Minimal Polynomials of Linear Recurrence Sequences - Matrices and Spaces of Matrices

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# Minimal Polynomials of Linear Recurrence Sequences

AUTHORS:

- William Stein

sage.matrix.berlekamp_massey.berlekamp_massey(a)

  Use the Berlekamp-Massey algorithm to find the minimal polynomial 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 \(\{a_r\}\) 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)

  OUTPUT:

  the minimal polynomial of the sequence, as a polynomial over the field in which the entries of \(a\) live

  Warning

  The result is only guaranteed to be correct on the full sequence if there exists a linear recurrence of length less than half the length of \(a\).

  EXAMPLES:

  Sage

      sage: from sage.matrix.berlekamp_massey import berlekamp_massey
      sage: berlekamp_massey([1,2,1,2,1,2])
      x^2 - 1
      sage: berlekamp_massey([GF(7)(1), 19, 1, 19])
      x^2 + 6
      sage: berlekamp_massey([2,2,1,2,1,191,393,132])
      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

  Python

      >>> from sage.all import *
      >>> from sage.matrix.berlekamp_massey import berlekamp_massey
      >>> berlekamp_massey([Integer(1),Integer(2),Integer(1),Integer(2),Integer(1),Integer(2)])
      x^2 - 1
      >>> berlekamp_massey([GF(Integer(7))(Integer(1)), Integer(19), Integer(1), Integer(19)])
      x^2 + 6
      >>> berlekamp_massey([Integer(2),Integer(2),Integer(1),Integer(2),Integer(1),Integer(191),Integer(393),Integer(132)])
      x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673
      >>> berlekamp_massey(prime_range(Integer(2), Integer(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