Python-FLINT

Python extension module wrapping FLINT (Fast Library for Number Theory) and Arb (arbitrary-precision ball arithmetic). Features:

• Integers, rationals, integers mod n
• Real and complex numbers with rigorous error tracking
• Polynomials, power series and matrices over all the above types
• Lots of mathematical functions

Documentation: https://python-flint.readthedocs.io/en/latest/

Repository: https://github.com/flintlib/python-flint/

Author: Fredrik Johansson [email protected]

Installation

Currently python-flint supports CPython versions 3.11-3.13 and 3.13t (free-threaded) and provides binaries on PyPI for the following platforms:

• Windows (x86-64)
• MacOS (x86-64, arm64)
• Linux (manylinux: x86-64, aarch64)

For these platforms python-flint can be installed simply with pip

pip install python-flint

Alternatively python-flint can be installed using conda

conda install -c conda-forge python-flint

Build from source

For other platforms or architectures installation needs to build from source. First install FLINT 3. Starting with python-flint 0.5.0 older versions of Flint such as 2.9 are not supported any more. Note that as of Flint 3 Arb no longer needs to be built separately as it is now merged into Flint.

As of e.g. Ubuntu 24.04 a new enough version of FLINT (at least version 3) can be installed from the Ubuntu repos like

sudo apt-get install libflint-dev

For older distros the version in the repos is too old and a newer version of FLINT needs to be built. See here for instructions on building FLINT:

• http://flintlib.org/

A script that builds and installs FLINT on Ubuntu can be found here:

• https://github.com/flintlib/python-flint/blob/master/bin/install_flint_ubuntu.sh

The latest release of Python-FLINT can then be built from source and installed using:

pip install --no-binary python-flint python-flint

Python-FLINT can also be installed from a git checkout or a source archive as follows:

pip install .

See the documentation for further notes on building and installing python-flint:

• https://python-flint.readthedocs.io/en/latest/build.html
• https://python-flint.readthedocs.io/en/latest/install.html

Examples

Import Python-FLINT:

>>> from flint import *

Number-theoretic functions:

>>> fmpz(1000).partitions_p()
24061467864032622473692149727991
>>> fmpq.bernoulli(64)
-106783830147866529886385444979142647942017/510

Polynomial arithmetic:

>>> a = fmpz_poly([1,2,3]); b = fmpz_poly([2,3,4]); a.gcd(a * b)
3*x^2 + 2*x + 1
>>> a = fmpz_poly(list(range(10001))); b = fmpz_poly(list(range(10000))); a.gcd(a * b).degree()
10000
>>> x = fmpz_poly([0,1]); ((1-x**2)*(1+x**3)**3*(1+x+2*x)).factor()
(-1, [(3*x + 1, 1), (x + (-1), 1), (x^2 + (-1)*x + 1, 3), (x + 1, 4)])

Matrix arithmetic:

>>> fmpz_mat([[1,1],[1,0]]) ** 10
[89, 55]
[55, 34]
>>> fmpq_mat.hilbert(10,10).det()
1/46206893947914691316295628839036278726983680000000000

Numerical evaluation:

>>> showgood(lambda: (arb.pi() * arb(163).sqrt()).exp() - 640320**3 - 744, dps=25)
-7.499274028018143111206461e-13
>>> showgood(lambda: (arb.pi() * 10**100 + arb(1)/1000).sin(), dps=25)
0.0009999998333333416666664683

Numerical integration:

>>> ctx.dps = 30
>>> acb.integral(lambda x, _: (-x**2).exp(), -100, 100) ** 2
[3.141592653589793238462643383 +/- 3.11e-28]

To do

• Write more tests and add missing docstrings
• Wrap missing flint types: matrices over finite fields, p-adic numbers, rational functions
• Build on the preliminary interface to FLINT's generic (gr) types.
• Make a nicer interface like ZZ(1) etc rather than fmpz_poly([1, 2]).
• Vector or array types (maybe)
• Many convenience methods
• Write generic implementations of functions missing for specific FLINT types
• Proper handling of special values in various places (throwing Python exceptions instead of aborting, etc.)
• Various automatic conversions
• Conversions to and from external types (numpy, sage, sympy, mpmath, gmpy)
• Improved printing and string input/output
• IPython hooks (TeX pretty-printing etc.)

