Python-flint is a Python extension module wrapping FLINT (Fast Library for Number Theory) and Arb (arbitrary-precision ball arithmetic). It supports integers, rationals, modular integers, real and complex ball arithmetic, polynomials and matrices over all these types and other mathematical functions.

JTransforms is a multithreaded FFT library written in pure Java. Currently, four types of transforms are available: Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), Discrete Sine Transform (DST) and Discrete Hartley Transform (DHT).

Giac/Xcas is a computer algebra system. It has a compatibility mode for maple, mupad and the TI89. It is available as a standalone program (graphic or text interfaces) or as a C++ library.

PariTwine is a glue library between the system for computer algebra and number theory PARI/GP and a number of other mathematics libraries, currently GMP, GNU MPFR, GNU MPC, FLINT and CMH.

libsemigroups is a C++14 library containing implementations of several algorithms for computing finite, and finitely presented, semigroups and monoids.

Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms. It provides an elegant API based on "expression templates". It is versatile: it supports all matrix sizes, all standard numeric types, various matrix decompositions and geometry features, and more.

This library provides a Python wrapper to SymEngine, a fast C++ symbolic manipulation library.

xtensor is a C++ library meant for numerical analysis with multi-dimensional array expressions.

xtensor provides:

• an extensible expression system enabling lazy broadcasting.

• an API following the idioms of the C++ standard library.

• tools to manipulate array expressions and build upon xtensor.

Normaliz is a tool for computations in affine monoids, vector configurations, rational polyhedra and rational cones. Normaliz now computes rational and algebraic polyhedra, i.e., polyhedra defined over real algebraic extensions of QQ.

Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms. It provides an elegant API based on "expression templates". It is versatile: it supports all matrix sizes, all standard numeric types, various matrix decompositions and geometry features, and more.

The Littlewood-Richardson Calculator (lrcalc) is a program designed to compute Littlewood-Richardson coefficients. It computes single Littlewood-Richardson coefficients, products of Schur functions, or skew Schur functions. In addition it computes products in the small quantum cohomology ring of a Grassmann variety. The software package also includes a program that performs fast computation of the more general multiplicative structure constants of Schubert polynomials.

This package provides Python bindings for the Littlewood-Richardson Calculator.

FORM is a symbolic manipulation system. It reads symbolic expressions from files and executes symbolic/algebraic transformations upon them. The answers are returned in a textual mathematical representation. The size of the considered expressions in FORM is only limited by the available disk space and not by the available RAM. This package also includes parform, a version of FORM parallelized using OpenMPI.

xtensor is a C++ library meant for numerical analysis with multi-dimensional array expressions.

xtensor provides:

• an extensible expression system enabling lazy broadcasting.

• an API following the idioms of the C++ standard library.

• tools to manipulate array expressions and build upon xtensor.

M4RI is a library for fast arithmetic with dense matrices over F2. The name M4RI comes from the first implemented algorithm: The Method of the Four Russians inversion algorithm published by Gregory Bard. This algorithm in turn is named after the Method of the Four Russians multiplication algorithm.

FORM is a symbolic manipulation system. It reads symbolic expressions from files and executes symbolic/algebraic transformations upon them. The answers are returned in a textual mathematical representation. The size of the considered expressions in FORM is only limited by the available disk space and not by the available RAM.

lmfit is a C library for Levenberg-Marquardt least-squares minimization and curve fitting. It is mature code, based on decades-old algorithms from the FORTRAN library MINPACK.

SymEngine is a standalone fast C++ symbolic manipulation library. Optional thin wrappers allow usage of the library from other languages.

FFLAS-FFPACK is a C++ template library for basic linear algebra operations over a finite field. FFLAS (Finite Field Linear Algebra Subprograms) provides the implementation of a subset of routines of the numerical BLAS; it also supports sparse matrix-vector products. FFPACK (Finite Field Linear Algebra Package) is inspired by the LAPACK library to provide functionalities of higher level, using the kernel of a BLAS. Additionally, it provides routines specific to exact linear algebra, such as the row echelon form.

Kiss FFT is a mixed-radix Fast Fourier Transform based up on the principle, 'Keep It Simple, Stupid.'

FLINT is a C library for number theory. It supports arithmetic with numbers, polynomials, power series and matrices over many base rings, including multiprecision integers and rationals, integers modulo n, p-adic numbers, finite fields (prime and non-prime order) and real and complex numbers (via the Arb extension library).

Operations that can be performed include conversions, arithmetic, GCDs, factoring, solving linear systems, and evaluating special functions. In addition, FLINT provides various low-level routines for fast arithmetic.

REDUCE is a portable general-purpose computer algebra system. It is a system for doing scalar, vector and matrix algebra by computer, which also supports arbitrary precision numerical approximation and interfaces to gnuplot to provide graphics. It can be used interactively for simple calculations but also provides a full programming language, with a syntax similar to other modern programming languages. REDUCE supports alternative user interfaces including Run-REDUCE, TeXmacs and GNU Emacs. This package provides the Codemist Standard Lisp (CSL) version of REDUCE. It uses the gnuplot program, if installed, to draw figures.

GAP is a system for computational discrete algebra, with particular emphasis on computational group theory. It provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects.

msolve is a C library implementing computer algebra algorithms for solving polynomial systems (with rational coefficients or coefficients in a prime field).

Currently, with msolve, you can basically solve multivariate polynomial systems. This encompasses:

• the computation of Groebner bases

• real root isolation of the solutions to polynomial systems

• the computation of the dimension and the degree of the solution set.