Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

CHAPTER
NINE
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 linearly 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 + · · · + b 0 a k = 0 (for all k ≥ 0), then g divides the
polynomial x j + ∑ j−1
i=0 b ix i .
INPUT:
•a - a list of even length of elements of a field (or domain)
OUTPUT:
•Polynomial - the minimal polynomial of the sequence (as a polynomial over the field in which the
entries of a live)
EXAMPLES:
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))
x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9
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