sagemathgh-35997: Improve getting matrix entries after permutation in…

… _palp_PM_max()

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<!-- Describe your changes here in detail --> Modify the _palp_PM_max()
to obtain the matrix entry after permutation without make copy of the
matrix. @mkoeppe
before
```
sage: %timeit -n 10 -r 3 o = lattice_polytope.cross_polytope(3);
o._palp_PM_max();
36.8 ms ± 5.19 ms per loop (mean ± std. dev. of 3 runs, 10 loops each)
```
after
```
sage: %timeit -n 10 -r 3 o = lattice_polytope.cross_polytope(3);
o._palp_PM_max();
4.88 ms ± 2.23 ms per loop (mean ± std. dev. of 3 runs, 10 loops each)

```
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### ⌛ Dependencies

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URL: sagemath#35997
Reported by: xuluze
Reviewer(s): Matthias Köppe

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=017780512407f8da87a091c485cf8c7162b5adaf
-md5=4e3e25464ee850c2593f16364bd6b206
-cksum=531215747
+sha1=d39abedb0ee7beb3d61b625d4627e10c347320f2
+md5=6a0dcd364b964e10b594339094d64a36
+cksum=2756767966

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-98520945e069c603400cf2ae902a843f7a1bda13
+259402d021bcc64831e1f24b26da34b76820b566

src/sage/geometry/lattice_polytope.py

@@ -3188,11 +3188,6 @@ def _palp_PM_max(self, check=False):
             sage: all(results)  # long time
             True
         """
-        def PGE(S, u, v):
-            if u == v:
-                return S.one()
-            return S((u, v), check=False)
-
         PM = self.vertex_facet_pairing_matrix()
         n_v = PM.ncols()
         n_f = PM.nrows()
@@ -3202,53 +3197,45 @@ def PGE(S, u, v):
         # and find all the ways of making the first row of PM_max
         def index_of_max(iterable):
             # returns the index of max of any iterable
-            m, x = 0, iterable[0]
-            for k, l in enumerate(iterable):
-                if l > x:
-                    m, x = k, l
-            return m
+            return max(enumerate(iterable), key=lambda x: x[1])[0]
 
         n_s = 1
-        permutations = {0 : [S_f.one(), S_v.one()]}
+        permutations = {0: [S_f.one(), S_v.one()]}
         for j in range(n_v):
-            m = index_of_max(
-                [(PM.with_permuted_columns(permutations[0][1]))[0][i]
-                 for i in range(j, n_v)])
+            m = index_of_max(PM[0, i] for i in range(j, n_v))
             if m > 0:
-                permutations[0][1] = PGE(S_v, j + 1, m + j + 1) * permutations[0][1]
+                permutations[0][1] = S_v((j + 1, m + j + 1), check=False) * permutations[0][1]
         first_row = list(PM[0])
 
