sage.{geometry,rings}: More # optional tags in doctests

src/sage/features/sagemath.py

@@ -122,6 +122,28 @@ def __init__(self):
                              [PythonModule('sage.groups.perm_gps.permgroup')])
 
 
+class sage__libs__pari(JoinFeature):
+    r"""
+    A :class:`sage.features.Feature` describing the presence of :mod:`sage.libs.pari`.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__libs__pari
+        sage: sage__libs__pari().is_present()                       # optional - sage.libs.pari
+        FeatureTestResult('sage.libs.pari', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__libs__pari
+            sage: isinstance(sage__libs__pari(), sage__libs__pari)
+            True
+        """
+        JoinFeature.__init__(self, 'sage.libs.pari',
+                             [PythonModule('sage.libs.pari.convert_sage')])
+
+
 class sage__plot(JoinFeature):
     r"""
     A :class:`~sage.features.Feature` describing the presence of :mod:`sage.plot`.
@@ -209,6 +231,27 @@ def __init__(self):
         PythonModule.__init__(self, 'sage.rings.real_double')
 
 
+class sage__rings__real_mpfr(PythonModule):
+    r"""
+    A :class:`~sage.features.Feature` describing the presence of :mod:`sage.rings.real_mpfr`.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__rings__real_mpfr
+        sage: sage__rings__real_mpfr().is_present()  # optional - sage.rings.real_mpfr
+        FeatureTestResult('sage.rings.real_mpfr', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__rings__real_mpfr
+            sage: isinstance(sage__rings__real_mpfr(), sage__rings__real_mpfr)
+            True
+        """
+        PythonModule.__init__(self, 'sage.rings.real_mpfr')
+
+
 class sage__symbolic(JoinFeature):
     r"""
     A :class:`~sage.features.Feature` describing the presence of :mod:`sage.symbolic`.
@@ -259,8 +302,10 @@ def all_features():
             sage__geometry__polyhedron(),
             sage__graphs(),
             sage__groups(),
+            sage__libs__pari(),
             sage__plot(),
             sage__rings__number_field(),
             sage__rings__padics(),
             sage__rings__real_double(),
+            sage__rings__real_mpfr(),
             sage__symbolic()]

src/sage/geometry/cone_catalog.py

@@ -374,8 +374,8 @@ def rearrangement(p, ambient_dim=None, lattice=None):
         sage: ambient_dim = ZZ.random_element(2,10).abs()
         sage: p = ZZ.random_element(1, ambient_dim)
         sage: K = cones.rearrangement(p, ambient_dim)
-        sage: P = SymmetricGroup(ambient_dim).random_element().matrix()
-        sage: all( K.contains(P*r) for r in K )
+        sage: P = SymmetricGroup(ambient_dim).random_element().matrix()     # optional - sage.groups
+        sage: all(K.contains(P*r) for r in K)                               # optional - sage.groups
         True
 
     The smallest ``p`` components of every element of the rearrangement
@@ -529,11 +529,11 @@ def schur(ambient_dim=None, lattice=None):
 
         sage: P = cones.schur(5)
         sage: Q = cones.nonnegative_orthant(5)
-        sage: G = ( g.change_ring(QQbar).normalized() for g in P )
-        sage: H = ( h.change_ring(QQbar).normalized() for h in Q )
-        sage: actual = max(arccos(u.inner_product(v)) for u in G for v in H)
-        sage: expected = 3*pi/4
-        sage: abs(actual - expected).n() < 1e-12
+        sage: G = ( g.change_ring(QQbar).normalized() for g in P )              # optional - sage.rings.number_fields
+        sage: H = ( h.change_ring(QQbar).normalized() for h in Q )              # optional - sage.rings.number_fields
+        sage: actual = max(arccos(u.inner_product(v)) for u in G for v in H)    # optional - sage.rings.number_fields
+        sage: expected = 3*pi/4                                                 # optional - sage.rings.number_fields
+        sage: abs(actual - expected).n() < 1e-12                                # optional - sage.rings.number_fields
         True
 
