gh-36679: `src/doc`: Update `# needs`
    
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Also some doctest cosmetics.

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In part cherry-picked from #35095.
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- [ ] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

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URL: #36679
Reported by: Matthias Köppe
Reviewer(s): David Coudert, Kwankyu Lee, Matthias Köppe

Author: Release Manager

src/doc/de/tutorial/tour_plotting.rst

@@ -1,3 +1,5 @@
+.. sage-doctest: needs sage.plot sage.symbolic
+
 .. _section-plot:
 
 Plotten

src/doc/en/developer/packaging_sage_library.rst

@@ -451,7 +451,7 @@ Apparently it does not in a very substantial way:
   merely a heuristic. Looking at the source of "entropy", through
   ``log`` from :mod:`sage.misc.functional`, a runtime dependency on
   symbolics comes in. In fact, for this reason, two doctests there are
-  already marked as ``# optional - sage.symbolic``.
+  already marked as ``# needs sage.symbolic``.
 
 So if packaged as **sagemath-coding**, now a domain expert would have
 to decide whether these dependencies on symbolics are strong enough to

src/doc/en/prep/Calculus.rst

@@ -1,4 +1,4 @@
-.. -*- coding: utf-8 -*-
+.. sage-doctest: needs sage.plot sage.symbolic
 
 .. linkall
 

src/doc/en/prep/Quickstarts/Differential-Equations.rst

@@ -1,4 +1,4 @@
-.. -*- coding: utf-8 -*-
+.. sage-doctest: needs sage.plot sage.symbolic
 
 .. linkall
 

src/doc/en/prep/Quickstarts/Statistics-and-Distributions.rst

@@ -142,13 +142,14 @@ the examples in ``r.kruskal_test?`` in the notebook.
 
 ::
 
-    sage: x=r([2.9, 3.0, 2.5, 2.6, 3.2])  # normal subjects                       # optional - rpy2
-    sage: y=r([3.8, 2.7, 4.0, 2.4])       # with obstructive airway disease       # optional - rpy2
-    sage: z=r([2.8, 3.4, 3.7, 2.2, 2.0])  # with asbestosis                       # optional - rpy2
-    sage: a = r([x,y,z])                  # make a long R vector of all the data  # optional - rpy2
-    sage: b = r.factor(5*[1]+4*[2]+5*[3]) # create something for R to tell        # optional - rpy2
-    ....:                                 #   which subjects are which
-    sage: a; b                            # show them                             # optional - rpy2
+    sage: # optional - rpy2
+    sage: x = r([2.9, 3.0, 2.5, 2.6, 3.2])      # normal subjects
+    sage: y = r([3.8, 2.7, 4.0, 2.4])           # with obstructive airway disease
+    sage: z = r([2.8, 3.4, 3.7, 2.2, 2.0])      # with asbestosis
+    sage: a = r([x,y,z])                        # make a long R vector of all the data
+    sage: b = r.factor(5*[1] + 4*[2] + 5*[3])   # create something for R to tell
+    ....:                                       #   which subjects are which
+    sage: a; b                                  # show them
      [1] 2.9 3.0 2.5 2.6 3.2 3.8 2.7 4.0 2.4 2.8 3.4 3.7 2.2 2.0
      [1] 1 1 1 1 1 2 2 2 2 3 3 3 3 3
     Levels: 1 2 3

src/doc/en/prep/Symbolics-and-Basic-Plotting.rst

@@ -1,4 +1,4 @@
-.. -*- coding: utf-8 -*-
+.. sage-doctest: needs sage.plot sage.symbolic
 
 .. linkall
 

src/doc/en/thematic_tutorials/geometry/polyhedra_tutorial.rst

@@ -91,7 +91,7 @@ and some rays.
 
 ::
 
-    sage: P1 = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]])
+    sage: P1 = Polyhedron(vertices=[[1, 0], [0, 1]], rays=[[1, 1]])
     sage: P1
     A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
 
@@ -118,9 +118,9 @@ We can also add a lineality space.
 
 ::
 
-    sage: P2 = Polyhedron(vertices = [[1/2, 0, 0], [0, 1/2, 0]],
-    ....:                 rays = [[1, 1, 0]],
-    ....:                 lines = [[0, 0, 1]])
+    sage: P2 = Polyhedron(vertices=[[1/2, 0, 0], [0, 1/2, 0]],
+    ....:                 rays=[[1, 1, 0]],
+    ....:                 lines=[[0, 0, 1]])
     sage: P2
     A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 2 vertices, 1 ray, 1 line
     sage: P2.plot()
@@ -144,7 +144,7 @@ The chosen ring depends on the input format.
 