         # Arrange other rows one by one and compare with first row
         for k in range(1, n_f):
             # Error for k == 1 already!
             permutations[n_s] = [S_f.one(), S_v.one()]
-            m = index_of_max(PM.with_permuted_columns(permutations[n_s][1])[k])
+            m = index_of_max(PM[k, permutations[n_s][1](j+1) - 1] for j in range(n_v))
             if m > 0:
-                permutations[n_s][1] = PGE(S_v, 1, m+1) * permutations[n_s][1]
-            d = ((PM.with_permuted_columns(permutations[n_s][1]))[k][0]
+                permutations[n_s][1] = S_v((1, m + 1), check=False) * permutations[n_s][1]
+            d = (PM[k, permutations[n_s][1](1) - 1]
                 - permutations[0][1](first_row)[0])
             if d < 0:
                 # The largest elt of this row is smaller than largest elt
                 # in 1st row, so nothing to do
                 continue
             # otherwise:
             for i in range(1, n_v):
-                m = index_of_max(
-                    [PM.with_permuted_columns(permutations[n_s][1])[k][j]
-                     for j in range(i, n_v)])
+                m = index_of_max(PM[k, permutations[n_s][1](j+1) - 1] for j in range(i,n_v))
                 if m > 0:
-                    permutations[n_s][1] = PGE(S_v, i + 1, m + i + 1) \
+                    permutations[n_s][1] = S_v((i + 1, m + i + 1), check=False) \
                                            * permutations[n_s][1]
                 if d == 0:
-                    d = (PM.with_permuted_columns(permutations[n_s][1])[k][i]
+                    d = (PM[k, permutations[n_s][1](i+1) - 1]
                         -permutations[0][1](first_row)[i])
                     if d < 0:
                         break
             if d < 0:
                 # This row is smaller than 1st row, so nothing to do
                 del permutations[n_s]
                 continue
-            permutations[n_s][0] = PGE(S_f, 1, k + 1) * permutations[n_s][0]
+            permutations[n_s][0] = S_f((1, k + 1), check=False) * permutations[n_s][0]
             if d == 0:
                 # This row is the same, so we have a symmetry!
                 n_s += 1
@@ -3258,9 +3245,9 @@ def index_of_max(iterable):
                 first_row = list(PM[k])
                 permutations = {0: permutations[n_s]}
                 n_s = 1
-        permutations = {k:permutations[k] for k in permutations if k < n_s}
+        permutations = {k: permutations[k] for k in permutations if k < n_s}
 
