Coverage for local/lib/python2.7/site-packages/sage/geometry/polyhedron/double_description.py : 99%

""" 

Double Description Algorithm for Cones 

This module implements the double description algorithm for extremal 

vertex enumeration in a pointed cone following [FP1996]_. With a 

little bit of preprocessing (see 

:mod:`~sage.geometry.polyhedron.double_description_inhomogeneous`) 

this defines a backend for polyhedral computations. But as far as this 

module is concerned, *inequality* always means without a constant term 

and the origin is always a point of the cone. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: alg = StandardAlgorithm(A); alg 

Pointed cone with inequalities 

(1, 0, 1) 

(0, 1, 1) 

(-1, -1, 1) 

sage: DD, _ = alg.initial_pair(); DD 

Double description pair (A, R) defined by 

[ 1 0 1] [ 2/3 -1/3 -1/3] 

A = [ 0 1 1], R = [-1/3 2/3 -1/3] 

[-1 -1 1] [ 1/3 1/3 1/3] 

The implementation works over any exact field that is embedded in 

`\RR`, for example:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(AA, [(1,0,1), (0,1,1), (-AA(2).sqrt(),-AA(3).sqrt(),1), 

....: (-AA(3).sqrt(),-AA(2).sqrt(),1)]) 

sage: alg = StandardAlgorithm(A) 

sage: alg.run().R 

[(-0.4177376677004119?, 0.5822623322995881?, 0.4177376677004119?), 

(-0.2411809548974793?, -0.2411809548974793?, 0.2411809548974793?), 

(0.07665629029830300?, 0.07665629029830300?, 0.2411809548974793?), 

(0.5822623322995881?, -0.4177376677004119?, 0.4177376677004119?)] 

""" 

#***************************************************************************** 

# Copyright (C) 2014 Volker Braun <vbraun.name@gmail.com> 

# 2015 Vincent Delecroix <20100.delecroix@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

#***************************************************************************** 

# TODO 

# 

# The adjacency check should use caching and the "combinatorial 

# criterion" instead of the "algebraic criterion", see [FP1996] 

# for definition. Since coefficient arithmetic is relatively expensive 

# we should avoid it as far as possible. 

# 

# Also, the variants of the double description algorithm described in 

# [FP1996] should be implemented. The design of this module is 

# such that variants of the basic algorithm should be easy to add as 

# subclasses of DoubleDescriptionPair and Problem. 

# ***************************************************************************** 

# Compare with PPL if the base ring is QQ. Can be left enabled since 

# we don't use the Python fallback for polyhedra over QQ unless you 

# construct one by hand. 

from __future__ import division, absolute_import 

VERIFY_RESULT = True 

import itertools 

from sage.misc.cachefunc import cached_method 

from sage.rings.all import QQ 

from sage.modules.free_module_element import vector 

from sage.matrix.matrix_space import MatrixSpace 

def random_inequalities(d, n): 

""" 

Random collections of inequalities for testing purposes. 

INPUT: 

- ``d`` -- integer. The dimension. 

- ``n`` -- integer. The number of random inequalities to generate. 

OUTPUT: 

A random set of inequalites as a :class:`StandardAlgorithm` instance. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import random_inequalities 

sage: P = random_inequalities(5, 10) 

sage: P.run().verify() 

""" 

from sage.matrix.constructor import random_matrix 

while True: 

A = random_matrix(QQ, n, d) 

if A.rank() == min(n, d) and not any(a == 0 for a in A.rows()): 

break 

return StandardAlgorithm(A) 

class DoubleDescriptionPair: 

def __init__(self, problem, A_rows, R_cols): 

r""" 

Base class for a double description pair `(A, R)` 

.. warning:: 

You should use the :meth:`Problem.initial_pair` or 

:meth:`Problem.run` to generate double description pairs 

for a set of inequalities, and not generate 

``DoubleDescriptionPair`` instances directly. 

INPUT: 

- ``problem`` -- instance of :class:`Problem`. 

- ``A_rows`` -- list of row vectors of the matrix `A`. These 

encode the inequalities. 

- ``R_cols`` -- list of column vectors of the matrix 

`R`. These encode the rays. 

