FLINT: Fast Library for Number Theory

Welcome to FLINT’s documentation! FLINT is a C library for doing number theory.

• Website: https://flintlib.org

• Source code on GitHub: https://github.com/flintlib/flint

• Issue tracker: https://github.com/flintlib/flint/issues

• Mailing list: https://groups.google.com/group/flint-devel

FLINT is free software distributed under the GNU Lesser General Public License (LGPL), version 3 or later.

Introduction

• Introduction
• Building, testing and installing
• Bug reporting
• Contributing to FLINT
• Contributors
• Examples
• Memory management
• Portability
• Threading

General utilities

• flint.h – global definitions
• profiler.h – performance profiling
• thread_pool.h – thread pool
• mpoly.h – support functions for multivariate polynomials
• machine_vectors.h – SIMD-accelerated operations on fixed-length vectors

Generic rings

• gr.h – generic structures and their elements
• gr.h (continued) – implementing rings
• gr.h (continued) – builtin domains and types
• gr_generic.h – basic algorithms and fallback implementations for generic elements
• gr_special.h – special arithmetic and transcendental functions
• gr_vec.h – vectors over generic rings
• gr_mat.h – dense matrices over generic rings
• gr_poly.h – dense univariate polynomials over generic rings
• gr_mpoly.h – sparse multivariate polynomials over generic rings

• Generic rings : detailed table of contents

Integers

• ulong_extras.h – arithmetic and number-theoretic functions for single-word integers
• fmpz.h – integers
• fmpz_vec.h – vectors of integers
• fmpz_factor.h – integer factorisation
• fmpz_mat.h – matrices over the integers
• fmpz_lll.h – LLL reduction
• fmpz_poly.h – univariate polynomials over the integers
• fmpz_poly_mat.h – matrices of polynomials over the integers
• fmpz_poly_factor.h – factorisation of polynomials over the integers
• fmpz_mpoly.h – multivariate polynomials over the integers
• fmpz_mpoly_factor.h – factorisation of multivariate polynomials over the integers
• long_extras.h – support functions for signed word arithmetic
• longlong.h – support functions for multi-word arithmetic
• mpn_extras.h – support functions for limb arrays
• aprcl.h – APRCL primality testing
• arith.h – arithmetic and special functions
• fft.h – Schoenhage-Strassen FFT
• fft_small.h – FFT modulo word-size primes
• qsieve.h – Quadratic sieve

• Integers : detailed table of contents

Rational numbers

• fmpq.h – rational numbers
• fmpq_vec.h – vectors over rational numbers
• fmpq_mat.h – matrices over the rational numbers
• fmpq_poly.h – univariate polynomials over the rational numbers
• fmpq_mpoly_factor.h – factorisation of multivariate polynomials over the rational numbers
• fmpq_mpoly.h – multivariate polynomials over the rational numbers
• fmpz_poly_q.h – rational functions over the rational numbers
• fmpz_mpoly_q.h – multivariate rational functions over Q

• Rational numbers : detailed table of contents

Integers mod n

• nmod.h – integers mod n (word-size n)
• nmod_vec.h – vectors over integers mod n (word-size n)
• nmod_mat.h – matrices over integers mod n (word-size n)
• nmod_poly.h – univariate polynomials over integers mod n (word-size n)
• nmod_poly_mat.h – matrices of univariate polynomials over integers mod n (word-size n)
• nmod_poly_factor.h – factorisation of univariate polynomials over integers mod n (word-size n)
• nmod_mpoly.h – multivariate polynomials over integers mod n (word-size n)
• nmod_mpoly_factor.h – factorisation of multivariate polynomials over integers mod n (word-size n)
• mpn_mod.h – integers mod n (packed multi-word n)
• fmpz_mod.h – arithmetic modulo integers
• fmpz_mod_vec.h – vectors over integers mod n
• fmpz_mod_mat.h – matrices over integers mod n
• fmpz_mod_poly.h – polynomials over integers mod n
• fmpz_mod_poly_factor.h – factorisation of polynomials over integers mod n
• fmpz_mod_mpoly.h – polynomials over the integers mod n
• fmpz_mod_mpoly_factor.h – factorisation of multivariate polynomials over the integers mod n

• Integers mod n : detailed table of contents

Groups and other structures

• perm.h – permutations
• qfb.h – binary quadratic forms
• dirichlet.h – Dirichlet characters
• dlog.h – discrete logarithms mod ulong primes
• bool_mat.h – matrices over booleans

Number fields and algebraic numbers

• nf.h – number fields
• nf_elem.h – number field elements
• fmpzi.h – Gaussian integers
• qqbar.h – algebraic numbers represented by minimal polynomials