-        b = PM.with_permuted_rows_and_columns(*permutations[0])[0]
+        b = tuple(PM[permutations[0][0](1) - 1, permutations[0][1](j+1) - 1] for j in range(n_v))
         # Work out the restrictions the current permutations
         # place on other permutations as a automorphisms
         # of the first row
@@ -3294,34 +3281,35 @@ def index_of_max(iterable):
                 # between 0 and S(0)
                 for s in range(l, n_f):
                     for j in range(1, S[0]):
-                        v = PM.with_permuted_rows_and_columns(
-                            *permutations_bar[n_p])[s]
-                        if v[0] < v[j]:
-                            permutations_bar[n_p][1] = PGE(S_v, 1, j + 1) * permutations_bar[n_p][1]
+                        v0 = PM[permutations_bar[n_p][0](s+1) - 1, permutations_bar[n_p][1](1) - 1]
+                        vj = PM[permutations_bar[n_p][0](s+1) - 1, permutations_bar[n_p][1](j+1) - 1]
+                        if v0 < vj:
+                            permutations_bar[n_p][1] = S_v((1, j + 1), check=False) * permutations_bar[n_p][1]
                     if ccf == 0:
-                        l_r[0] = PM.with_permuted_rows_and_columns(
-                                 *permutations_bar[n_p])[s][0]
-                        permutations_bar[n_p][0] = PGE(S_f, l + 1, s + 1) * permutations_bar[n_p][0]
+                        l_r[0] = PM[permutations_bar[n_p][0](s+1) - 1, permutations_bar[n_p][1](1) - 1]
+                        if s != l:
+                            permutations_bar[n_p][0] = S_f((l + 1, s + 1), check=False) * permutations_bar[n_p][0]
                         n_p += 1
                         ccf = 1
                         permutations_bar[n_p] = copy(permutations[k])
                     else:
-                        d1 = PM.with_permuted_rows_and_columns(
-                             *permutations_bar[n_p])[s][0]
+                        d1 = PM[permutations_bar[n_p][0](s+1) - 1, permutations_bar[n_p][1](1) - 1]
                         d = d1 - l_r[0]
                         if d < 0:
                             # We move to the next line
                             continue
                         elif d==0:
                             # Maximal values agree, so possible symmetry
-                            permutations_bar[n_p][0] = PGE(S_f, l + 1, s + 1) * permutations_bar[n_p][0]
+                            if s != l:
+                                permutations_bar[n_p][0] = S_f((l + 1, s + 1), check=False) * permutations_bar[n_p][0]
                             n_p += 1
                             permutations_bar[n_p] = copy(permutations[k])
                         else:
                             # We found a greater maximal value for first entry.
                             # It becomes our new reference:
                             l_r[0] = d1
-                            permutations_bar[n_p][0] = PGE(S_f, l + 1, s + 1) * permutations_bar[n_p][0]
+                            if s != l:
+                                permutations_bar[n_p][0] = S_f((l + 1, s + 1), check=False) * permutations_bar[n_p][0]
                             # Forget previous work done
                             cf = 0
                             permutations_bar = {0:copy(permutations_bar[n_p])}
@@ -3343,18 +3331,16 @@ def index_of_max(iterable):
                         s -= 1
                         # Find the largest value in this symmetry block
                         for j in range(c + 1, h):
-                            v = PM.with_permuted_rows_and_columns(
-                                *permutations_bar[s])[l]
-                            if (v[c] < v[j]):
-                                permutations_bar[s][1] = PGE(S_v, c + 1, j + 1) * permutations_bar[s][1]
+                            vc = PM[(permutations_bar[s][0])(l+1) - 1, (permutations_bar[s][1])(c+1) - 1]
+                            vj = PM[(permutations_bar[s][0])(l+1) - 1, (permutations_bar[s][1])(j+1) - 1]
+                            if (vc < vj):
+                                permutations_bar[s][1] = S_v((c + 1, j + 1), check=False) * permutations_bar[s][1]
                         if ccf == 0:
                             # Set reference and carry on to next permutation
-                            l_r[c] = PM.with_permuted_rows_and_columns(
-                                     *permutations_bar[s])[l][c]
+                            l_r[c] = PM[(permutations_bar[s][0])(l+1) - 1, (permutations_bar[s][1])(c+1) - 1]
                             ccf = 1
                         else:
-                            d1 = PM.with_permuted_rows_and_columns(
-                                *permutations_bar[s])[l][c]
+                            d1 = PM[(permutations_bar[s][0])(l+1) - 1, (permutations_bar[s][1])(c+1) - 1]
                             d = d1 - l_r[c]
                             if d < 0:
                                 n_p -= 1
@@ -3383,7 +3369,7 @@ def index_of_max(iterable):
                 # the restrictions the last worked out
                 # row imposes.
                 c = 0
-                M = (PM.with_permuted_rows_and_columns(*permutations[0]))[l]
+                M = tuple(PM[permutations[0][0](l+1) - 1, permutations[0][1](j+1) - 1] for j in range(n_v))
                 while c < n_v:
                     s = S[c] + 1
                     S[c] = c + 1
@@ -5196,11 +5182,10 @@ def _palp_canonical_order(V, PM_max, permutations):
          in 2-d lattice M, (1,3,2,4))
     """
     n_v = PM_max.ncols()
-    n_f = PM_max.nrows()
     S_v = SymmetricGroup(n_v)
     p_c = S_v.one()
-    M_max = [max([PM_max[i][j] for i in range(n_f)]) for j in range(n_v)]
-    S_max = [sum([PM_max[i][j] for i in range(n_f)]) for j in range(n_v)]
+    M_max = [max(row[j] for row in PM_max.rows()) for j in range(n_v)]
+    S_max = sum(PM_max)
     for i in range(n_v):
         k = i
         for j in range(i + 1, n_v):
@@ -5213,13 +5198,15 @@ def _palp_canonical_order(V, PM_max, permutations):
             p_c = S_v((1 + i, 1 + k), check=False) * p_c
     # Create array of possible NFs.
     permutations = [p_c * l[1] for l in permutations.values()]
-    Vs = [(V.column_matrix().with_permuted_columns(sig).hermite_form(), sig)
+    Vmatrix = V.column_matrix()
+    Vs = [(Vmatrix.with_permuted_columns(sig).hermite_form(), sig)
           for sig in permutations]
     Vmin = min(Vs, key=lambda x:x[0])
-    vertices = [V.module()(_) for _ in Vmin[0].columns()]
+    Vmodule = V.module()
+    vertices = [Vmodule(_) for _ in Vmin[0].columns()]
     for v in vertices:
         v.set_immutable()
-    return (PointCollection(vertices, V.module()), Vmin[1])
+    return (PointCollection(vertices, Vmodule), Vmin[1])
 
 
 def _palp_convert_permutation(permutation):