Elements of free modules#

AUTHORS:

William Stein
Josh Kantor
Thomas Feulner (2012-11): Added FreeModuleElement.hamming_weight() and FreeModuleElement_generic_sparse.hamming_weight()
Jeroen Demeyer (2015-02-24): Implement fast Cython methods get_unsafe and set_unsafe similar to other places in Sage (trac ticket #17562)

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: sage.structure.element.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

additive_order()#

Return the additive order of self.

EXAMPLES:

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

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

sage: v = vector(Integers(30), [6, 15]); v
(6, 15)
sage: v.additive_order()
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)])
sage: m.apply_map(lambda x: x^2)
(1, x^2, sin(x + 1)^2)
sage: m.apply_map(sin)
(sin(1), sin(x), sin(sin(x + 1)))

sage: m = vector(ZZ, 9, range(9))
sage: k.<a> = GF(9)
sage: m.apply_map(k)
(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
(0, 1, 2, 0, 1, 2, 0, 1, 2)
sage: n.parent()
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)])
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})
sage: dm = d.column()
sage: sm = s.column()
sage: all([d.is_dense(), dm.is_dense(), s.is_sparse(), sm.is_sparse()])
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])
sage: w = v.conjugate(); w
(2.3 + 5.4*I, -1.7 - 3.6*I)
sage: w.parent()
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: 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: 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])
sage: v.curl()
Traceback (most recent call last):
...
ValueError: Unable to determine ordered variable names for Symbolic Ring
sage: v.curl([x, y, z])
(0, 0, 2)

Note that callable vectors have well defined variable orderings:

sage: v(x, y, z) = (-y, x, 0)
sage: v.curl()
(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

See also sage.misc.functional.norm()

EXAMPLES:

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

The default is the usual Euclidean norm.

sage: v.norm()
sqrt(14)
sage: v.norm(2)
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: 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: 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()
(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()
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 trac ticket #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: 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()
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: 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: 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
array([                  0,                   1, 9223372036854775807]...)
sage: wn.dtype
dtype('int64')
sage: w.dot_product(w)
85070591730234615847396907784232501250
sage: wn.dot(wn)        # overflow
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)
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)
sage: 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)
sage: B = v.plot()
sage: A+B # should just show one vector
Graphics object consisting of 2 graphics primitives

Examples of the plot types:

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)
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
Graphics3d Object

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

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

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

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

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

sage: plot(vector([1]))
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)
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)
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)])
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})
sage: dm = d.row()
sage: sm = s.row()
sage: all([d.is_dense(), dm.is_dense(), s.is_sparse(), sm.is_sparse()])
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
(1/2, 2/5, 0)
sage: v.set(2, pi); v
(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: 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:

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

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

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)
sage: 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#

Bases: sage.modules.free_module_element.FreeModuleElement

A generic dense element of a free module.

function(*args)#

Return a vector over a callable symbolic expression ring.

EXAMPLES:

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: 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#

Bases: sage.modules.free_module_element.FreeModuleElement

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)
sage: next(v.items())
(0, 1)
sage: list(v.items())
[(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)
sage: next(v.items())
(0, 1)
sage: list(v.items())
[(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().

vector(object)

vector(ring, object)

vector(object, ring)

vector(ring, degree, object)

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: 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: 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, long, real, or complex, they will first be coerced to the appropriate Sage objects. This fixes trac ticket #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
(23.4, 2.0, 2.0 + 7.0*I, 1.0)
sage: v[1].parent()
Complex Double Field

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

sage: K.<sqrt3> = QuadraticField(3)
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: 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)
sage: v = vector(c); v
(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: 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):
...
ValueError: vector is immutable; please change a copy instead (use copy())

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)
sage: prepare(c, None)
([2.0, 3.0], Real Double Field)

This checks a bug listed at trac ticket #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: w4 = vector(FiniteField(17).random_element() for _ in range(10))
sage: w5 = vector(RR.random_element() for _ in range(10))
sage: [v1, v2, v3, v4, v5] == [w1, w2, w3, w4, 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:

vector(object)

vector(ring, object)

vector(object, ring)

vector(ring, degree, object)

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

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

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)

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

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: 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, long, real, or complex, they will first be coerced to the appropriate Sage objects. This fixes trac ticket #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
(23.4, 2.0, 2.0 + 7.0*I, 1.0)
sage: v[1].parent()
Complex Double Field

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

sage: K.<sqrt3> = QuadraticField(3)
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: 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)
sage: v = vector(c); v
(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: 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:

zero_vector(degree)

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)
Traceback (most recent call last):
...
TypeError: first argument must be a ring