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.

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.

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.

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.

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.

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.

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.

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.

The la4j library is a Java library that provides Linear Algebra primitives (matrices and vectors) and algorithms. The key features of the la4j library are:

• No dependencies and tiny size

• Fluent object-oriented/functional API

• Sparse (CRS, CCS) and dense (1D/2D arrays) matrices

• Linear systems solving (Gaussian, Jacobi, Zeidel, Square Root, Sweep and other)

• Matrices decomposition (Eigenvalues/Eigenvectors, SVD, QR, LU, Cholesky and other)

• MatrixMarket/CSV IO formats support for matrices and vectors

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.

The CM software implements the construction of ring class fields of imaginary quadratic number fields and of elliptic curves with complex multiplication via floating point approximations, and the elliptic curve primality proving algorithm (ECPP). It consists of libraries that can be called from within a C program and of executable command line applications.

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.

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).

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.

PyNormaliz provides an interface to Normaliz via libNormaliz. It offers the complete functionality of Normaliz, and can be used interactively from Python.

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.

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.

Gappa is a tool intended to help verifying and formally proving properties on numerical programs dealing with floating-point or fixed-point arithmetic. It has been used to write robust floating-point filters for CGAL and it is used to certify elementary functions in CRlibm. While Gappa is intended to be used directly, it can also act as a backend prover for the Why3 software verification platform or as an automatic tactic for the Coq proof assistant.

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). This CMake build offers the file FFTW3LibraryDepends.cmake required by some dependent packages, absent in the gnu build version.

GMP-ECM factors integers using the elliptic curve method (ECM) as well as the P-1 and P+1 algorithms. It provides a library and a stand-alone binary.

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

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.

fpylll is a Python wrapper for fplll.