 ::
 
-    sage: P3 = Polyhedron(vertices = [[0.5, 0], [0, 0.5]])
+    sage: P3 = Polyhedron(vertices=[[0.5, 0], [0, 0.5]])
     sage: P3
     A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 2 vertices
     sage: P3.parent()
@@ -163,22 +163,22 @@ The following example demonstrates the limitations of :code:`RDF`.
 
 ::
 
-    sage: D = polytopes.dodecahedron()                                        # optional - sage.rings.number_field
-    sage: D                                                                   # optional - sage.rings.number_field
+    sage: D = polytopes.dodecahedron()                                                  # needs sage.rings.number_field
+    sage: D                                                                             # needs sage.rings.number_field
     A 3-dimensional polyhedron
      in (Number Field in sqrt5 with defining polynomial x^2 - 5
          with sqrt5 = 2.236067977499790?)^3
      defined as the convex hull of 20 vertices
 
-    sage: vertices_RDF = [n(v.vector(),digits=6) for v in D.vertices()]       # optional - sage.rings.number_field
-    sage: D_RDF = Polyhedron(vertices=vertices_RDF, base_ring=RDF)            # optional - sage.rings.number_field
+    sage: vertices_RDF = [n(v.vector(),digits=6) for v in D.vertices()]                 # needs sage.rings.number_field
+    sage: D_RDF = Polyhedron(vertices=vertices_RDF, base_ring=RDF)                      # needs sage.rings.number_field
     doctest:warning
     ...
     UserWarning: This polyhedron data is numerically complicated; cdd
     could not convert between the inexact V and H representation
     without loss of data. The resulting object might show
     inconsistencies.
-    sage: D_RDF = Polyhedron(vertices=sorted(vertices_RDF), base_ring=RDF)    # optional - sage.rings.number_field
+    sage: D_RDF = Polyhedron(vertices=sorted(vertices_RDF), base_ring=RDF)              # needs sage.rings.number_field
     Traceback (most recent call last):
     ...
     ValueError: *Error: Numerical inconsistency is found.  Use the GMP exact arithmetic.
@@ -199,11 +199,12 @@ It is also possible to define a polyhedron over algebraic numbers.
 
 ::
 
-    sage: sqrt_2 = AA(2)^(1/2)                                                # optional - sage.rings.number_field
-    sage: cbrt_2 = AA(2)^(1/3)                                                # optional - sage.rings.number_field
-    sage: timeit('Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]])')         # optional - sage.rings.number_field  # random
+    sage: # needs sage.rings.number_field
+    sage: sqrt_2 = AA(2)^(1/2)
+    sage: cbrt_2 = AA(2)^(1/3)
+    sage: timeit('Polyhedron(vertices=[[sqrt_2, 0], [0, cbrt_2]])')     # random
     5 loops, best of 3: 43.2 ms per loop
-    sage: P4 = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]]); P4          # optional - sage.rings.number_field
+    sage: P4 = Polyhedron(vertices=[[sqrt_2, 0], [0, cbrt_2]]); P4
     A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
 
 .. end of output
@@ -212,11 +213,12 @@ There is another way to create a polyhedron over algebraic numbers:
 
 ::
 
-    sage: K.<a> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))                # optional - sage.rings.number_field
-    sage: L.<b> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))                # optional - sage.rings.number_field
-    sage: timeit('Polyhedron(vertices = [[a, 0], [0, b]])')                   # optional - sage.rings.number_field  # random
+    sage: # needs sage.rings.number_field
+    sage: K.<a> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
+    sage: L.<b> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))
+    sage: timeit('Polyhedron(vertices=[[a, 0], [0, b]])')               # random
     5 loops, best of 3: 39.9 ms per loop
-    sage: P5 = Polyhedron(vertices = [[a, 0], [0, b]]); P5                    # optional - sage.rings.number_field
+    sage: P5 = Polyhedron(vertices=[[a, 0], [0, b]]); P5
     A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
 
 .. end of output
@@ -225,10 +227,11 @@ If the base ring is known it may be a good option to use the proper :meth:`sage.
 
 ::
 
-    sage: J = K.composite_fields(L)[0]                                        # optional - sage.rings.number_field
-    sage: timeit('Polyhedron(vertices = [[J(a), 0], [0, J(b)]])')             # optional - sage.rings.number_field  # random
+    sage: # needs sage.rings.number_field
+    sage: J = K.composite_fields(L)[0]
+    sage: timeit('Polyhedron(vertices=[[J(a), 0], [0, J(b)]])')         # random
     25 loops, best of 3: 9.8 ms per loop
-    sage: P5_comp = Polyhedron(vertices = [[J(a), 0], [0, J(b)]]); P5_comp    # optional - sage.rings.number_field
+    sage: P5_comp = Polyhedron(vertices=[[J(a), 0], [0, J(b)]]); P5_comp
     A 1-dimensional polyhedron
      in (Number Field in ab with defining polynomial
          x^6 - 6*x^4 - 4*x^3 + 12*x^2 - 24*x - 4
@@ -242,9 +245,9 @@ It is not possible to define a polyhedron over it:
 
