Remove unnecessary constraint for berlekamp_massey function

src/sage/matrix/berlekamp_massey.py

@@ -29,15 +29,15 @@
     of a linear recurrence sequence `a`.
 
     The minimal polynomial of a linear recurrence `\{a_r\}` is
-    by definition the unique monic polynomial `g`, such that if
+    by definition the monic polynomial `g`, such that if
     `\{a_r\}` satisfies a linear recurrence
     `a_{j+k} + b_{j-1} a_{j-1+k} + \cdots + b_0 a_k=0`
     (for all `k\geq 0`), then `g` divides the
     polynomial `x^j + \sum_{i=0}^{j-1} b_i x^i`.
 
     INPUT:
 
-    - ``a`` -- list of even length of elements of a field (or domain)
+    - ``a`` -- list of elements of a field (or domain)
 
     OUTPUT:
 
@@ -46,7 +46,7 @@
 
     .. WARNING::
 
-         The result is only guaranteed to be correct on the full
+         The result is only guaranteed to be unique on the full
          sequence if there exists a linear recurrence of length less
          than half the length of `a`.
 
@@ -61,34 +61,33 @@
         x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673
         sage: berlekamp_massey(prime_range(2, 38))                                      # needs sage.libs.pari
         x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9
+        sage: berlekamp_massey([1,2,3,4,5])
+        x^2 - 2*x + 1
 
     TESTS::
 
         sage: berlekamp_massey("banana")
         Traceback (most recent call last):
         ...
         TypeError: argument must be a list or tuple
-        sage: berlekamp_massey([1,2,5])
-        Traceback (most recent call last):
-        ...
-        ValueError: argument must have an even number of terms
     """
     if not isinstance(a, (list, tuple)):
         raise TypeError("argument must be a list or tuple")
-    if len(a) % 2:
-        raise ValueError("argument must have an even number of terms")
-
-    M = len(a) // 2
 
     try:
         K = a[0].parent().fraction_field()
     except AttributeError:
         K = sage.rings.rational_field.RationalField()
 
     R, x = K['x'].objgen()
-    f0, f1 = R(a), x**(2 * M)
-    s0, s1 = 1, 0
-    while f1.degree() >= M:
-        f0, (q, f1) = f1, f0.quo_rem(f1)
-        s0, s1 = s1, s0 - q * s1
-    return s1.reverse().monic()
+    f0, f1 = R(a[::-1]), x**len(a)
+    if f0.is_zero():
+        return R.one()
+    s0, s1 = 0, 1
+    k = len(a)
+    while True:
+        f1, (q, f0) = f0, f1.quo_rem(f0)
+        s0, s1 = s1, s0 + q * s1
+        if f0.is_zero() or f0.degree() - f1.degree() < -k + 2 * q.degree():
+            return s1.monic()
+        k -= 2 * q.degree()

src/sage/matrix/matrix2.pyx

@@ -6157,7 +6157,7 @@ cdef class Matrix(Matrix1):
             [3 4 5]
             [6 7 8]
             sage: t.wiedemann(0)
-            x^2 - 12*x - 18
+            x^3 - 12*x^2 - 18*x
             sage: t.charpoly()                                                          # needs sage.libs.pari
             x^3 - 12*x^2 - 18*x
         """

src/sage/rings/cfinite_sequence.py

@@ -1193,13 +1193,12 @@
             ...
             ValueError: sequence too short for guessing
 
-        With Berlekamp-Massey, if an odd number of values is given, the last one is dropped.
-        So with an odd number of values the result may not generate the last value::
+        Using Berlekamp-Massey::
 
-            sage: r = C.guess([1,2,4,8,9], algorithm='bm'); r
-            C-finite sequence, generated by -1/2/(x - 1/2)
+            sage: r = C.guess([0,1,1,1], algorithm='bm'); r
+            C-finite sequence, generated by -x/(x - 1)
             sage: r[0:5]
-            [1, 2, 4, 8, 16]
+            [0, 1, 1, 1, 1]
 
         Using pari::
 
@@ -1214,14 +1213,14 @@
             from sage.matrix.berlekamp_massey import berlekamp_massey
             if len(sequence) < 2:
                 raise ValueError('sequence too short for guessing')
-            R = PowerSeriesRing(QQ, 'x')
-            if len(sequence) % 2:
-                sequence.pop()
-            l = len(sequence) - 1
-            denominator = S(berlekamp_massey(sequence).reverse())
-            numerator = R(S(sequence) * denominator, prec=l).truncate()
-
-            return CFiniteSequence(numerator / denominator)
+            x = S.gen()
+            rden = berlekamp_massey(sequence)
+            rnum = S(sequence[:rden.degree()][::-1]) * rden // x**(rden.degree())
+            if rnum.is_zero():
+                return CFiniteSequence(S(0), rden.reverse())
+            num = x**(rden.degree() - rnum.degree() - 1) * rnum.reverse()
+            den = rden.reverse()
+            return CFiniteSequence(num / den)
 
         if algorithm == 'pari':
             if len(sequence) < 6: