Algebraeon

Algebraeon is a computer algebra system written purely in Rust. It implements algorithms for working with matrices, polynomials, algebraic numbers, factorizations, etc. The focus is on exact algebraic computations over approximate numerical solutions. Algebraeon is in early stages of development and the API subject to change. Algebraeon uses Malachite under the hood for arbitrary sized integer and rational numbers.

See the user guide to get started.

The latest published version of Algebraeon is hosted on crates.io here and the formal documentation is here.

Examples

Factoring Integers

To factor large integers using Algebraeon

use std::str::FromStr;
use algebraeon::{nzq::Natural, rings::number::natural::factorization::factor};

let n = Natural::from_str("706000565581575429997696139445280900").unwrap();
let f = factor(n.clone()).unwrap();
println!("{} = {}", n, f);
/*
Output:
    706000565581575429997696139445280900 = 2^2 × 5^2 × 6988699669998001 × 1010203040506070809
*/

Algebraeon implements Lenstra elliptic-curve factorization for quickly finding prime factors with around 20 digits.

Factoring Polynomials

Factor the polynomials $x^2 - 5x + 6$ and $x^{15} - 1$.

use algebraeon::rings::{polynomial::*, structure::*};
use algebraeon::nzq::Integer;

let x = &Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(2) - 5*x + 6).into_verbose();
println!("f(λ) = {}", f.factor().unwrap());
/*
Output:
    f(λ) = 1 * ((-2)+λ) * ((-3)+λ)
*/

let f = (x.pow(15) - 1).into_verbose();
println!("f(λ) = {}", f.factor().unwrap());
/*
Output:
    f(λ) = 1 * ((-1)+λ) * (1+λ+λ^2) * (1+λ+λ^2+λ^3+λ^4) * (1+(-1)λ+λ^3+(-1)λ^4+λ^5+(-1)λ^7+λ^8)
*/

so

x^2 - 5x + 6 = (x-2)(x-3)

x^{15}-1 = (x-1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^8-x^7+x^5-x^4+x^3-x+1)

Linear Systems of Equations

Find the general solution to the linear system

a \begin{pmatrix}3 \\ 4 \\ 1\end{pmatrix} + b \begin{pmatrix}2 \\ 1 \\ 2\end{pmatrix} + c \begin{pmatrix}1 \\ 3 \\ -1\end{pmatrix} = \begin{pmatrix}5 \\ 5 \\ 3\end{pmatrix}

for integers $a$, $b$ and $c$.

use algebraeon::rings::linear::matrix::Matrix;
use algebraeon::nzq::Integer;
let x = Matrix::<Integer>::from_rows(vec![vec![3, 4, 1], vec![2, 1, 2], vec![1, 3, -1]]);
let y = Matrix::<Integer>::from_rows(vec![vec![5, 5, 3]]);
let s = x.row_solution_lattice(y);
s.pprint();
/*
Output:
    Start Affine Lattice
    Offset
    ( 2    0    -1 )
    Start Linear Lattice
    ( 1    -1    -1 )
    End Linear Lattice
    End Affine Lattice
*/

so the general solution is all $a$, $b$, $c$ such that

\begin{pmatrix}a \\ b \\ c\end{pmatrix} = \begin{pmatrix}2 \\ 0 \\ -1\end{pmatrix} + t\begin{pmatrix}1 \\ -1 \\ -1\end{pmatrix}

for some integer $t$.

Complex Root Isolation

Find all complex roots of the polynomial $$f(x) = x^5 + x^2 - x + 1$$

use algebraeon::rings::{polynomial::*, structure::*};
use algebraeon::nzq::Integer;

let x = &Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(5) + x.pow(2) - x + 1).into_verbose();
// Find the complex roots of f(x)
for root in f.all_complex_roots() {
    println!("root {} of degree {}", root, root.degree());
}
/*
Output:
    root ≈-1.328 of degree 3
    root ≈0.662-0.559i of degree 3
    root ≈0.662+0.559i of degree 3
    root -i of degree 2
    root i of degree 2
*/

Despite the output, the roots found are not numerical approximations. Rather, they are stored internally as exact algebraic numbers by using isolating boxes in the complex plane.

Contributing

Contributions are welcome. There are two primary ways to contribute:

Using the issue tracker

Use the issue tracker if you have questions, feature requests, bugs reports, etc.

Changing the code-base

You should fork this repository, make changes, and submit a pull request. Submitted code should, where applicable, have associated unit tests.

Algebraeon is organized as a cargo workspace. Run cargo test in the root directory to build and run all tests.

A suggested workflow for testing new features:

• Create a new binary in examples/src/bin, for example my_main.rs.
• To run, use cargo run --bin my_main in the root directory.
• Test any changes to the codebase with unit tests and/or using my_main.rs.

CLA

If you wish to contribute code we ask that you sign a CLA. The CLA allows us to relicense future versions of Algebraeon, including your contribution, under any licences the Free Software Foundation classifies as a Free Software Licence and which are approved by the Open Source Initiative as Open Source licences. It does not allow us to relicense of your contribution under any other more restrictive licences. You can sign the CLA when you submit a pull request.