Introduction

What is 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.

Source Code and Contributing

The project is open source and is hosted on GitHub here. Contributions are welcome and should be made via GitHub.

Stability

The API is subject to large breaking changes at this time. I hope to stabilize things more in the not too distant future.

Crates

Algebraeon is published to crates.io under an umbrella crate algebraeon and re-exports the sub-crates:

• algebraeon-sets
• algebraeon-nzq
• algebraeon-groups
• algebraeon-rings
• algebraeon-geometry

Algorithms

To give a flavour of what Algebraeon can do some of the algorithms it implements are listed:

• Euclids algorithm for GCD and the extended version for obtaining Bezout coefficients.
• Polynomial GCD computations using subresultant pseudo-remainder sequences.
• AKS algorithm for natural number primality testing.
• Matrix algorithms including:
  • Putting a matrix into Hermite normal form. In particular putting it into echelon form.
  • Putting a matrix into Smith normal form.
  • Gram–Schmidt algorithm for orthogonalization and orthonormalization.
  • Putting a matrix into Jordan normal.
  • Finding the general solution to a linear or affine system of equations.
• Polynomial factoring algorithms including:
  • Kronecker's method for factoring polynomials over the integers (slow).
  • Berlekamp-Zassenhaus algorithm for factoring polynomials over the integers.
  • Berlekamp's algorithm for factoring polynomials over finite fields.
  • Cantor–Zassenhaus algorithm for factoring polynomials over finite fields.
  • Trager's algorithm for factoring polynomials over algebraic number fields.
• Expressing symmetric polynomials in terms of elementary symmetric polynomials.
• Computations with algebraic numbers:
  • Real root isolation and arithmetic.
  • Complex root isolation and arithmetic.
• Computations with multiplication tables for small finite groups.
• Todd-Coxeter algorithm for the enumeration of finite index cosets of a finitely generated groups.