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 fredrik.johansson@gmail.com

Installation

Currently python-flint supports CPython versions 3.10-3.13. For Windows (x86-64) or OSX (x86-64 or arm64) or Linux (x86-64 manylinux_2_17) there are CPython binary wheels for python-flint on PyPI. 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

It is also possible to use python-flint with some PyPy versions. Binary wheels are not provided for this on PyPI but can be installed with conda.

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/setup.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: finite fields, p-adic numbers, rational functions
• 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	Not yet	3.10-3.13	3.0-3.2?	3.0-3.1?
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 free-threaded builds are tested in CI but wheels are not provided on PyPI. There are no known issues related to using python-flint in a PEP 703 free-threaded build but it is likely that mutating objects from multiple threads is not safe.

Binary wheels are not yet provided for Linux aarch64 (gh-105) or for Windows on ARM but may be added when CI runners for Linux/Windows are available.

CHANGELOG

Next release (0.7.0)...

Contributors (0.7.0):

• Jake Moss (JM)
• Giacomo Pope (GP)
• Joris Roos (JR)
• Edgar Costa (EC)
• Oscar Benjamin (OB)

Highlights (0.7.0):

• gh-97, gh-182: Add fq_default and fq_default_poly for finite fields and univariate polynomials over finite fields. (GP)
• gh-132, gh-164, gh-190, gh-192: 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-129 Use meson/meson-python instead of setuptools as the build system for parallel builds and better detection of build and dependency requirements. (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-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.
• 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-149 Bump Flint version to 3.1.3-p1 (Flint 3.0.0 - 3.1.3-p1 is supported but the wheels are built with 3.1.3-p1). (OB)
• 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+.