1=None)#

Returns a vector or free module element with a specified number of zeros.

CALL FORMATS:

1. zero_vector(degree)

2. zero_vector(ring, degree)

INPUT:

• degree - the number of zero entries in the vector or free module element

• ring - default ZZ - the base ring of the vector space or module containing the constructed zero vector

OUTPUT:

A vector or free module element with degree entries, all equal to zero and belonging to the ring if specified. If no ring is given, a free module element over ZZ is returned.

EXAMPLES:

A zero vector over the field of rationals.

sage: v = zero_vector(QQ, 5); v
(0, 0, 0, 0, 0)
sage: v.parent()
Vector space of dimension 5 over Rational Field


A free module zero element.

sage: w = zero_vector(Integers(6), 3); w
(0, 0, 0)
sage: w.parent()
Ambient free module of rank 3 over Ring of integers modulo 6


If no ring is given, the integers are used.

sage: u = zero_vector(9); u
(0, 0, 0, 0, 0, 0, 0, 0, 0)
sage: u.parent()
Ambient free module of rank 9 over the principal ideal domain Integer Ring


Non-integer degrees produce an error.

sage: zero_vector(5.6)
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer


Negative degrees also give an error.

sage: zero_vector(-3)
Traceback (most recent call last):
...
ValueError: rank (=-3) must be nonnegative


Garbage instead of a ring will be recognized as such.

sage: zero_vector(x^2, 5)                                                       # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: first argument must be a ring