     The dual of the Schur cone is the "downward monotonic cone"
@@ -566,7 +566,7 @@ def schur(ambient_dim=None, lattice=None):
         True
         sage: x = V.random_element()
         sage: y = V.random_element()
-        sage: majorized_by(x,y) == ( (y-x) in S )
+        sage: majorized_by(x,y) == ( (y-x) in S )                               # optional - sage.rings.number_fields
         True
 
     If a ``lattice`` was given, it is actually used::

src/sage/geometry/hyperplane_arrangement/affine_subspace.py

@@ -33,8 +33,8 @@
     True
     sage: c < b
     False
-    sage: A = AffineSubspace([8,38,21,250], VectorSpace(GF(19),4))
-    sage: A
+    sage: A = AffineSubspace([8,38,21,250], VectorSpace(GF(19),4))                  # optional - sage.libs.pari
+    sage: A                                                                         # optional - sage.libs.pari
     Affine space p + W where:
        p = (8, 0, 2, 3)
        W = Vector space of dimension 4 over Finite Field of size 19
@@ -384,9 +384,9 @@ def intersection(self, other):
             []
             sage: A.intersection(C).intersection(B)
 
-            sage: D = AffineSubspace([1,2,3], VectorSpace(GF(5),3))
-            sage: E = AffineSubspace([3,4,5], VectorSpace(GF(5),3))
-            sage: D.intersection(E)
+            sage: D = AffineSubspace([1,2,3], VectorSpace(GF(5),3))                 # optional - sage.libs.pari
+            sage: E = AffineSubspace([3,4,5], VectorSpace(GF(5),3))                 # optional - sage.libs.pari
+            sage: D.intersection(E)                                                 # optional - sage.libs.pari
             Affine space p + W where:
               p = (3, 4, 0)
               W = Vector space of dimension 3 over Finite Field of size 5

src/sage/geometry/hyperplane_arrangement/arrangement.py

@@ -60,30 +60,30 @@
 The default base field is `\QQ`, the rational numbers.  Finite fields are also
 supported::
 
-    sage: H.<x,y,z> = HyperplaneArrangements(GF(5))
-    sage: a = H([(1,2,3), 4], [(5,6,7), 8]);  a
+    sage: H.<x,y,z> = HyperplaneArrangements(GF(5))                             # optional - sage.libs.pari
+    sage: a = H([(1,2,3), 4], [(5,6,7), 8]);  a                                 # optional - sage.libs.pari
     Arrangement <y + 2*z + 3 | x + 2*y + 3*z + 4>
 
 Number fields are also possible::
 
     sage: x = var('x')
-    sage: NF.<a> = NumberField(x**4 - 5*x**2 + 5,embedding=1.90)
-    sage: H.<y,z> = HyperplaneArrangements(NF)
-    sage: A = H([[(-a**3 + 3*a, -a**2 + 4), 1], [(a**3 - 4*a, -1), 1],
+    sage: NF.<a> = NumberField(x**4 - 5*x**2 + 5, embedding=1.90)                                           # optional - sage.rings.number_field
+    sage: H.<y,z> = HyperplaneArrangements(NF)                                                              # optional - sage.rings.number_field
+    sage: A = H([[(-a**3 + 3*a, -a**2 + 4), 1], [(a**3 - 4*a, -1), 1],                                      # optional - sage.rings.number_field
     ....:        [(0, 2*a**2 - 6), 1], [(-a**3 + 4*a, -1), 1],
     ....:        [(a**3 - 3*a, -a**2 + 4), 1]])
-    sage: A
+    sage: A                                                                                                 # optional - sage.rings.number_field
     Arrangement of 5 hyperplanes of dimension 2 and rank 2
-    sage: A.base_ring()
+    sage: A.base_ring()                                                                                     # optional - sage.rings.number_field
     Number Field in a with defining polynomial x^4 - 5*x^2 + 5 with a = 1.902113032590308?
 