TESTS:: 

sage: from sage.geometry.polyhedron.double_description import \ 

....: DoubleDescriptionPair, Problem 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: alg = Problem(A) 

sage: DoubleDescriptionPair(alg, 

....: [(1, 0, 1), (0, 1, 1), (-1, -1, 1)], 

....: [(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3), (-1/3, -1/3, 1/3)]) 

Double description pair (A, R) defined by 

[ 1 0 1] [ 2/3 -1/3 -1/3] 

A = [ 0 1 1], R = [-1/3 2/3 -1/3] 

[-1 -1 1] [ 1/3 1/3 1/3] 

""" 

self.problem = problem 

self.A = list(A_rows) 

self.R = list(R_cols) 

self.one = problem._field.one() 

self.zero = problem._field.zero() 

# a cache for scalar products (see the method zero_set) 

self.zero_set_cache = {} 

def _make_new(self, A_rows, R_cols): 

r""" 

Construct a new double description pair. 

INPUT: 

- ``A_rows`` -- list of row vectors of the matrix `A`. These 

encode the inequalities. 

- ``R_cols`` -- list of column vectors of the matrix 

`R`. These encode the rays. 

OUTPUT: 

A new double description pair of the same (sub)class of 

:class:`DoubleDescriptionProblem`. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import \ 

....: DoubleDescriptionPair, StandardAlgorithm 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: DD = StandardAlgorithm(A).run() 

sage: DDnew = DD._make_new(DD.A, DD.R); DDnew 

Double description pair (A, R) defined by 

[ 1 0 1] [ 2/3 -1/3 -1/3] 

A = [ 0 1 1], R = [-1/3 2/3 -1/3] 

[-1 -1 1] [ 1/3 1/3 1/3] 

sage: DDnew is DD 

False 

sage: DDnew.__class__ is DD.__class__ 

True 

""" 

return self.__class__(self.problem, A_rows, R_cols) 

def __repr__(self): 

r""" 

Return string representation. 

OUTPUT: 

String. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import \ 

....: DoubleDescriptionPair, StandardAlgorithm 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: DD = StandardAlgorithm(A).run() 

sage: DD.__repr__() 

'Double description pair (A, R) defined by\n [ 1 0 1] 

[ 2/3 -1/3 -1/3]\nA = [ 0 1 1], R = [-1/3 2/3 -1/3]\n 

[-1 -1 1] [ 1/3 1/3 1/3]' 

""" 

from sage.typeset.ascii_art import ascii_art 

from sage.matrix.constructor import matrix 

s = ascii_art('Double description pair (A, R) defined by') 

A = ascii_art(matrix(self.A)) 

A._baseline = (len(self.A) // 2) 

A = ascii_art('A = ') + A 

R = ascii_art(matrix(self.R).transpose()) 

if len(self.R) > 0: 

R._baseline = (len(self.R[0]) // 2) 

else: 

R._baseline = 0 

R = ascii_art('R = ') + R 

return str(s * (A + ascii_art(', ') + R)) 

def inner_product_matrix(self): 

""" 

Return the inner product matrix between the rows of `A` 

and the columns of `R`. 

OUTPUT: 

A matrix over the base ring. There is one row for each row of 

`A` and one column for each column of `R`. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: alg = StandardAlgorithm(A) 

sage: DD, _ = alg.initial_pair() 

sage: DD.inner_product_matrix() 

[1 0 0] 

[0 1 0] 

[0 0 1] 

""" 

from sage.matrix.constructor import matrix 

return matrix(self.problem.base_ring(), [[a.inner_product(r) for r in self.R] for a in self.A]) 

def cone(self): 

""" 

Return the cone defined by `A`. 

This method is for debugging only. Assumes that the base ring 

is `\QQ`. 

OUTPUT: 

The cone defined by the inequalities as a 

:func:`~sage.geometry.polyhedron.constructor.Polyhedron`, 

using the PPL backend. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: DD, _ = StandardAlgorithm(A).initial_pair() 

sage: DD.cone().Hrepresentation() 

(An inequality (-1, -1, 1) x + 0 >= 0, 

An inequality (0, 1, 1) x + 0 >= 0, 

An inequality (1, 0, 1) x + 0 >= 0) 

""" 

from sage.geometry.polyhedron.constructor import Polyhedron 

assert self.problem.base_ring() == QQ # required for PPL backend 

if not self.A: 

return Polyhedron(vertices=[[0] * self.problem.dim()], backend='ppl') 

else: 

ieqs = [[0] + list(a) for a in self.A] 

return Polyhedron(ieqs=ieqs, base_ring=self.problem.base_ring(), backend='ppl') 

def verify(self): 

r""" 

Validate the double description pair. 

