Minimal Polynomials of Linear Recurrence Sequences¶

AUTHORS:

William Stein

sage.matrix.berlekamp_massey.berlekamp_massey(a)[source]¶

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