Introduction — FLINT

FLINT (Fast Library for Number Theory) is a C library for computations in number theory and computer algebra. It provides arithmetic in basic rings and fields, among them integers, rationals, integers modulo n, finite fields, p-adic numbers and number fields, together with univariate and multivariate polynomials, power series and matrices, with factorisation, linear algebra and number theoretic functions. It also provides arbitrary precision real and complex arithmetic with rigorous error bounds (ball arithmetic), special functions, exact real and complex numbers, and a generic interface to all of these rings. FLINT is highly optimised, and it is portable: it is written in C11, with assembly and SIMD code for x86-64 and ARM64.

FLINT is also a research project in the algorithms of computer algebra. The algorithm pages of this documentation describe the methods that are implemented, with references to the literature.

Contents of the library#

The library is divided into modules, one per header file. The home page lists them by topic. In outline:

Integers and rationals: fmpz (integers of any size, stored in one word when small), fmpq, and integer factorisation and primality proving.
Word-size arithmetic: ulong_extras (number theory on words) and nmod (arithmetic modulo a word-size n).
Polynomials and matrices over \mathbb{Z}, \mathbb{Q}, \mathbb{Z}/n\mathbb{Z}, finite fields, p-adic numbers and number fields: for example fmpz_poly, nmod_poly, fq_poly, fmpz_mat, nmod_mat and the multivariate fmpz_mpoly and nmod_mpoly.
Real and complex numbers: the ball arithmetic of arb and acb with polynomials, matrices and special functions (the former Arb library), floating-point types (arf, nfloat, mp_real), and exact real and complex numbers in ca and qqbar (the former Calcium library).
Generic rings: the gr interface, through which the same code works over any of the rings above, with generic polynomials, matrices and series in gr_poly, gr_mat and gr_series; the guide Getting started with generic rings introduces it.

Many operations automatically use several threads when the user allows it; see Threads.

Organisation#

A module foo consists of the header src/foo.h and the directory src/foo, which holds one source file per function or small group of functions, a test directory with one test program t-*.c per function or group, and in some modules profile and tune directories with benchmark and tuning programs. For example, fmpz_poly_add is in src/fmpz_poly/add.c and is tested by src/fmpz_poly/test/t-add.c. Programs using FLINT include the headers as <flint/foo.h> and link against the single library libflint.

This documentation has four parts: the User guide (this part), the reference documentation of each module (every public function with its prototype; the module pages also contain guides to using the module), the algorithm pages, and the examples and tutorials. Conventions shared by all modules (types, naming, memory management, aliasing) are described in Types, naming and aliasing and the following pages.

Requirements#

FLINT requires

GMP, version 6.2.1 or later (https://gmplib.org/);
MPFR, version 4.1.0 or later (https://mpfr.org/);
a C compiler supporting C11, such as GCC or Clang, and on Windows also MSVC;
for building, GNU Make (with the Autotools when building from a git checkout), or CMake on Windows.

MPIR, a fork of GMP that FLINT supported before version 3.0, is no longer supported. Optional dependencies are a BLAS library, the Boehm–Demers–Weiser garbage collector, and NTL (only for testing the conversion functions of the C++ header NTL-interface.h). FLINT is optimised in particular for x86-64 and ARM64 processors, and also runs on other 64-bit and 32-bit platforms. See Building, testing and installing.

History and authors#

FLINT was started by William Hart and David Harvey in 2007, and William Hart led the project until 2022. FLINT 2, released in 2011, was a complete rewrite by William Hart, Sebastian Pancratz and Fredrik Johansson; the 2.x series added, among much else, finite fields (Mike Hansen) and multivariate polynomials (Daniel Schultz). FLINT 3, released in 2023, merged the formerly separate libraries Arb (ball arithmetic) and Calcium (exact real and complex numbers), started by Fredrik Johansson in 2012 and 2020, Antic (algebraic number theory), started by William Hart in 2013, and Generic-Rings. FLINT is currently maintained by Fredrik Johansson (project leader) and Albin Ahlbäck. Many other people have contributed; see Contributors and the contributors listed for each release in History.

License#

FLINT is free software, distributed under the GNU Lesser General Public License (LGPL), version 3 or later. A copy of the license is included in the source distribution (files COPYING and COPYING.LESSER). FLINT versions 2.6 to 3.0 were distributed under the LGPL version 2.1 or later, and earlier versions under the GPL version 2 or later.

Citing FLINT#

If FLINT contributes to published research, please cite it as "The FLINT team. FLINT: Fast Library for Number Theory, Version 3.x.y, https://flintlib.org", with the version that was used. Specific algorithms often have their own papers, given in the references of the corresponding pages.