Nemo is a computer algebra package for the Julia programming language, maintained by William Hart, Tommy Hofmann, Claus Fieker, Fredrik Johansson with additional code by Oleksandr Motsak, Marek Kaluba and other contributors.
- https://github.com/Nemocas/Nemo.jl (Source code)
- https://nemocas.github.io/Nemo.jl/stable/ (Online documentation)
The features of Nemo so far include:
- Multiprecision integers and rationals
- Integers modulo n
- p-adic numbers
- Finite fields (prime and non-prime order)
- Number field arithmetic
- Algebraic numbers
- Exact real and complex numbers
- Arbitrary precision real and complex balls
- Univariate and multivariate polynomials and matrices over the above
Nemo depends on AbstractAlgebra.jl which provides Nemo with generic routines for:
- Univariate and multivariate polynomials
- Absolute and relative power series
- Laurent series
- Fraction fields
- Residue rings
- Matrices and linear algebra
- Young Tableaux
- Permutation groups
To use Nemo we require Julia 1.6 or higher. Please see https://julialang.org/downloads/ for instructions on how to obtain julia for your system.
At the Julia prompt simply type
julia> using Pkg; Pkg.add("Nemo")
Here are some examples of using Nemo.
This example computes recursive univariate polynomials.
julia> using Nemo julia> R, x = polynomial_ring(ZZ, "x") (Univariate polynomial ring in x over ZZ, x) julia> S, y = polynomial_ring(R, "y") (Univariate polynomial ring in y over univariate polynomial ring, y) julia> T, z = polynomial_ring(S, "z") (Univariate polynomial ring in z over univariate polynomial ring, z) julia> f = x + y + z + 1 z + y + x + 1 julia> p = f^30; # semicolon suppresses output julia> @time q = p*(p+1); 0.161733 seconds (79.42 k allocations: 2.409 MiB)
Here is an example using generic recursive ring constructions.
julia> using Nemo julia> R, x = finite_field(7, 11, "x") (Finite field of degree 11 over GF(7), x) julia> S, y = polynomial_ring(R, "y") (Univariate polynomial ring in y over GF(7^11), y) julia> T = residue_ring(S, y^3 + 3x*y + 1) Residue ring of univariate polynomial ring modulo y^3 + 3*x*y + 1 julia> U, z = polynomial_ring(T, "z") (Univariate polynomial ring in z over residue ring, z) julia> f = (3y^2 + y + x)*z^2 + ((x + 2)*y^2 + x + 1)*z + 4x*y + 3; julia> g = (7y^2 - y + 2x + 7)*z^2 + (3y^2 + 4x + 1)*z + (2x + 1)*y + 1; julia> s = f^12; julia> t = (s + g)^12; julia> @time resultant(s, t) 0.059095 seconds (391.89 k allocations: 54.851 MiB, 5.22% gc time) (x^10 + 4*x^8 + 6*x^7 + 3*x^6 + 4*x^5 + x^4 + 6*x^3 + 5*x^2 + x)*y^2 + (5*x^10 + x^8 + 4*x^7 + 3*x^5 + 5*x^4 + 3*x^3 + x^2 + x + 6)*y + 2*x^10 + 6*x^9 + 5*x^8 + 5*x^7 + x^6 + 6*x^5 + 5*x^4 + 4*x^3 + x + 3
Here is an example using matrices.
julia> using Nemo julia> R, x = polynomial_ring(ZZ, "x") (Univariate polynomial ring in x over ZZ, x) julia> S = matrix_space(R, 40, 40) Matrix space of 40 rows and 40 columns over univariate polynomial ring in x over ZZ julia> M = rand(S, 2:2, -20:20); julia> @time det(M); 0.080976 seconds (132.28 k allocations: 23.341 MiB, 4.11% gc time)
And here is an example with power series.
julia> using Nemo julia> R, x = QQ["x"] (Univariate polynomial ring in x over QQ, x) julia> S, t = power_series_ring(R, 100, "t") (Univariate power series ring over univariate polynomial ring, t + O(t^101)) julia> u = t + O(t^100) t + O(t^100) julia> @time divexact((u*exp(x*u)), (exp(u)-1)); 0.412813 seconds (667.49 k allocations: 33.966 MiB, 90.26% compilation time)
Nemo depends on various C libraries which are installed using binaries by default. With julia version >= 1.3, the use of these binaries can be overridden by putting the following into the file
[e134572f-a0d5-539d-bddf-3cad8db41a82] FLINT = "/prefix/for/libflint" [d9960996-1013-53c9-9ba4-74a4155039c3] Arb = "/prefix/for/libarb" [e21ec000-9f72-519e-ba6d-10061e575a27] Antic = "/prefix/for/libantic"
(If only a specific library should be overridden, only the specific entry should be added.)
Enabling a threaded version of flint can be done by setting the environment variable
NEMO_THREADED=1. To set the actual number of threads, use