Real and complex numbers

• Feature overview
• Using ball arithmetic
• Technical conventions and potential issues
• Arb example programs
• mag.h – fixed-precision unsigned floating-point numbers for bounds
• nfloat.h – packed floating-point numbers with n-word precision
• arf.h – arbitrary-precision floating-point numbers
• acf.h – complex floating-point numbers
• arb.h – real numbers
• acb.h – complex numbers
• arb_poly.h – polynomials over the real numbers
• acb_poly.h – polynomials over the complex numbers
• arb_fmpz_poly.h – extra methods for integer polynomials
• acb_dft.h – Discrete Fourier transform
• arb_mat.h – matrices over the real numbers
• acb_mat.h – matrices over the complex numbers
• acb_hypgeom.h – hypergeometric functions of complex variables
• arb_hypgeom.h – hypergeometric functions of real variables
• acb_elliptic.h – elliptic integrals and functions of complex variables
• acb_modular.h – modular forms of complex variables
• acb_theta.h – Riemann theta functions
• acb_dirichlet.h – Dirichlet L-functions, Riemann zeta and related functions
• bernoulli.h – support for Bernoulli numbers
• hypgeom.h – support for hypergeometric series
• partitions.h – computation of the partition function
• arb_calc.h – calculus with real-valued functions
• acb_calc.h – calculus with complex-valued functions
• arb_fpwrap.h – floating-point wrappers of Arb mathematical functions
• fmpz_extras.h – extra methods for FLINT integers
• General formulas and bounds
• Algorithms for mathematical constants
• Algorithms for the gamma function
• Algorithms for the Hurwitz zeta function
• Algorithms for polylogarithms
• Algorithms for hypergeometric functions
• Algorithms for the arithmetic-geometric mean

• Real and complex numbers (Arb) : detailed table of contents

Exact real and complex numbers

• Introduction
• Calcium example programs
• calcium.h – global definitions
• ca.h – exact real and complex numbers
• ca_vec.h – vectors of real and complex numbers
• ca_poly.h – dense univariate polynomials over the real and complex numbers
• ca_mat.h – matrices over the real and complex numbers
• ca_ext.h – real and complex extension numbers
• ca_field.h – extension fields
• fexpr.h – flat-packed symbolic expressions
• fexpr_builtin.h – builtin symbols

Finite fields

• fq.h – finite fields
• fq_default.h – unified finite fields
• fq_vec.h – vectors over finite fields
• fq_mat.h – matrices over finite fields
• fq_default_mat.h – matrices over finite fields
• fq_poly.h – univariate polynomials over finite fields
• fq_default_poly.h – univariate polynomials over finite fields
• fq_poly_factor.h – factorisation of univariate polynomials over finite fields
• fq_default_poly_factor.h – factorisation of univariate polynomials over finite fields
• fq_embed.h – Computing isomorphisms and embeddings of finite fields

• fq_nmod.h – finite fields (word-size characteristic)
• fq_nmod_vec.h – vectors over finite fields (word-size characteristic)
• fq_nmod_mat.h – matrices over finite fields (word-size characteristic)
• fq_nmod_poly.h – univariate polynomials over finite fields (word-size characteristic)
• fq_nmod_poly_factor.h – factorisation of univariate polynomials over finite fields (word-size characteristic)
• fq_nmod_embed.h – Computing isomorphisms and embeddings of finite fields
• fq_nmod_mpoly.h – multivariate polynomials over finite fields of word-sized characteristic
• fq_nmod_mpoly_factor.h – factorisation of multivariate polynomials over finite fields of word-sized characteristic

• fq_zech.h – finite fields (Zech logarithm representation)
• fq_zech_vec.h – vectors over finite fields (Zech logarithm representation)
• fq_zech_mat.h – matrices over finite fields (Zech logarithm representation)
• fq_zech_poly.h – univariate polynomials over finite fields (Zech logarithm representation)
• fq_zech_poly_factor.h – factorisation of univariate polynomials over finite fields (Zech logarithm representation)
• fq_zech_embed.h – Computing isomorphisms and embeddings of finite fields

p-adic numbers

• padic.h – p-adic numbers
• padic_poly.h – polynomials over p-adic numbers
• padic_mat.h – matrices over p-adic numbers
• qadic.h – unramified extensions over p-adic numbers

Floating-point support code

• double_extras.h – support functions for double arithmetic
• double_interval.h – double-precision interval arithmetic and helpers
• d_vec.h – double precision vectors
• d_mat.h – double precision matrices
• mpfr_vec.h – vectors of MPFR floating-point numbers
• mpfr_mat.h – matrices of MPFR floating-point numbers

Interfaces

• flint_ctypes - Python interface

References

• References

Version history

• History and changes