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 functions for 1D and 2D Discrete Cosine Transform (DCT), Discrete Sine Transform (DST) and Discrete Hartley Transform (DHT).

Singular is a computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry and singularity theory.

The CMH software computes Igusa (genus 2) class polynomials, which parameterize the CM points in the moduli space of 2-dimensional abelian varieties, i.e. Jacobians of hyperelliptic curves. It can also be used to compute theta constants at arbitrary precision.

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.

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).

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 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.

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.

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

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.

Symmetrica is a library for combinatorics. It has support for the representation theory of the symmetric group and related groups, combinatorics of tableaux, symmetric functions and polynomials, Schubert polynomials, and the representation theory of Hecke algebras of type A_n.

Givaro is a C++ library implementing the basic arithmetic of various algebraic objects: prime fields, extension fields, finite fields, finite rings, polynomials, algebraic numbers, arbitrary precision integers and rationals (C++ wrappers over gmp), fixed precision integers. It also provides data-structures and templated classes for the manipulation of compound objects, such as vectors, matrices and univariate polynomials.

The gf2x library provides arithmetic of polynomials over finite fields of characteristic 2. It implements the multiplication, squaring and greatest common divisor operations.

Clac is a command line, stack-based calculator with postfix notation that displays the stack contents at all times. As you type, the stack changes are reflected immediately.

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.

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.

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.

fplll contains implementations of several lattice algorithms. The implementation relies on floating-point orthogonalization, and LLL is central to the code, hence the name.

It includes implementations of floating-point LLL reduction algorithms, offering different speed/guarantees ratios. It contains a wrapper choosing the estimated best sequence of variants in order to provide a guaranteed output as fast as possible. In the case of the wrapper, the succession of variants is oblivious to the user.

It includes an implementation of the BKZ reduction algorithm, including the BKZ-2.0 improvements (extreme enumeration pruning, pre-processing of blocks, early termination). Additionally, Slide reduction and self dual BKZ are supported.

It also includes a floating-point implementation of the Kannan-Fincke-Pohst algorithm that finds a shortest non-zero lattice vector. For the same task, the GaussSieve algorithm is also available in fplll. Finally, it contains a variant of the enumeration algorithm that computes a lattice vector closest to a given vector belonging to the real span of the lattice.

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.

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.

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.

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.