 Notation (iii): a list or tuple of hyperplanes::
 
-    sage: H.<x,y,z> = HyperplaneArrangements(GF(5))
-    sage: k = [x+i for i in range(4)];  k
+    sage: H.<x,y,z> = HyperplaneArrangements(GF(5))                             # optional - sage.libs.pari
+    sage: k = [x+i for i in range(4)];  k                                       # optional - sage.libs.pari
     [Hyperplane x + 0*y + 0*z + 0, Hyperplane x + 0*y + 0*z + 1,
      Hyperplane x + 0*y + 0*z + 2, Hyperplane x + 0*y + 0*z + 3]
-    sage: H(k)
+    sage: H(k)                                                                  # optional - sage.libs.pari
     Arrangement <x | x + 1 | x + 2 | x + 3>
 
 Notation (iv): using the library of arrangements::
@@ -264,10 +264,10 @@
     sage: a = hyperplane_arrangements.semiorder(3)
     sage: a.has_good_reduction(5)
     True
-    sage: b = a.change_ring(GF(5))
+    sage: b = a.change_ring(GF(5))                                              # optional - sage.libs.pari
     sage: pa = a.intersection_poset()
-    sage: pb = b.intersection_poset()
-    sage: pa.is_isomorphic(pb)
+    sage: pb = b.intersection_poset()                                           # optional - sage.libs.pari
+    sage: pa.is_isomorphic(pb)                                                  # optional - sage.libs.pari
     True
     sage: a.face_vector()
     (0, 12, 30, 19)
@@ -356,7 +356,6 @@
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 
 from sage.geometry.hyperplane_arrangement.hyperplane import AmbientVectorSpace, Hyperplane
-from sage.combinat.posets.posets import Poset
 
 
 class HyperplaneArrangementElement(Element):
@@ -1129,7 +1128,7 @@ def change_ring(self, base_ring):
 
             sage: H.<x,y> = HyperplaneArrangements(QQ)
             sage: A = H([(1,1), 0], [(2,3), -1])
-            sage: A.change_ring(FiniteField(2))
+            sage: A.change_ring(FiniteField(2))                                     # optional - sage.libs.pari
             Arrangement <y + 1 | x + y>
 
         TESTS:
@@ -1165,25 +1164,30 @@ def n_regions(self):
 
             sage: H.<x,y> = HyperplaneArrangements(QQ)
             sage: A = H([(1,1), 0], [(2,3), -1], [(4,5), 3])
-            sage: B = A.change_ring(FiniteField(7))
-            sage: B.n_regions()
+            sage: B = A.change_ring(FiniteField(7))                                 # optional - sage.libs.pari
+            sage: B.n_regions()                                                     # optional - sage.libs.pari
             Traceback (most recent call last):
             ...
             TypeError: base field must have characteristic zero
 
         Check that :trac:`30749` is fixed::
 
