fix comment formatting

Author: TheBlupper

src/sage/matrix/matrix_integer_dense.pyx

@@ -3346,15 +3346,15 @@ cdef class Matrix_integer_dense(Matrix_dense):
             A = IntegerMatrix.from_matrix(self)
             return LLL.is_reduced(A, delta=delta, eta=eta)
         elif algorithm == 'sage':
-            # this is pretty slow
+            # This is pretty slow
             import sage.modules.misc
             G, mu = sage.modules.misc.gram_schmidt(self.rows())
-            #For any $i>j$, we have $|mu_{i, j}| <= \eta$
+            # For any $i>j$, we have $|mu_{i, j}| <= \eta$
             for e in mu.list():
                 if e.abs() > eta:
                     return False
 
-            #For any $i<d$, we have $\delta |b_i^*|^2 <= |b_{i+1}^* + mu_{i+1, i} b_i^* |^2$
+            # For any $i<d$, we have $\delta |b_i^*|^2 <= |b_{i+1}^* + mu_{i+1, i} b_i^* |^2$
             norms = [G[i].norm()**2 for i in range(len(G))]
             for i in range(1,self.nrows()):
                 if norms[i] < (delta - mu[i,i-1]**2) * norms[i-1]:

src/sage/modules/free_module_integer.py

@@ -839,7 +839,7 @@ def approximate_closest_vector(self, t, delta=None, algorithm='embedding', *args
             sage: L.closest_vector(t)
             (1326, 1349, 1339, 1345)
 
-            sage: # checking that the other algorithms work
+            sage: # Checking that the other algorithms work
             sage: L.approximate_closest_vector(t, algorithm='nearest_plane')
             (1326, 1349, 1339, 1345)
             sage: L.approximate_closest_vector(t, algorithm='rounding_off')
@@ -848,7 +848,7 @@ def approximate_closest_vector(self, t, delta=None, algorithm='embedding', *args
         if delta is None:
             delta = ZZ(99)/ZZ(100)
 
-        # bound checks on delta are performed in is_LLL_reduced
+        # Bound checks on delta are performed in is_LLL_reduced
         if not self._reduced_basis.is_LLL_reduced(delta=delta):
             self.LLL(*args, delta=delta, **kwargs)
 
@@ -859,11 +859,10 @@ def approximate_closest_vector(self, t, delta=None, algorithm='embedding', *args
             L = matrix(QQ, B.nrows()+1, B.ncols()+1)
             L.set_block(0, 0, B)
             L.set_block(B.nrows(), 0, matrix(t))
-            weight = (B[-1]*B[-1]).isqrt()+1  # norm of the largest vector
+            weight = (B[-1]*B[-1]).isqrt()+1  # Norm of the largest vector
             L[-1, -1] = weight
 
-            # the vector should be the last row but we iterate
-            # just in case
+            # The vector should be the last row but we iterate just in case
             for v in reversed(L.LLL(delta=delta, *args, **kwargs).rows()):
                 if abs(v[-1]) == weight:
                     return t - v[:-1]*v[-1].sign()