 ::
 
-    sage: sqrt_2s = sqrt(2)                                                   # optional - sage.symbolic
-    sage: cbrt_2s = 2^(1/3)                                                   # optional - sage.symbolic
-    sage: Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]])                 # optional - sage.symbolic
+    sage: sqrt_2s = sqrt(2)                                                             # needs sage.symbolic
+    sage: cbrt_2s = 2^(1/3)                                                             # needs sage.symbolic
+    sage: Polyhedron(vertices=[[sqrt_2s, 0], [0, cbrt_2s]])                             # needs sage.symbolic
     Traceback (most recent call last):
     ...
     ValueError: no default backend for computations with Symbolic Ring
@@ -389,7 +392,7 @@ inequalities and equalities as objects.
 
 ::
 
-    sage: P3_QQ = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], base_ring=QQ)
+    sage: P3_QQ = Polyhedron(vertices=[[0.5, 0], [0, 0.5]], base_ring=QQ)
     sage: HRep = P3_QQ.Hrepresentation()
     sage: H1 = HRep[0]; H1
     An equation (2, 2) x - 1 == 0
@@ -527,7 +530,7 @@ In order to use a specific backend, we specify the :code:`backend` parameter.
 
 ::
 
-    sage: P1_cdd = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]], backend='cdd')
+    sage: P1_cdd = Polyhedron(vertices=[[1, 0], [0, 1]], rays=[[1, 1]], backend='cdd')
     sage: P1_cdd
     A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
 
@@ -567,7 +570,7 @@ The :code:`cdd` backend accepts also entries in :code:`RDF`:
 
 ::
 
-    sage: P3_cdd = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], backend='cdd')
+    sage: P3_cdd = Polyhedron(vertices=[[0.5, 0], [0, 0.5]], backend='cdd')
     sage: P3_cdd
     A 1-dimensional polyhedron in RDF^2 defined as the convex hull of 2 vertices
 
@@ -577,12 +580,12 @@ but not algebraic or symbolic values:
 
 ::
 
-    sage: P4_cdd = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='cdd')            # optional - sage.rings.number_field
+    sage: P4_cdd = Polyhedron(vertices=[[sqrt_2, 0], [0, cbrt_2]], backend='cdd')       # needs sage.rings.number_field
     Traceback (most recent call last):
     ...
     ValueError: No such backend (=cdd) implemented for given basering (=Algebraic Real Field).
 
-    sage: P5_cdd = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='cdd')          # optional - sage.symbolic
+    sage: P5_cdd = Polyhedron(vertices=[[sqrt_2s, 0], [0, cbrt_2s]], backend='cdd')     # needs sage.symbolic
     Traceback (most recent call last):
     ...
     ValueError: No such backend (=cdd) implemented for given basering (=Symbolic Ring).
@@ -656,8 +659,8 @@ An example with quadratic field:
 
 ::
 
-    sage: V = polytopes.dodecahedron().vertices_list()                                    # optional - sage.rings.number_field
-    sage: Polyhedron(vertices=V, backend='polymake')               # optional - jupymake  # optional - sage.rings.number_field
+    sage: V = polytopes.dodecahedron().vertices_list()                                  # needs sage.rings.number_field
+    sage: Polyhedron(vertices=V, backend='polymake')    # optional - jupymake           # needs sage.rings.number_field
     A 3-dimensional polyhedron
      in (Number Field in sqrt5 with defining polynomial x^2 - 5
      with sqrt5 = 2.236067977499790?)^3
@@ -681,7 +684,7 @@ examples.
 
 ::
 
-    sage: type(D)                                                                         # optional - sage.rings.number_field
+    sage: type(D)                                                                       # needs sage.rings.number_field
     <class 'sage.geometry.polyhedron.parent.Polyhedra_field_with_category.element_class'>
 
 .. end of output
@@ -691,13 +694,14 @@ backend :code:`field` is called.
 
 ::
 
-    sage: P4.parent()                                                                     # optional - sage.rings.number_field
+    sage: # needs sage.rings.number_field
+    sage: P4.parent()
     Polyhedra in AA^2
-    sage: P5.parent()                                                                     # optional - sage.rings.number_field
+    sage: P5.parent()
     Polyhedra in AA^2
-    sage: type(P4)                                                                        # optional - sage.rings.number_field
+    sage: type(P4)
     <class 'sage.geometry.polyhedron.parent.Polyhedra_field_with_category.element_class'>
-    sage: type(P5)                                                                        # optional - sage.rings.number_field
+    sage: type(P5)
     <class 'sage.geometry.polyhedron.parent.Polyhedra_field_with_category.element_class'>
 
 .. end of output
@@ -709,13 +713,15 @@ The fourth backend is :code:`normaliz` and is an optional Sage package.
 
