prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  Unoptimal memory complexity of `sage.matrix.berlekamp`
  The code here is unoptimal:
  
  https://github.com/sagemath/sage/blob/6695becb762aebab78ef47d0fb12eae52be5d79d/src/sage/matrix/berlekamp_massey.py#L90-L98
  
  For example, the following code uses a lot of memory:
  
  ```python
  sage: from sage.matrix.berlekamp_massey import berlekamp_massey
  sage: p = next_prime(2**64)
  sage: ls = [GF(p).random_element() for _ in range(20000)]
  sage: berlekamp_massey(ls);
  ```
  
  To be more specific, the dictionaries are not necessarily and only `f[j - 2]` and `f[j - 1]` are used every time, same for `s`. So they can be stored as temporary variables.
  
  ### Additional Information
  
  I am fixing it.
  
  ### Checklist
  
  - [X] I have searched the existing issues for a bug report that matches the one I want to file, without success.
  - [X] I have read the documentation and troubleshoot guide
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `src/sage/matrix/berlekamp_massey.py`
  Content:
  ```
  1 """
  2 Minimal Polynomials of Linear Recurrence Sequences
  3 
  4 AUTHORS:
  5 
  6 - William Stein
  7 """
  8 # ****************************************************************************
  9 #       Copyright (C) 2005 William Stein <wstein@gmail.com>
  10 #
  11 #  Distributed under the terms of the GNU General Public License (GPL)
  12 #
  13 #    This code is distributed in the hope that it will be useful,
  14 #    but WITHOUT ANY WARRANTY; without even the implied warranty of
  15 #    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  16 #    General Public License for more details.
  17 #
  18 #  The full text of the GPL is available at:
  19 #
  20 #                  https://www.gnu.org/licenses/
  21 # ****************************************************************************
  22 
  23 import sage.rings.rational_field
  24 
  25 
  26 def berlekamp_massey(a):
  27     r"""
  28     Use the Berlekamp-Massey algorithm to find the minimal polynomial
  29     of a linear recurrence sequence `a`.
  30 
  31     The minimal polynomial of a linear recurrence `\{a_r\}` is
  32     by definition the unique monic polynomial `g`, such that if
  33     `\{a_r\}` satisfies a linear recurrence
  34     `a_{j+k} + b_{j-1} a_{j-1+k} + \cdots + b_0 a_k=0`
  35     (for all `k\geq 0`), then `g` divides the
  36     polynomial `x^j + \sum_{i=0}^{j-1} b_i x^i`.
  37 
  38     INPUT:
  39 
  40     -  ``a`` -- a list of even length of elements of a field (or domain)
  41 
  42     OUTPUT:
  43 
  44     the minimal polynomial of the sequence, as a polynomial over the
  45     field in which the entries of `a` live
  46 
  47     .. WARNING::
  48 
  49          The result is only guaranteed to be correct on the full
  50          sequence if there exists a linear recurrence of length less
  51          than half the length of `a`.
  52 
  53     EXAMPLES::
  54 
  55         sage: from sage.matrix.berlekamp_massey import berlekamp_massey
  56         sage: berlekamp_massey([1,2,1,2,1,2])
  57         x^2 - 1
  58         sage: berlekamp_massey([GF(7)(1), 19, 1, 19])
  59         x^2 + 6
  60         sage: berlekamp_massey([2,2,1,2,1,191,393,132])
  61         x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673
  62         sage: berlekamp_massey(prime_range(2, 38))                                      # needs sage.libs.pari
  63         x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9
  64 
  65     TESTS::
  66 
  67         sage: berlekamp_massey("banana")
  68         Traceback (most recent call last):
  69         ...
  70         TypeError: argument must be a list or tuple
  71         sage: berlekamp_massey([1,2,5])
  72         Traceback (most recent call last):
  73         ...
  74         ValueError: argument must have an even number of terms
  75     """
  76     if not isinstance(a, (list, tuple)):
  77         raise TypeError("argument must be a list or tuple")
  78     if len(a) % 2:
  79         raise ValueError("argument must have an even number of terms")
  80 
  81     M = len(a) // 2
  82 
  83     try:
  84         K = a[0].parent().fraction_field()
  85     except AttributeError:
  86         K = sage.rings.rational_field.RationalField()
  87     R = K['x']
  88     x = R.gen()
  89 
  90     f = {-1: R(a), 0: x**(2 * M)}
  91     s = {-1: 1, 0: 0}
  92     j = 0
  93     while f[j].degree() >= M:
  94         j += 1
  95         qj, f[j] = f[j - 2].quo_rem(f[j - 1])
  96         s[j] = s[j - 2] - qj * s[j - 1]
  97     t = s[j].reverse()