Compatibility table

Generally each release of python-flint will be compatible with a range of Python versions as described in SPEC 0. Since python-flint 0.5.0 the minimum supported Flint version is 3.0 and each release of python-flint supports all versions of Flint >=3.0 available at the time of release.

Compatible versions (note that 0.7.0 is not yet released):

python-flint	Release date	CPython	FLINT	Cython
0.7.0	16th Mar 2025	3.11-3.13	3.0-3.2	3.0.11-3.1.0a1
0.6.0	1st Feb 2024	3.9-3.12	3.0 only	3.0 only

As of python-flint 0.7.0, CPython 3.13 PEP 703 free-threaded (no-GIL) builds of python-flint are provided. In the the free-threaded build, mutating matrices or polynomials from multiple threads can lead to memory corruption. Provided matrices or polynomials are not mutated when shared across threads there are no known issues with the free-threaded build but these should be considered experimental.

CHANGELOG

Next release (0.8.0)...

0.7.0

Contributors (0.7.0):

• Jake Moss (JM)
• Giacomo Pope (GP)
• Joris Roos (JR)
• Edgar Costa (EC)
• Frédéric Chapoton (FC)
• Oscar Benjamin (OB)
• Tom Hubrecht (TH)

Highlights (0.7.0):

• gh-270, PyPI packages are now built with FLINT 3.2.0 (previously 3.0.1 was used). All versions from FLINT 3.0.0 to FLINT 3.2.0 are compatible with python-flint but some features require newer FLINT versions and the PyPI packages now use FLINT 3.2.0.
• gh-97, gh-182: Add fq_default and fq_default_poly for finite fields and univariate polynomials over finite fields. This exposes all of the different implementations of finite fields (fq_zech, fq_nmod etc) via the fq_default interface. (GP)
• gh-132, gh-164, gh-190, gh-191: gh-192: gh-216: gh-225: gh-228: Add fmpz_mpoly, fmpq_mpoly, nmod_poly and fmpz_mod_poly types for multivariate polynomials with integer, rational or integers mod n coefficients. (JM)
• gh-142 Add acb_theta module for the numerical evaluation of theta functions (only available for Flint >= 3.1). (EC)
• gh-218 gh-254 gh-255 An experimental interface for FLINT's generic rings has been added. This provides access to many of FLINT's types that are not yet wrapped by python-flint such as Gaussian integer, number fields, qqbar, calcium, as well as both univariate and multivariate polynomials and series over these rings (no matrices yet though). (OB and TH)
• gh-129 gh-208 Use meson/meson-python instead of setuptools as the build system for parallel builds and better detection of build and dependency requirements. (OB)
• gh-201 gh-202 The documentation has been updated and is now at readthedocs. (OB) gh-235 Nightly wheels for python-flint can now be installed from the [Anaconda Scientific Python Nightly Wheels index] (https://anaconda.org/scientific-python-nightly-wheels/python-flint). gh-259 Add PyPI wheels for Linux aarch64 (Linux on ARM CPU). (OB)

Compatibility break (0.7.0):

• gh-189 As of python-flint 0.7.0 fmpq_poly.factor() now returns primitive rather than monic factors i.e. 2*x + 1 rather than x + 1/2. This ensures consistency between all poly types including between fmpq_poly and fmpq_mpoly. (OB)

Other changes (0.7.0):

