Exact Linear Algebra for SAGE - William Stein - University of ...

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Notes Remark Echelon Forms of Matrices Computing Echelon Forms Decomposing Spaces Under the Action of Matrix There is also an approach called FFLAS-FFPACK that works just like above, but for the small finite field matrices, does the matrices using a general floating point BLAS (=Basic Linear Algebra System), e.g., ATLAS. This can result in additional speedups, but requires that a BLAS be installed. This is not yet implemented in SAGE, because SAGE does not include an optimized BLAS. William Stein Exact Linear Algebra for SAGE
Echelon Forms of Matrices Computing Echelon Forms Decomposing Spaces Under the Action of Matrix Rational Reconstruction Rational reconstruction is a process that allows one to sometimes lift an integer modulo m uniquely to a bounded rational number. Algorithm (Rational Reconstruction) Given an integer a ≥ 0 and an integer m > 1, this algorithm efficiently computes the numerator n and denominator d of the unique rational number n/d, if it exists, with r m |n|, d ≤ and n ≡ ad (mod m), (2.1) 2 or reports that there is no such number. 1 [Reduce mod m] Let a be the least integer between 0 and m − 1 that is congruent to a modulo m. 2 [Trivial cases] If a = 0 or a = 1, return a. 3 [Initialize] Let b = p m/2, u = m, v = a, and set U = (1, 0, u) and V = (0, 1, v). Use the notation U i and V i to refer to the ith entries of U, V , for i = 0, 1, 2. 4 [Iterate] Do the following as long as |V 2| > b: Set q = ⌊U 2/V 2⌋, set T = U − qV , set U = V and V = T . 5 [Numerator and Denominator] Set d = |V 1| and n = V 2. 6 [Good?] If d ≤ b and gcd(n, d) = 1 return n/d, otherwise report that there is no rational number as in (2.1). William Stein Exact Linear Algebra for SAGE
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