Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: A._multiply_classical(B) == A._multiply_strassen(B, 3) #indirect doctest True AUTHORS: •David Harvey •Simon King (2011-07): Improve memory efficiency; trac ticket #11610 sage.matrix.strassen.test(n, m, R, c=2) INPUT: •n - integer •m - integer •R - ring •c - integer (optional, default:2) EXAMPLES: sage: from sage.matrix.strassen import test sage: for n in range(5): print n, test(2*n,n,Frac(QQ[’x’]),2) 0 True 1 True 2 True 3 True 4 True 274 Chapter 8. Generic Asymptotically Fast Strassen Algorithms
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 + 37/9 275
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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