• gh-269 All univariate and multivariate polynomial types have is_zero, is_one and is_constant methods. All polynomial types now consistently handle negative powers where possible.
• gh-261 Add fmpz_mat.fflu for fraction-free LU decomposition of an integer matrix.
• gh-251 Add mpmath-style precision context managers for arb extraprec, extradps, workprec and workdps. (TH)
• gh-250 Add fmpq.gcd() method.
• gh-215 gh-219 The FLINT binding declarations are now fully generated automatically from the FLINT docs. (OB)
• gh-203 gh-204 gh-205 gh-206 gh-207 gh-211 gh-212 gh-271 Various linting fixes and codebase improvements (FC and GP).
• gh-189 All scalar and poly types now have sqrt. All poly types now have factor_squarefree and leading_coefficient methods. Exception types raised in a number of places were changed to DomainError for better consistency. (OB)
• gh-196 Supported Python versions are 3.10-3.13 (3.9 dropped). CI Testing added for 3.13 free-threaded CPython.
• gh-194 Add version checking for build requirements. (OB)
• gh-180 Add equal_trunc, add_trunc, sub_trunc, mul_low, mul_mod and pow_trunc methods to fmpz_mod_poly. (GP)
• gh-177 Remove old Py2 code for compatibility with Cython 3.1. (OB)
• gh-176 Fix the error messages from fmpq constructor. (OB)
• gh-174 Add pow_mod and compose_mod methods to nmod_poly and fmpz_mod_poly. Also add some missing methods to nmod_poly that other poly types already have. (GP)
• gh-172 Add fmpz_is_square. (JR)
• gh-168 Make comparisons consistent between different types. Add is_one and is_zero for all poly types. (OB)
• gh-161 Add acb.lerch_phi to compute the Lerch transcendent. (OB)
• gh-160 Add bits to arb and acb, add log_base to arb. (JR)
• gh-148 Remove debug symbols to make smaller Linux binaries. (OB)
• gh-144 Add rel_one_accuracy_bits to arb and acb. (EC)
• gh-137 Add erfinv and erfcinv for arb. (JR)
• gh-119 Add compatibility with Flint 3.1. (OB)

0.6.0

• gh-112, gh-111, gh-110, gh-108: Add pyproject.toml and build dependencies. This means that python-flint can be built from source without --no-build-isolation.
• gh-109: Use exact division for non-field domains. Now fmpz(6)/fmpz(3) returns an exact result fmpz(2) or raises an error if an exact result is not possible. Similar changes for fmpz_poly/fmpz, fmpz_mat/fmpz, and for polynomial division with fmpz_poly, fmpq_poly, nmod_poly and fmpz_mod_poly.
• gh-106: Add fmpz_mod_mat for matrices of integers mod n where n is larger than word sized.
• gh-104: Bump Flint from 3.0.0 to 3.0.1

0.5.0

Important compatibility changes:

• gh-80, gh-94, gh-98: Switch from Flint 2.9 to Flint 3.
• gh-100: Supports Python 3.12 by using setuptools instead of numpy.distutils.

New features:

• gh-87: Adds fmpz_mod_poly type for polynomials over fmpz_mod.
• gh-85: Adds discrete logarithms to fmpz_mod.
• gh-83: Introduces the fmpz_mod type for multi-precision integer mods.

Bug fixes:

• gh-93: Fixes a bug with pow(int, int, fmpz) which previously gave incorrect results.
• gh-78, gh-79: minor fixes for the nmod type.

0.4.4

• gh-75, gh-77: finish bulk of the work in refactoring python-flint into submodules
• gh-72: The roots method of arb_poly is not supported. Use either the complex_roots method or acb_roots(p).roots() to get the old behaviour of returning the complex roots. The roots method on fmpz_poly and fmpq_poly now return integer and rational roots respectively. To access complex roots on these types, use the complex_roots method. For acb_poly, both roots and complex_roots behave the same
• gh-71: Include files in sdist and fix issue gh-70
• gh-67: Continue refactoring job to introduce submodules into python-flint

0.4.3

• gh-63: The roots method of arb_poly, and nmod_poly is no longer supported. Use acb_roots(p).roots() to get the old behaviour of returning the roots as acb. Note that the roots method of fmpz_poly and fmpq_poly currently returns the complex roots of the polynomial.
• gh-61: Start refactoring job to introduce submodules into python-flint

0.4.2

• gh-57: Adds manylinux wheels

0.4.1

• gh-47: Removes Linux wheels, updates instructions for building from source.

0.4.0

• gh-45: Adds wheels for Windows, OSX and manylinux but the Linux wheels are broken.

License

Python-FLINT is licensed MIT. FLINT and Arb are LGPL v2.1+.