This method used the PPL backend to check that the double 

description pair is valid. An assertion is triggered if it is 

not. Does nothing if the base ring is not `\QQ`. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import \ 

....: DoubleDescriptionPair, Problem 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: alg = Problem(A) 

sage: DD = DoubleDescriptionPair(alg, 

....: [(1, 0, 3), (0, 1, 1), (-1, -1, 1)], 

....: [(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3), (-1/3, -1/3, 1/3)]) 

sage: DD.verify() 

Traceback (most recent call last): 

... 

assert A_cone == R_cone 

AssertionError 

""" 

from sage.geometry.polyhedron.constructor import Polyhedron 

if self.problem.base_ring() is not QQ: 

return 

A_cone = self.cone() 

R_cone = Polyhedron(vertices=[[self.zero] * self.problem.dim()], rays=self.R, 

base_ring=self.problem.base_ring(), backend='ppl') 

assert A_cone == R_cone 

assert A_cone.n_inequalities() <= len(self.A) 

assert R_cone.n_rays() == len(self.R) 

def R_by_sign(self, a): 

""" 

Classify the rays into those that are positive, zero, and negative on `a`. 

INPUT: 

- ``a`` -- vector. Coefficient vector of a homogeneous inequality. 

OUTPUT: 

A triple consisting of the rays (columns of `R`) that are 

positive, zero, and negative on `a`. In that order. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: DD, _ = StandardAlgorithm(A).initial_pair() 

sage: DD.R_by_sign(vector([1,-1,0])) 

([(2/3, -1/3, 1/3)], [(-1/3, -1/3, 1/3)], [(-1/3, 2/3, 1/3)]) 

sage: DD.R_by_sign(vector([1,1,1])) 

([(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3)], [], [(-1/3, -1/3, 1/3)]) 

""" 

pos = [] 

nul = [] 

neg = [] 

for r in self.R: 

sgn = a * r 

if sgn == self.zero: 

nul.append(r) 

elif sgn > self.zero: 

pos.append(r) 

else: 

neg.append(r) 

return pos, nul, neg 

def zero_set(self, ray): 

""" 

Return the zero set (active set) `Z(r)`. 

INPUT: 

- ``ray`` -- a ray vector. 

OUTPUT: 

A set containing the inequality vectors that are zero on ``ray``. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: DD, _ = Problem(A).initial_pair() 

sage: r = DD.R[0]; r 

(2/3, -1/3, 1/3) 

sage: DD.zero_set(r) 

{(-1, -1, 1), (0, 1, 1)} 

""" 

if ray not in self.zero_set_cache: 

self.zero_set_cache[ray] = (0, set()) 

n, t = self.zero_set_cache[ray] 

if n != len(self.A): 

t.update(self.A[i] for i in range(n,len(self.A)) if self.A[i].inner_product(ray) == self.zero) 

self.zero_set_cache[ray] = (len(self.A), t) 

return t 

def is_extremal(self, ray): 

""" 

Test whether the ray is extremal. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)]) 

sage: DD = StandardAlgorithm(A).run() 

sage: DD.is_extremal(DD.R[0]) 

True 

""" 

from sage.matrix.constructor import matrix 

A_Zray = matrix(self.problem.base_ring(), list(self.zero_set(ray))) 

return A_Zray.rank() == self.problem.dim() - 1 

@cached_method 

def matrix_space(self, nrows, ncols): 

r""" 

Return a matrix space of size ``nrows`` and ``ncols`` over the base ring 

of ``self``. 

These matrix spaces are cached to avoid their creation in the very 

demanding :meth:`add_inequality` and more precisely :meth:`are_adjacent`. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)]) 

sage: DD, _ = Problem(A).initial_pair() 

sage: DD.matrix_space(2,2) 

Full MatrixSpace of 2 by 2 dense matrices over Rational Field 

sage: DD.matrix_space(3,2) 

Full MatrixSpace of 3 by 2 dense matrices over Rational Field 

sage: K.<sqrt2> = QuadraticField(2) 

sage: A = matrix([[1,sqrt2],[2,0]]) 

sage: DD, _ = Problem(A).initial_pair() 

sage: DD.matrix_space(1,2) 

Full MatrixSpace of 1 by 2 dense matrices over Number Field in sqrt2 

with defining polynomial x^2 - 2 

""" 

return MatrixSpace(self.problem.base_ring(), nrows, ncols) 

def are_adjacent(self, r1, r2): 

""" 

Return whether the two rays are adjacent. 

INPUT: 

- ``r1``, ``r2`` -- two rays. 

