errata, improve random tests

Author: lftabera

src/sage/rings/polynomial/polynomial_element.pyx

@@ -2077,17 +2077,6 @@ cdef class Polynomial(CommutativeAlgebraElement):
             0
             sage: g._mul_generic(h) - K([h*c for c in g.list()])
             0
-
-        Compare with external libraries::
-
-            sage: K = ZZ['x']
-            sage: L = []
-            sage: for i in range(10):
-            ...     f = K.random_element(randint(10,100),100)
-            ...     g = K.random_element(randint(10,100),100)
-            ...     L.append(f*g - f._mul_generic(g))
-            sage: L
-            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
         """
         if self is right:
             return self._square_generic()
@@ -2326,17 +2315,6 @@ cdef class Polynomial(CommutativeAlgebraElement):
             sage: h2 = f._mul_karatsuba(g,randint(0,10))
             sage: h1 == h2
             True
-
-        Compare with external libraries::
-
-            sage: K = ZZ['x']
-            sage: L = []
-            sage: for i in range(10):
-            ...     f = K.random_element(randint(10,100),100)
-            ...     g = K.random_element(randint(10,100),100)
-            ...     L.append(f*g - f._mul_karatsuba(g, randint(0,32)))
-            sage: L
-            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
         """
         if self.is_zero():
             return self
@@ -6601,7 +6579,7 @@ cdef do_karatsuba_different_size(left, right, Py_ssize_t K_threshold):
     Here, we use Fibonacci numbers that need deepest recursion in this method.
 
         sage: K = ZZ[x]
-        sage: f = K.random_element(28)
+        sage: f = K.random_element(21)
         sage: g = K.random_element(34)
         sage: f*g - f._mul_karatsuba(g,0)
         0
@@ -6676,7 +6654,7 @@ cdef do_karatsuba(left, right, Py_ssize_t K_threshold,Py_ssize_t start_l, Py_ssi
         - num_elts: the length of the polynomials.
 
     Thus, the actual polynomials we want to multiply are represented by the
-    slices: left[ start_l: start_l+num_elts ], left[ right_l: right_l+num_elts ].
+    slices: left[ start_l: start_l+num_elts ], right[ right_l: right_l+num_elts ].
     We use this representation in order to avoid creating slices of lists and
     create smaller lists.
 

src/sage/rings/tests.py

@@ -294,3 +294,21 @@ def test_random_arith(level=MAX_LEVEL, trials=1):
         if i >= trials:
             return
 
+@random_testing
+def test_karatsuba_multiplication(degree1, degree2, verbose=False):
+    """
+    Test univariate karatsuba multiplication against an external library (FLINT).
+
+    EXAMPLES::
+
+    sage: sage.rings.tests.test_karatsuba_multiplication(1000,1000)
+    """
+    from sage.all import ZZ, randint
+    threshold = randint(0,min(degree1,degree2))
+    R = ZZ['x']
+    for i in range(10):
+        f = R.random_element(degree1,100)
+        g = R.random_element(degree2,100)
+        if g*f - f._mul_karatsuba(g,threshold) != 0:
+            raise ValueError("Multiplication failed")
+    return