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sage: P.<b1,b2,b3> = QQ[]
sage: A = Matrix(P, 3, 4, [[1,1,2,b1], [1,0,1,b2], [2,1,3,b3]])
sage: A
[ 1  1  2 b1]
[ 1  0  1 b2]
[ 2  1  3 b3]
sage: A.echelon_form('row_reduction')
[            1             0             1            b2]
[            0             1             1       b1 - b2]
[            0             0             0 -b1 - b2 + b3]

Hi,

20,000 x 20,000 is not large and you should be able to do it in a few seconds on a modern CPU:

http://m4ri.sagemath.org/performance.html

However, Sage doesn't expose the fast code we have for this in M4RI:, cf.

http://m4ri.sagemath.org/doxygen/solve_8h.html

That's why it's not as fast as it should be. Sorry, we should fix this.

Do you mean this?

sage: R.<x,y,z> = GF(127)[]
sage: p = R.random_element(degree=3)
sage: Q = [R.random_element() for _ in range(R.ngens())]
sage: r = p.reduce(Ideal(Q))
sage: M = (p - r).lift(Q)
sage: p -r == sum([M[i]*Q[i] for i in range(len(M))])
True