elementary_vectors.functions_dd¶

Double description

EXAMPLES:

We consider the following matrix:

sage: M = matrix([[1,2,0,0,3],[0,1,-1,2,1]])
sage: M
[ 1  2  0  0  3]
[ 0  1 -1  2  1]

Next, we compute the elementary vectors of this matrix:

sage: from elementary_vectors import *
sage: evs = elementary_vectors(M)
sage: evs
[(-2, 1, 1, 0, 0),
 (4, -2, 0, 1, 0),
 (-1, -1, 0, 0, 1),
 (0, 0, -2, -1, 0),
 (3, 0, -1, 0, -1),
 (-6, 0, 0, -1, 2),
 (0, 3, 1, 0, -2),
 (0, -6, 0, 1, 4)]

We compute the corresponding sign vectors:

sage: from sign_vectors import *
sage: svs = [sign_vector(v) for v in evs]
sage: svs
[(-++00), (+-0+0), (--00+), (00--0), (+0-0-), (-00-+), (0++0-), (0-0++)]

Now, we define a vector a. We will use this vector as a third row in our matrix:

sage: a = vector([1, 1, 1, 0, 0])
sage: Ma = M.stack(a)
sage: Ma
[ 1  2  0  0  3]
[ 0  1 -1  2  1]
[ 1  1  1  0  0]

That way, we describe a different vector space. The corresponding elementary vectors can be computed as before:

sage: evs_a = elementary_vectors(Ma)
sage: evs_a
[(-4, 2, 2, 0, 0), (-6, 6, 0, -2, -2), (-6, 0, 6, 2, 2), (0, -6, 6, 4, 4)]

Similarly, we obtain the following sign vectors:

sage: [sign_vector(v) for v in evs_a]
[(-++00), (-+0--), (-0+++), (0-+++)]

A different approach is to use the double description method. First, we compute two lists of sign vectors:

sage: from elementary_vectors.functions_dd import *
sage: E0, Ep = dd_input(M, svs, a)
sage: E0
[(-++00)]
sage: Ep
[(+-0+0), (++00-), (00++0), (+0-0-), (+00+-), (0++0-), (0+0--)]

Then, we use the computed lists, to compute the new list of sign vectors by applying double description:

sage: dd(E0, Ep)
[(-++00), (+-0++), (-0+++), (0-+++)]

There, is also a convenient command that computed this list of sign vectors:

sage: double_description(M, a)
[(-++00), (+-0++), (-0+++), (0-+++)]

Functions

`cocircuits_iterative`(ai)	Compute the sign vectors corresponding to given vectors.
`dd`(E0, Ep, **kwargs)	Compute the sign vectors corresponding to given lists of sign vectors.
`dd_input`(M, svs, a)	INPUT:
`determine_sign`(X, a[, M])	Determine the sign of the corresponding scalar product.
`double_description`(M, a)	INPUT:

elementary_vectors.functions_dd.cocircuits_iterative(ai)¶

Compute the sign vectors corresponding to given vectors.

INPUT:

ai – a list of vectors

OUTPUT: Compute iteratively the sign vectors corresponding to the elementary vectors determined by the given vectors ai.

EXAMPLES:

sage: from elementary_vectors.functions_dd import cocircuits_iterative
sage: ai = [vector([1,-2,0,2,2]), vector([0,1,4,4,1]), vector([1,0,-1,0,0])]
sage: cocircuits_iterative(ai)
[(0+0-+), (+-+-0), (+0+-+), (+-+0-)]

elementary_vectors.functions_dd.dd(E0, Ep, **kwargs)¶

Compute the sign vectors corresponding to given lists of sign vectors.

INPUT:

E0 – a list of sign vectors
Ep – a list of sign vectors

OUTPUT: TODO

See also

dd_input() double_description() elementary_vectors()

elementary_vectors.functions_dd.dd_input(M, svs, a)¶

INPUT:

M – a matrix
svs – a list of sign vectors
a – a vector

OUTPUT: a tuple (E0, Ep) where E0 are sign vectors such that the corresponding elementary vectors v satisfy a v = 0 and Ep are sign vectors such that the corresponding elementary vectors v satisfy a v > 0.

elementary_vectors.functions_dd.determine_sign(X, a, M=None)¶

Determine the sign of the corresponding scalar product.

INPUT:

X – a sign vector
a – a vector
M – a matrix (default: None)

OUTPUT: First, the vector v corresponding to the sign vector sv is computed. Then, the sign of the scalar product a v is returned. If the sign can not be determined, the matrix M is used to find an appropriate sign vector for X. In this case, if M is not specified, an exception is raised.

Note

This might not work if X is not a cocircuit of the oriented matroid corresponding to the kernel of M.

EXAMPLES:

sage: from sign_vectors import *
sage: from elementary_vectors.functions_dd import *
sage: M = matrix([[1, 1, 2, 3], [2, -1, 0, 0]])
sage: X = sign_vector("++-0")
sage: a = vector([1, 0, 0, 0])
sage: determine_sign(X, a, M)
1
sage: a = vector([0, 0, 0, 1])
sage: determine_sign(X, a)
0
sage: a = vector([1, 1, 0, 0])
sage: determine_sign(X, a)
1
sage: a = vector([1, -1, 0, 0])
sage: determine_sign(X, a, M)
-1
sage: determine_sign(-X, a, M)
1

elementary_vectors.functions_dd.double_description(M, a)¶

INPUT:

M – a matrix
a – a vector

OUTPUT: a list of elementary vectors v of data satisfying a v = 0.

EXAMPLES:

We consider the following matrix:

sage: M = matrix([[1,2,0,0,3],[0,1,-1,2,1]])
sage: M
[ 1  2  0  0  3]
[ 0  1 -1  2  1]

Next, we compute the elementary vectors of this matrix:

sage: from elementary_vectors import *
sage: evs = elementary_vectors(M)
sage: evs
[(-2, 1, 1, 0, 0),
 (4, -2, 0, 1, 0),
 (-1, -1, 0, 0, 1),
 (0, 0, -2, -1, 0),
 (3, 0, -1, 0, -1),
 (-6, 0, 0, -1, 2),
 (0, 3, 1, 0, -2),
 (0, -6, 0, 1, 4)]

We compute the corresponding sign vectors:

sage: from sign_vectors import *
sage: svs = [sign_vector(v) for v in evs]
sage: svs
[(-++00), (+-0+0), (--00+), (00--0), (+0-0-), (-00-+), (0++0-), (0-0++)]

Now, we define a vector a. We will use this vector as a third row in our matrix:

sage: a = vector([1, 1, 1, 0, 0])
sage: Ma = M.stack(a)
sage: Ma
[ 1  2  0  0  3]
[ 0  1 -1  2  1]
[ 1  1  1  0  0]

That way, we describe a different vector space. The corresponding elementary vectors can be computed as before:

sage: evs_a = elementary_vectors(Ma)
sage: evs_a
[(-4, 2, 2, 0, 0), (-6, 6, 0, -2, -2), (-6, 0, 6, 2, 2), (0, -6, 6, 4, 4)]

Similarly, we obtain the following sign vectors:

sage: [sign_vector(v) for v in evs_a]
[(-++00), (-+0--), (-0+++), (0-+++)]

A different approach is to use the double description method:

sage: double_description(M, a)
[(-++00), (+-0++), (-0+++), (0-+++)]

The output is identical up to multiplies.

elementary_vectors.functions_dd¶

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