gh-40281: Remove programmatic usage of analyse()
    
`analyse` is meant to be used by the user, it is (very likely, because
of all the extra explanation) slower than `common_parent`.

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preview.

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URL: #40281
Reported by: user202729
Reviewer(s):

Author: Release Manager

src/sage/geometry/polyhedron/parent.py

@@ -896,10 +896,7 @@ def _coerce_base_ring(self, other):
         if not other_ring.is_exact():
             other_ring = RDF  # the only supported floating-point numbers for now
 
-        cm_map, cm_ring = get_coercion_model().analyse(self.base_ring(), other_ring)
-        if cm_ring is None:
-            raise TypeError(f'Could not coerce type {other} into ZZ, QQ, or RDF.')
-        return cm_ring
+        return get_coercion_model().common_parent(self.base_ring(), other_ring)
 
     def _coerce_map_from_(self, X):
         r"""

src/sage/structure/coerce.pyx

@@ -808,6 +808,9 @@ cdef class CoercionModel:
         elements or parents). If the parent of the result can be determined
         then it will be returned.
 
+        For programmatic usages, use :meth:`canonical_coercion` and
+        :meth:`common_parent` instead.
+
         EXAMPLES::
 
             sage: cm = sage.structure.element.get_coercion_model()
@@ -938,6 +941,9 @@ cdef class CoercionModel:
         the actual morphism and action objects (rather than their string
         representations), then this is the function to use.
 
+        For programmatic usages, use :meth:`canonical_coercion` and
+        :meth:`common_parent` instead.
+
         EXAMPLES::
 
             sage: cm = sage.structure.element.get_coercion_model()