OUTPUT: 

Boolean. Whether the two rays are adjacent. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)]) 

sage: DD = StandardAlgorithm(A).run() 

sage: DD.are_adjacent(DD.R[0], DD.R[1]) 

True 

sage: DD.are_adjacent(DD.R[0], DD.R[2]) 

True 

sage: DD.are_adjacent(DD.R[0], DD.R[3]) 

False 

""" 

Z = self.zero_set(r1).intersection(self.zero_set(r2)) 

if not Z: 

return self.problem.dim() == 2 

Z = list(Z) 

# here we try to create a matrix as fast as possible 

# since the generic matrix constructor is very slow (trac #18231) 

A_Z12 = self.matrix_space(len(Z), len(Z[0])).matrix(Z, coerce=False) 

return A_Z12.rank() == self.problem.dim() - 2 

def dual(self): 

""" 

Return the dual. 

OUTPUT: 

For the double description pair `(A, R)` this method returns 

the dual double description pair `(R^T, A^T)` 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)]) 

sage: DD, _ = Problem(A).initial_pair() 

sage: DD 

Double description pair (A, R) defined by 

[ 0 1 0] [0 1 0] 

A = [ 1 0 0], R = [1 0 0] 

[ 0 -1 1] [1 0 1] 

sage: DD.dual() 

Double description pair (A, R) defined by 

[0 1 1] [ 0 1 0] 

A = [1 0 0], R = [ 1 0 -1] 

[0 0 1] [ 0 0 1] 

""" 

return self._make_new(self.R, self.A) 

def first_coordinate_plane(self): 

""" 

Restrict to the first coordinate plane. 

OUTPUT: 

A new double description pair with the constraint `x_0 = 0` 

added. 

EXAMPLES:: 

sage: A = matrix([(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: DD, _ = StandardAlgorithm(A).initial_pair() 

sage: DD 

Double description pair (A, R) defined by 

A = [ 1 1], R = [ 1/2 -1/2] 

[-1 1] [ 1/2 1/2] 

sage: DD.first_coordinate_plane() 

Double description pair (A, R) defined by 

[ 1 1] 

A = [-1 1], R = [ 0] 

[-1 0] [1/2] 

[ 1 0] 

""" 

R = self.problem.base_ring() 

d = self.problem.dim() 

a_neg = vector(R, [-self.one] + [self.zero] * (d - 1)) 

a_pos = vector(R, [+self.one] + [self.zero] * (d - 1)) 

new = self._make_new(self.A, self.R) 

new.add_inequality(a_neg) 

new.add_inequality(a_pos) 

return new 

class Problem: 

pair_class = DoubleDescriptionPair 

def __init__(self, A): 

""" 

Base class for implementations of the double description algorithm 

It does not make sense to instantiate the base class directly, 

it just provides helpers for implementations. 

INPUT: 

- ``A`` -- a matrix. The rows of the matrix are interpreted as 

homogeneous inequalities `A x \geq 0`. Must have maximal rank. 

TESTS:: 

sage: A = matrix([(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: Problem(A) 

Pointed cone with inequalities 

(1, 1) 

(-1, 1) 

""" 

assert A.rank() == A.ncols() # implementation assumes maximal rank 

if A.is_mutable(): 

A = A.__copy__() 

A.set_immutable() 

self._A = A 

self._field = A.base_ring().fraction_field() 

@cached_method 

def A(self): 

""" 

Return the rows of the defining matrix `A`. 

OUTPUT: 

The matrix `A` whose rows are the inequalities. 

EXAMPLES:: 

sage: A = matrix([(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: Problem(A).A() 

((1, 1), (-1, 1)) 

""" 

rows = [a.change_ring(self._field) for a in self._A.rows()] 

for a in rows: a.set_immutable() 

return tuple(rows) 

def A_matrix(self): 

""" 

Return the defining matrix `A`. 

OUTPUT: 

Matrix whose rows are the inequalities. 

EXAMPLES:: 

sage: A = matrix([(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: Problem(A).A_matrix() 

[ 1 1] 

[-1 1] 

""" 

return self._A 

def base_ring(self): 

""" 

Return the base field. 

OUTPUT: 

A field. 

EXAMPLES:: 

sage: A = matrix(AA, [(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: Problem(A).base_ring() 

Algebraic Real Field 

""" 

return self._field 

@cached_method 

def dim(self): 

""" 

Return the ambient space dimension. 

OUTPUT: 

Integer. The ambient space dimension of the cone. 

EXAMPLES:: 

sage: A = matrix(QQ, [(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: Problem(A).dim() 

2 

""" 

return self._A.ncols() 

def __repr__(self): 

r""" 

Return a string representation. 