             sage: R.<y> = QQ[]
-            sage: v1 = AA.polynomial_root(AA.common_polynomial(y^2 - 3), RIF(RR(1.7320508075688772), RR(1.7320508075688774)))
-            sage: v2 = QQbar.polynomial_root(AA.common_polynomial(y^4 - y^2 + 1), CIF(RIF(RR(0.8660254037844386), RR(0.86602540378443871)), RIF(-RR(0.50000000000000011), -RR(0.49999999999999994))))
-            sage: my_vectors = (vector(AA, [-v1, -1, 1]), vector(AA, [0, 2, 1]), vector(AA,[v1, -1, 1]), vector(AA, [1, 0, 0]), vector(AA, [1/2, AA(-1/2*v2^3 + v2),0]), vector(AA, [-1/2, AA(-1/2*v2^3 + v2), 0]))
-            sage: H = HyperplaneArrangements(AA, names='xyz')
-            sage: x,y,z = H.gens()
-            sage: A = H(backend="normaliz")  # optional - pynormaliz
-            sage: for v in my_vectors:        # optional - pynormaliz
+            sage: v1 = AA.polynomial_root(AA.common_polynomial(y^2 - 3),                                    # optional - sage.rings.number_field
+            ....:                         RIF(RR(1.7320508075688772), RR(1.7320508075688774)))
+            sage: v2 = QQbar.polynomial_root(AA.common_polynomial(y^4 - y^2 + 1),                           # optional - sage.rings.number_field
+            ....:                            CIF(RIF(RR(0.8660254037844386), RR(0.86602540378443871)),
+            ....:                                RIF(-RR(0.50000000000000011), -RR(0.49999999999999994))))
+            sage: my_vectors = (vector(AA, [-v1, -1, 1]), vector(AA, [0, 2, 1]), vector(AA, [v1, -1, 1]),   # optional - sage.rings.number_field
+            ....:               vector(AA, [1, 0, 0]), vector(AA, [1/2, AA(-1/2*v2^3 + v2),0]),
+            ....:               vector(AA, [-1/2, AA(-1/2*v2^3 + v2), 0]))
+            sage: H = HyperplaneArrangements(AA, names='xyz')                                               # optional - sage.rings.number_field
+            sage: x,y,z = H.gens()                                                                          # optional - sage.rings.number_field
+            sage: A = H(backend="normaliz")                                     # optional - pynormaliz     # optional - sage.rings.number_field
+            sage: for v in my_vectors:                                          # optional - pynormaliz     # optional - sage.rings.number_field
             ....:     a, b, c = v
             ....:     A = A.add_hyperplane(a*x + b*y + c*z)
-            sage: A.n_regions()              # optional - pynormaliz
+            sage: A.n_regions()                                                 # optional - pynormaliz     # optional - sage.rings.number_field
             24
         """
         if self.base_ring().characteristic() != 0:
@@ -1211,8 +1215,8 @@ def n_bounded_regions(self):
 
             sage: H.<x,y> = HyperplaneArrangements(QQ)
             sage: A = H([(1,1),0], [(2,3),-1], [(4,5),3])
-            sage: B = A.change_ring(FiniteField(7))
-            sage: B.n_bounded_regions()
+            sage: B = A.change_ring(FiniteField(7))                             # optional - sage.libs.pari
+            sage: B.n_bounded_regions()                                         # optional - sage.libs.pari
             Traceback (most recent call last):
             ...
             TypeError: base field must have characteristic zero
@@ -1243,14 +1247,14 @@ def has_good_reduction(self, p):
         EXAMPLES::
 
             sage: a = hyperplane_arrangements.semiorder(3)
-            sage: a.has_good_reduction(5)
+            sage: a.has_good_reduction(5)                                       # optional - sage.libs.pari
             True
-            sage: a.has_good_reduction(3)
+            sage: a.has_good_reduction(3)                                       # optional - sage.libs.pari
             False
-            sage: b = a.change_ring(GF(3))
+            sage: b = a.change_ring(GF(3))                                      # optional - sage.libs.pari
             sage: a.characteristic_polynomial()
             x^3 - 6*x^2 + 12*x
-            sage: b.characteristic_polynomial()  # not equal to that for a
+            sage: b.characteristic_polynomial()  # not equal to that for a      # optional - sage.libs.pari
             x^3 - 6*x^2 + 10*x
         """
         if self.base_ring() != QQ:
@@ -1495,9 +1499,9 @@ def essentialization(self):
             Hyperplane arrangements in 1-dimensional linear space over
             Rational Field with coordinate x
 
-            sage: H.<x,y> = HyperplaneArrangements(GF(2))
-            sage: C = H([(1,1),1], [(1,1),0])
-            sage: C.essentialization()
+            sage: H.<x,y> = HyperplaneArrangements(GF(2))                       # optional - sage.libs.pari
+            sage: C = H([(1,1),1], [(1,1),0])                                   # optional - sage.libs.pari
+            sage: C.essentialization()                                          # optional - sage.libs.pari
             Arrangement <y | y + 1>
 
             sage: h = hyperplane_arrangements.semiorder(4)
@@ -1587,9 +1591,9 @@ def sign_vector(self, p):
 
         TESTS::
 