 ::
 
-    sage: P1_normaliz = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]], backend='normaliz')  # optional - pynormaliz
-    sage: type(P1_normaliz)                                                                           # optional - pynormaliz
+    sage: # optional - pynormaliz
+    sage: P1_normaliz = Polyhedron(vertices=[[1, 0], [0, 1]], rays=[[1, 1]],
+    ....:                          backend='normaliz')
+    sage: type(P1_normaliz)
     <class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_normaliz_with_category.element_class'>
-    sage: P2_normaliz = Polyhedron(vertices = [[1/2, 0, 0], [0, 1/2, 0]],                             # optional - pynormaliz
-    ....:                 rays = [[1, 1, 0]],
-    ....:                 lines = [[0, 0, 1]], backend='normaliz')
-    sage: type(P2_normaliz)                                                                           # optional - pynormaliz
+    sage: P2_normaliz = Polyhedron(vertices=[[1/2, 0, 0], [0, 1/2, 0]],
+    ....:                          rays=[[1, 1, 0]],
+    ....:                          lines=[[0, 0, 1]], backend='normaliz')
+    sage: type(P2_normaliz)
     <class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_normaliz_with_category.element_class'>
 
 .. end of output
@@ -724,7 +730,7 @@ This backend does not work with :code:`RDF` or other inexact fields.
 
 ::
 
-    sage: P3_normaliz = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], backend='normaliz')             # optional - pynormaliz
+    sage: P3_normaliz = Polyhedron(vertices=[[0.5, 0], [0, 0.5]], backend='normaliz')   # optional - pynormaliz
     Traceback (most recent call last):
     ...
     ValueError: No such backend (=normaliz) implemented for given basering (=Real Double Field).
@@ -738,12 +744,14 @@ the computation is done using an embedded number field.
 
 ::
 
-    sage: P4_normaliz = Polyhedron(vertices = [[sqrt_2, 0], [0, cbrt_2]], backend='normaliz')       # optional - pynormaliz
-    sage: P4_normaliz                                                                               # optional - pynormaliz
+    sage: # optional - pynormaliz
+    sage: P4_normaliz = Polyhedron(vertices=[[sqrt_2, 0], [0, cbrt_2]],
+    ....:                          backend='normaliz')
+    sage: P4_normaliz
     A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
-
-    sage: P5_normaliz = Polyhedron(vertices = [[sqrt_2s, 0], [0, cbrt_2s]], backend='normaliz')     # optional - pynormaliz
-    sage: P5_normaliz                                                                               # optional - pynormaliz
+    sage: P5_normaliz = Polyhedron(vertices=[[sqrt_2s, 0], [0, cbrt_2s]],
+    ....:                          backend='normaliz')
+    sage: P5_normaliz
     A 1-dimensional polyhedron in (Symbolic Ring)^2 defined as the convex hull of 2 vertices
 
 .. end of output
@@ -753,12 +761,14 @@ The backend :code:`normaliz` provides other methods such as
 
 ::
 
-    sage: P6 = Polyhedron(vertices = [[0, 0], [3/2, 0], [3/2, 3/2], [0, 3]], backend='normaliz')  # optional - pynormaliz
-    sage: IH = P6.integral_hull(); IH                                                             # optional - pynormaliz
+    sage: # optional - pynormaliz
+    sage: P6 = Polyhedron(vertices=[[0, 0], [3/2, 0], [3/2, 3/2], [0, 3]],
+    ....:                 backend='normaliz')
+    sage: IH = P6.integral_hull(); IH
     A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices
-    sage: P6.plot(color='blue')+IH.plot(color='red')                                              # optional - pynormaliz
+    sage: P6.plot(color='blue') + IH.plot(color='red')
     Graphics object consisting of 12 graphics primitives
-    sage: P1_normaliz.integral_hull()                                                             # optional - pynormaliz
+    sage: P1_normaliz.integral_hull()
     A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
 
 .. end of output
@@ -788,7 +798,7 @@ polytope is already defined!
 
 ::
 
-    sage: A = polytopes.buckyball(); A  # can take long                       # optional - sage.rings.number_field
+    sage: A = polytopes.buckyball(); A  # can take long                                 # needs sage.rings.number_field
     A 3-dimensional polyhedron
      in (Number Field in sqrt5 with defining polynomial x^2 - 5
          with sqrt5 = 2.236067977499790?)^3