  98     return ~(t[t.degree()]) * t  # make monic  (~ is inverse in python)
  99
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/src/sage/matrix/berlekamp_massey.py b/src/sage/matrix/berlekamp_massey.py
  --- a/src/sage/matrix/berlekamp_massey.py
  +++ b/src/sage/matrix/berlekamp_massey.py
  @@ -84,15 +84,11 @@
           K = a[0].parent().fraction_field()
       except AttributeError:
           K = sage.rings.rational_field.RationalField()
  -    R = K['x']
  -    x = R.gen()
  -
  -    f = {-1: R(a), 0: x**(2 * M)}
  -    s = {-1: 1, 0: 0}
  -    j = 0
  -    while f[j].degree() >= M:
  -        j += 1
  -        qj, f[j] = f[j - 2].quo_rem(f[j - 1])
  -        s[j] = s[j - 2] - qj * s[j - 1]
  -    t = s[j].reverse()
  -    return ~(t[t.degree()]) * t  # make monic  (~ is inverse in python)
  +
  +    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()
  
verification_info: {"golden_diff": "diff --git a/src/sage/matrix/berlekamp_massey.py b/src/sage/matrix/berlekamp_massey.py\n--- a/src/sage/matrix/berlekamp_massey.py\n+++ b/src/sage/matrix/berlekamp_massey.py\n@@ -84,15 +84,11 @@\n         K = a[0].parent().fraction_field()\n     except AttributeError:\n         K = sage.rings.rational_field.RationalField()\n-    R = K['x']\n-    x = R.gen()\n-\n-    f = {-1: R(a), 0: x**(2 * M)}\n-    s = {-1: 1, 0: 0}\n-    j = 0\n-    while f[j].degree() >= M:\n-        j += 1\n-        qj, f[j] = f[j - 2].quo_rem(f[j - 1])\n-        s[j] = s[j - 2] - qj * s[j - 1]\n-    t = s[j].reverse()\n-    return ~(t[t.degree()]) * t  # make monic  (~ is inverse in python)\n+\n+    R, x = K['x'].objgen()\n+    f0, f1 = R(a), x**(2 * M)\n+    s0, s1 = 1, 0\n+    while f1.degree() >= M:\n+        f0, (q, f1) = f1, f0.quo_rem(f1)\n+        s0, s1 = s1, s0 - q * s1\n+    return s1.reverse().monic()\n", "issue": "Unoptimal memory complexity of `sage.matrix.berlekamp`\nThe code here is unoptimal:\r\n\r\nhttps://github.com/sagemath/sage/blob/6695becb762aebab78ef47d0fb12eae52be5d79d/src/sage/matrix/berlekamp_massey.py#L90-L98\r\n\r\nFor example, the following code uses a lot of memory:\r\n\r\n```python\r\nsage: from sage.matrix.berlekamp_massey import berlekamp_massey\r\nsage: p = next_prime(2**64)\r\nsage: ls = [GF(p).random_element() for _ in range(20000)]\r\nsage: berlekamp_massey(ls);\r\n```\r\n\r\nTo be more specific, the dictionaries are not necessarily and only `f[j - 2]` and `f[j - 1]` are used every time, same for `s`. So they can be stored as temporary variables.\r\n\r\n### Additional Information\r\n\r\nI am fixing it.\r\n\r\n### Checklist\r\n\r\n- [X] I have searched the existing issues for a bug report that matches the one I want to file, without success.\r\n- [X] I have read the documentation and troubleshoot guide\n", "before_files": [{"content": "\"\"\"\nMinimal Polynomials of Linear Recurrence Sequences\n\nAUTHORS:\n\n- William Stein\n\"\"\"\n# ****************************************************************************\n#       Copyright (C) 2005 William Stein <wstein@gmail.com>\n#\n#  Distributed under the terms of the GNU General Public License (GPL)\n#\n#    This code is distributed in the hope that it will be useful,\n#    but WITHOUT ANY WARRANTY; without even the implied warranty of\n#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU\n#    General Public License for more details.\n#\n#  The full text of the GPL is available at:\n#\n#                  https://www.gnu.org/licenses/\n# ****************************************************************************\n\nimport sage.rings.rational_field\n\n\ndef berlekamp_massey(a):\n    r\"\"\"\n    Use the Berlekamp-Massey algorithm to find the minimal polynomial\n    of a linear recurrence sequence `a`.\n\n    The minimal polynomial of a linear recurrence `\\{a_r\\}` is\n    by definition the unique monic polynomial `g`, such that if\n    `\\{a_r\\}` satisfies a linear recurrence\n    `a_{j+k} + b_{j-1} a_{j-1+k} + \\cdots + b_0 a_k=0`\n    (for all `k\\geq 0`), then `g` divides the\n    polynomial `x^j + \\sum_{i=0}^{j-1} b_i x^i`.\n\n    INPUT:\n\n    -  ``a`` -- a list of even length of elements of a field (or domain)\n\n    OUTPUT:\n\n    the minimal polynomial of the sequence, as a polynomial over the\n    field in which the entries of `a` live\n\n    .. WARNING::\n\n         The result is only guaranteed to be correct on the full\n         sequence if there exists a linear recurrence of length less\n         than half the length of `a`.