-            sage: H.<x,y> = HyperplaneArrangements(GF(3))
-            sage: A = H(x, y)
-            sage: A.sign_vector([1, 2])
+            sage: H.<x,y> = HyperplaneArrangements(GF(3))                   # optional - sage.libs.pari
+            sage: A = H(x, y)                                               # optional - sage.libs.pari
+            sage: A.sign_vector([1, 2])                                     # optional - sage.libs.pari
             Traceback (most recent call last):
             ...
             ValueError: characteristic must be zero
@@ -1992,6 +1996,8 @@ def poset_of_regions(self, B=None, numbered_labels=True):
             sage: A.poset_of_regions(B=base_region)
             Finite poset containing 14 elements
         """
+        from sage.combinat.posets.posets import Poset
+
         # We use RX to keep track of indexes and R to keep track of which regions
         # we've already hit. This poset is graded, so we can go one set at a time
         RX = self.regions()
@@ -2480,7 +2486,7 @@ def face_semigroup_algebra(self, field=None, names='e'):
             sage: (e3 + 2*e4) * (e1 - e7)
             e4 - e6
 
-            sage: U3 = a.face_semigroup_algebra(field=GF(3)); U3
+            sage: U3 = a.face_semigroup_algebra(field=GF(3)); U3                        # optional - sage.libs.pari
             Finite-dimensional algebra of degree 13 over Finite Field of size 3
 
         TESTS:
@@ -2550,8 +2556,8 @@ def region_containing_point(self, p):
         TESTS::
 
             sage: A = H([(1,1),0], [(2,3),-1], [(4,5),3])
-            sage: B = A.change_ring(FiniteField(7))
-            sage: B.region_containing_point((1,2))
+            sage: B = A.change_ring(FiniteField(7))                         # optional - sage.libs.pari
+            sage: B.region_containing_point((1,2))                          # optional - sage.libs.pari
             Traceback (most recent call last):
             ...
             ValueError: base field must have characteristic zero
@@ -2991,8 +2997,8 @@ def matroid(self):
         intersection lattice::
 
             sage: f = sum([list(M.flats(i)) for i in range(M.rank()+1)], [])
-            sage: PF = Poset([f, lambda x,y: x < y])
-            sage: PF.is_isomorphic(A.intersection_poset())
+            sage: PF = Poset([f, lambda x, y: x < y])                                   # optional - sage.combinat
+            sage: PF.is_isomorphic(A.intersection_poset())                              # optional - sage.combinat
             True
         """
         if not self.is_central():

src/sage/geometry/linear_expression.py

@@ -440,7 +440,7 @@ def evaluate(self, point):
             9
             sage: ex([1,1])    # syntactic sugar
             9
-            sage: ex([pi, e])
+            sage: ex([pi, e])                                           # optional - sage.symbolic
             2*pi + 3*e + 4
         """
         try:

src/sage/geometry/polyhedral_complex.py

@@ -1,4 +1,4 @@
-# -*- coding: utf-8 -*-
+# sage.doctest: optional - sage.graphs
 r"""
 Finite polyhedral complexes
 

src/sage/geometry/polyhedron/backend_number_field.py

@@ -119,11 +119,11 @@ def _init_from_Vrepresentation(self, vertices, rays, lines,
 
         Check that the coordinates of a vertex get simplified in the Symbolic Ring::
 
-            sage: p = Polyhedron(ambient_dim=2, base_ring=SR, backend='number_field')
+            sage: p = Polyhedron(ambient_dim=2, base_ring=SR, backend='number_field')                               # optional - sage.symbolic
             sage: from sage.geometry.polyhedron.backend_number_field import Polyhedron_number_field
-            sage: Polyhedron_number_field._init_from_Vrepresentation(p, [(0,1/2),(sqrt(2),0),(4,5/6)], [], []); p
+            sage: Polyhedron_number_field._init_from_Vrepresentation(p, [(0,1/2),(sqrt(2),0),(4,5/6)], [], []); p   # optional - sage.symbolic
             A 2-dimensional polyhedron in (Symbolic Ring)^2 defined as the convex hull of 3 vertices
-            sage: p.vertices()[0][0]
+            sage: p.vertices()[0][0]                                                                                # optional - sage.symbolic
             0
         """
         (vertices, rays, lines), internal_base_ring \