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.

GiNaC is a C++ library for symbolic computation. Contrary to other CAS it does not try to provide extensive algebraic capabilities and a simple programming language but instead accepts a given language (C++) and extends it by a set of algebraic capabilities.

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.

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.

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).

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.

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.

IML is a C library implementing algorithms for computing exact solutions to dense systems of linear equations over the integers. Currently, IML provides the following functionality:

• Nonsingular rational system solving: compute the unique rational solution X to the system AX=B, where A and B are integer matrices, A nonsingular.

• Compute the right nullspace or kernel of an integer matrix.

• Certified linear system solving: compute a minimal denominator solution x to a system Ax=b, where b is an integer vector and A is an integer matrix with arbitrary shape and rank profile.

In addition, IML provides some low level routines for a variety of mod p matrix operations: computing the row-echelon form, determinant, rank profile, and inverse of a mod p matrix. These mod p routines are not general purpose; they require that p satisfy some preconditions based on the dimension of the input matrix (usually p should be prime and should be no more than about 20 bits long).

This package provides a simple Python interface to the Singular computer algebra system.

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

Sollya is a computer program whose purpose is to provide an environment for safe floating-point code development. It is particularly targeted to the automated implementation of mathematical floating-point libraries (libm). Amongst other features, it offers a certified infinity norm, an automatic polynomial implementer, and a fast Remez algorithm.

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.

NTL is a C++ library providing data structures and algorithms for manipulating signed, arbitrary length integers, and for vectors, matrices, and polynomials over the integers and over finite fields.

JLargeArrays is a Java library of one-dimensional arrays that can store up to 263 elements.

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.

bc is an arbitrary precision numeric processing language. It includes an interactive environment for evaluating mathematical statements. Its syntax is similar to that of C, so basic usage is familiar. It also includes "dc", a reverse-polish calculator.

Spectra stands for Sparse Eigenvalue Computation Toolkit as a Redesigned ARPACK. It is a C++ library for large scale eigenvalue problems, built on top of Eigen. It is implemented as a header-only C++ library and can be easily embedded in C++ projects that require calculating eigenvalues of large matrices.

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

FFTW is a C subroutine library for computing the discrete Fourier transform (DFT) in one or more dimensions, of arbitrary input size, and of both real and complex data (as well as of even/odd data---i.e. the discrete cosine/ sine transforms or DCT/DST). Single-precision version.

PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorisations, algebraic number theory, elliptic curves...), but it also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers, etc., and a lot of transcendental functions. PARI is also available as a C library to allow for faster computations.

GP2C, the GP to C compiler, translates GP scripts to PARI programs.

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.

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.

PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorisations, algebraic number theory, elliptic curves...), but it also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers, etc., and a lot of transcendental functions. PARI is also available as a C library to allow for faster computations.

LinBox is a C++ template library for exact linear algebra computation with dense, sparse, and structured matrices over the integers and over finite fields.