Suppose I have, e.g., 250 linear equations with Rational{BigInt} coefficients, with 150 variables. If the system is inconsistent I want to know about it, otherwise I want the exact solution.
Is there an easier way than writing my own Gaussian elimination?
This won’t run often, so I’m not concerned with long runtimes.
Have a look at Nemo.jl; it does exact linear algebra.
Just for sake of completeness, either using AbstractAlgebra or (more efficiently) Nemo, it can be done as follows:
julia> using AbstractAlgebra # will also work with using Nemo;
julia> A = QQ[1 0; 3 0; 5 0]; # create an AbstractAlgebra matrix
julia> v = QQ[1; 2; 3]; # create a 3x1 matrix
julia> can_solve_with_solution(A, v) # solve Ax = v
(false, [1; 0])
julia> A = QQ[1 2; 3 4; 5 6];
julia> can_solve_with_solution(A, v) # solve Ax = v
(true, [0; 1//2])
julia> fl, x = can_solve_with_solution(A, v) # solve Ax = v
(true, [0; 1//2])
julia> A*x == v
true
One can also do can_solve_with_solution(A, v; side = :left) to solve xA = v.
TBH I didn’t even realize that can_solve_with_solution existed, instead i used AbstractAlgebra.rref 
hello thofma,
this interests me very much. does a QQMatrix has some elements of the type BigInt ? I want to solve a very big system A.X=Y with Rational{BigInt}. Thanks !
And, is there a fast way to convert a Matrix{Rational{BigInt}} to a QQMatrix, and back ? I could copy it term by term, eventually …
thanks a lot 
Yes, there is a fast way. Here is an example, where one solves A*X = Y where X and Y are of type Vector.
julia> AA = Rational{BigInt}[1 2; 3 4]
2×2 Matrix{Rational{BigInt}}:
1 2
3 4
julia> A = matrix(QQ, AA)
[1 2]
[3 4]
julia> YY = Rational{BigInt}[5, 6]
2-element Vector{Rational{BigInt}}:
5
6
julia> Y = QQ.(YY)
2-element Vector{QQFieldElem}:
5
6
julia> fl, X = can_solve_with_solution(A, Y; side = :right) # solves A * X = Y for X
(true, QQFieldElem[-4, 9//2])
julia> Rational{BigInt}.(X)
2-element Vector{Rational{BigInt}}:
-4
9//2
Here is an example, where X and Y are themselves of type Matrix:
julia> YY = Rational{BigInt}[5 6; 7 8]
2×2 Matrix{Rational{BigInt}}:
5 6
7 8
julia> Y = matrix(QQ, YY)
[5 6]
[7 8]
julia> fl, X = can_solve_with_solution(A, Y; side = :right) # solves A * X = Y for X
(true, [-3 -4; 4 5])
julia> XX = Matrix{Rational{BigInt}}(X)
2×2 Matrix{Rational{BigInt}}:
-3 -4
4 5
Depending on how what you mean with “very big”, it might take a while to solve your system, but give it a try!
Ok thanks, everything works well I am new to Julia, and I don’t understand “constructors” well !
“Big Matrix” means only that I might need some BigInt, in any case the coefficients explode
Thanks a lot !
Gustave
Hello
I actually I should use Matrix{Complex{Rational{Int}}} . Is there way to do so with Nemo ? Or maybe with AbstractAlgebra ? Thanks !
finally I did the code with Complex{Float64}. But I like Algebra, so that your answer interests me !!!
Bye
Sorry for the late reply. Yes, this is also possible:
julia> QQi = Nemo.QQiField()
Gaussian rational field
julia> AA = Complex{Rational{BigInt}}[1 2im; 3 4im]
2×2 Matrix{Complex{Rational{BigInt}}}:
1//1+0//1*im 0//1+2//1*im
3//1+0//1*im 0//1+4//1*im
julia> A = matrix(QQi, AA)
[1 2*im]
[3 4*im]
julia> YY = Complex{Rational{BigInt}}[5im, 6]
2-element Vector{Complex{Rational{BigInt}}}:
0//1 + 5//1*im
6//1 + 0//1*im
julia> Y = QQi.(YY)
2-element Vector{Nemo.QQiFieldElem}:
5*im
6
julia> fl, X = can_solve_with_solution(A, Y; side = :right) # solves A * X = Y for X
(true, Nemo.QQiFieldElem[6 - 10*im, 15//2 + 3*im])
julia> XX = [Complex{Rational{BigInt}}(real(y), imag(y)) for y in X]
2-element Vector{Complex{Rational{BigInt}}}:
6//1 - 10//1*im
15//2 + 3//1*im
OK thanks, it’s perfect ! I gonna try it tomorrow
Have a nice day
Gustave
It works perfectly well, thank you !!!