OUTPUT: 

String. 

EXAMPLES:: 

sage: A = matrix(QQ, [(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: Problem(A).__repr__() 

'Pointed cone with inequalities\n(1, 1)\n(-1, 1)' 

""" 

return 'Pointed cone with inequalities\n' + '\n'.join(map(str, self.A())) 

def initial_pair(self): 

""" 

Return an initial double description pair. 

Picks an initial set of rays by selecting a basis. This is 

probably the most efficient way to select the initial set. 

INPUT: 

- ``pair_class`` -- subclass of 

:class:`DoubleDescriptionPair`. 

OUTPUT: 

A pair consisting of a :class:`DoubleDescriptionPair` instance 

and the tuple of remaining unused inequalities. 

EXAMPLES:: 

sage: A = matrix([(-1, 1), (-1, 2), (1/2, -1/2), (1/2, 2)]) 

sage: from sage.geometry.polyhedron.double_description import Problem 

sage: DD, remaining = Problem(A).initial_pair() 

sage: DD.verify() 

sage: remaining 

[(1/2, -1/2), (1/2, 2)] 

""" 

pivot_rows = self.A_matrix().pivot_rows() 

A0 = [self.A()[pivot] for pivot in pivot_rows] 

Ac = [self.A()[i] for i in range(len(self.A())) if i not in pivot_rows] 

from sage.matrix.constructor import identity_matrix, matrix 

I = identity_matrix(self.base_ring(), self.dim()) 

R = matrix(self.base_ring(), A0).solve_right(I) 

return self.pair_class(self, A0, R.columns()), list(Ac) 

class StandardDoubleDescriptionPair(DoubleDescriptionPair): 

""" 

Double description pair for the "Standard Algorithm". 

See :class:`StandardAlgorithm`. 

TESTS:: 

sage: A = matrix([(-1, 1, 0), (-1, 2, 1), (1/2, -1/2, -1)]) 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: DD, _ = StandardAlgorithm(A).initial_pair() 

""" 

def add_inequality(self, a): 

""" 

Add the inequality ``a`` to the matrix `A` of the double description. 

INPUT: 

- ``a`` -- vector. An inequality. 

EXAMPLES:: 

sage: A = matrix([(-1, 1, 0), (-1, 2, 1), (1/2, -1/2, -1)]) 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: DD, _ = StandardAlgorithm(A).initial_pair() 

sage: DD.add_inequality(vector([1,0,0])) 

sage: DD 

Double description pair (A, R) defined by 

[ -1 1 0] [ 1 1 0 0] 

A = [ -1 2 1], R = [ 1 1 1 1] 

[ 1/2 -1/2 -1] [ 0 -1 -1/2 -2] 

[ 1 0 0] 

""" 

R_pos, R_nul, R_neg = self.R_by_sign(a) 

if not R_neg: 

return 

R_new = [] 

for rp, rn in itertools.product(R_pos, R_neg): 

if not self.are_adjacent(rp, rn): 

continue 

r = a.inner_product(rp) * rn - a.inner_product(rn) * rp 

r.set_immutable() 

R_new.append(r) 

self.R = R_pos + R_nul + R_new 

self.A.append(a) 

class StandardAlgorithm(Problem): 

""" 

Standard implementation of the double description algorithm 

See [FP1996]_ for the definition of the "Standard 

Algorithm". 

EXAMPLES:: 

sage: A = matrix(QQ, [(1, 1), (-1, 1)]) 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: DD = StandardAlgorithm(A).run() 

sage: DD.R # the extremal rays 

[(1/2, 1/2), (-1/2, 1/2)] 

""" 

pair_class = StandardDoubleDescriptionPair 

def run(self): 

""" 

Run the Standard Algorithm. 

OUTPUT: 

A double description pair `(A, R)` of all inequalities as a 

:class:`DoubleDescriptionPair`. By virtue of the double 

description algorithm, the columns of `R` are the extremal 

rays. 

EXAMPLES:: 

sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm 

sage: A = matrix(QQ, [(0,1,0), (1,0,0), (0,-1,1), (-1,0,1)]) 

sage: StandardAlgorithm(A).run() 

Double description pair (A, R) defined by 

[ 0 1 0] [0 0 1 1] 

A = [ 1 0 0], R = [1 0 1 0] 

[ 0 -1 1] [1 1 1 1] 

[-1 0 1] 

""" 

DD, remaining = self.initial_pair() 

for a in remaining: 

DD.add_inequality(a) 

if VERIFY_RESULT: 

DD.verify() 

return DD