\n\n    EXAMPLES::\n\n        sage: from sage.matrix.berlekamp_massey import berlekamp_massey\n        sage: berlekamp_massey([1,2,1,2,1,2])\n        x^2 - 1\n        sage: berlekamp_massey([GF(7)(1), 19, 1, 19])\n        x^2 + 6\n        sage: berlekamp_massey([2,2,1,2,1,191,393,132])\n        x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673\n        sage: berlekamp_massey(prime_range(2, 38))                                      # needs sage.libs.pari\n        x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9\n\n    TESTS::\n\n        sage: berlekamp_massey(\"banana\")\n        Traceback (most recent call last):\n        ...\n        TypeError: argument must be a list or tuple\n        sage: berlekamp_massey([1,2,5])\n        Traceback (most recent call last):\n        ...\n        ValueError: argument must have an even number of terms\n    \"\"\"\n    if not isinstance(a, (list, tuple)):\n        raise TypeError(\"argument must be a list or tuple\")\n    if len(a) % 2:\n        raise ValueError(\"argument must have an even number of terms\")\n\n    M = len(a) // 2\n\n    try:\n        K = a[0].parent().fraction_field()\n    except AttributeError:\n        K = sage.rings.rational_field.RationalField()\n    R = K['x']\n    x = R.gen()\n\n    f = {-1: R(a), 0: x**(2 * M)}\n    s = {-1: 1, 0: 0}\n    j = 0\n    while f[j].degree() >= M:\n        j += 1\n        qj, f[j] = f[j - 2].quo_rem(f[j - 1])\n        s[j] = s[j - 2] - qj * s[j - 1]\n    t = s[j].reverse()\n    return ~(t[t.degree()]) * t  # make monic  (~ is inverse in python)\n", "path": "src/sage/matrix/berlekamp_massey.py"}], "after_files": [{"content": "\"\"\"\nMinimal Polynomials of Linear Recurrence Sequences\n\nAUTHORS:\n\n- William Stein\n\"\"\"\n# ****************************************************************************\n#       Copyright (C) 2005 William Stein <wstein@gmail.com>\n#\n#  Distributed under the terms of the GNU General Public License (GPL)\n#\n#    This code is distributed in the hope that it will be useful,\n#    but WITHOUT ANY WARRANTY; without even the implied warranty of\n#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU\n#    General Public License for more details.\n#\n#  The full text of the GPL is available at:\n#\n#                  https://www.gnu.org/licenses/\n# ****************************************************************************\n\nimport sage.rings.rational_field\n\n\ndef berlekamp_massey(a):\n    r\"\"\"\n    Use the Berlekamp-Massey algorithm to find the minimal polynomial\n    of a linear recurrence sequence `a`.\n\n    The minimal polynomial of a linear recurrence `\\{a_r\\}` is\n    by definition the unique monic polynomial `g`, such that if\n    `\\{a_r\\}` satisfies a linear recurrence\n    `a_{j+k} + b_{j-1} a_{j-1+k} + \\cdots + b_0 a_k=0`\n    (for all `k\\geq 0`), then `g` divides the\n    polynomial `x^j + \\sum_{i=0}^{j-1} b_i x^i`.\n\n    INPUT:\n\n    -  ``a`` -- a list of even length of elements of a field (or domain)\n\n    OUTPUT:\n\n    the minimal polynomial of the sequence, as a polynomial over the\n    field in which the entries of `a` live\n\n    .. WARNING::\n\n         The result is only guaranteed to be correct on the full\n         sequence if there exists a linear recurrence of length less\n         than half the length of `a`.\n\n    EXAMPLES::\n\n        sage: from sage.matrix.berlekamp_massey import berlekamp_massey\n        sage: berlekamp_massey([1,2,1,2,1,2])\n        x^2 - 1\n        sage: berlekamp_massey([GF(7)(1),19,1,19])\n        x^2 + 6\n        sage: berlekamp_massey([2,2,1,2,1,191,393,132])\n        x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673\n        sage: berlekamp_massey(prime_range(2,38))\n        x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9\n\n    TESTS::\n\n        sage: berlekamp_massey(\"banana\")\n        Traceback (most recent call last):\n        ...\n        TypeError: argument must be a list or tuple\n        sage: berlekamp_massey([1,2,5])\n        Traceback (most recent call last):\n        ...\n        ValueError: argument must have an even number of terms\n    \"\"\"\n    if not isinstance(a, (list, tuple)):\n        raise TypeError(\"argument must be a list or tuple\")\n    if len(a) % 2:\n        raise ValueError(\"argument must have an even number of terms\")\n\n    M = len(a) // 2\n\n    try:\n        K = a[0].parent().fraction_field()\n    except AttributeError:\n        K = sage.rings.rational_field.RationalField()\n\n    R, x = K['x'].objgen()\n    f0, f1 = R(a), x**(2 * M)\n    s0, s1 = 1, 0\n    while f1.degree() >= M:\n        f0, (q, f1) = f1, f0.quo_rem(f1)\n        s0, s1 = s1, s0 - q * s1\n    return s1.reverse().monic()\n", "path": "src/sage/matrix/berlekamp_massey.py"}]}