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 3.10-3.14 and 3.14t (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. 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:

The requirement for Cython 3.1 is only for CPython's free-threaded build. Otherwise Cython 3.0 is fine. Cython 3.2 is required for a stable ABI build of python-flint.

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. There are some other known issues with the free-threaded build so it should be considered experimental.

CHANGELOG

0.9.0

Contributors (0.9.0):

• Rémy Oudompheng (RO)
• Agriya Khetarpal (AK)
• Oscar Benjamin (OB)
• Daniel Simmons-Marengo (DSM)
• ForeverHaibara (FH)
• Jeroen Hanselman (JH)
• Sam Schiavone (SS)

Changes (0.9.0):

• gh-322, Add mul_low and pow_trunc methods to fmpz_poly, fmpq_poly and nmod_poly. (RO)
• gh-318, Add emscripten build in CI. Polynomial factors and roots are now sorted into a consistent order for nmod_poly and fq_default_poly. Some tests are fixed so that they pass on 32-bit systems. (AK, OB)
• gh-312, Add discriminant method to fmpz_poly, fmpq_poly and nmod_poly. (RO)
• gh-336, Fixed a bug in arb.neg() which caused it to return its input without negating it. (DSM)
• gh-339, Add fmpq.__float__ method so that float(fmpq) and complex(fmpq) work. (OB)
• gh-324, Faster conversion from int to fmpz and back. (RO).
• gh-326, Add support for building with the Python stable ABI. (OB)
• gh-341, Disable hashing for inexact arb values. (DSM)
• gh-347, Fix storing the variable name for fq_default. (OB)
• gh-350, Add support for Cython 3.2. (OB)
• gh-352, Fix a crash in fmpz_mat.lll() for empty matrices. (OB)
• gh-359, Sort factorisations of all mpoly types. (OB)
• gh-367, Add type stubs and tests for many more classes. (OB)
• gh-370, Fix unused_gens() for the zero polynomial. (OB)
• gh-374, Fixed a bug in nmod.__hash__. (FH)
• gh-378, Build Windows extension modules with MSVC. (OB)
• gh-379, Add Windows ARM64 wheels. (OB)
• gh-380, Support building against FLINT 3.4.0. (OB)
• gh-393, Use fmpz_mod_mat_det() when building against FLINT 3.1 or newer. (OB)
• gh-401, Upload Pyodide wheels to PyPI. (OB)
• gh-389, gh-402, Support building against FLINT 3.5.0. (OB)
• gh-392, Add interface to acb_theta_jet (JH, SS)
• gh-409, Support building against FLINT 3.6.0. (OB)

0.8.0

Contributors (0.8.0):

• Oscar Benjamin (OB)
• Robert Dougherty-Bliss (RDB)
• Rémy Oudompheng (RO)
• Agriya Khetarpal (AK)

Notes (0.8.0):

• This mostly a maintenance release with some bug fixes, dependency updates and a few smaller features.
• Since GitHub Actions is retiring its MacOS x86-64 runners python-flint 0.8.0 is likely the last release to provide prebuilt wheels for MacOS x86-64 (MacOS arm64 remains fully supported). It is likely that future versions will still work fine on MacOS x86-64 but would require building from source and will not be tested in python-flint's CI. MacOS arm64 wheels will still be provided and tested.

Changes (0.8.0):

• gh-302, gh-283, gh-284, Wheels now ship MPFR 4.2.2 and FLINT 3.3.1. Cython 3.1 is now supported for building (and required for the freethreaded build). Wheels are provided for CPython 3.14 and 3.14t (free-threaded) and PyPy 3.11. (OB)
• gh-310, Add truncate, left_shift and right_shift methods to fmpz_poly, fmpq_poly, nmod_poly, acb_poly, arb_poly to match other univariate polynomial types. (RO)
• gh-287, gh-293, gh-305, gh-307, gh-309, Add type annotations for fmpz, fmpq, nmod, fmpz_mod, fq_default, fmpz_poly, fmpq_poly, nmod_poly, fmpz_mod_poly, fq_default_poly, fmpz_mpoly, fmpq_mpoly, nmod_mpoly, fmpz_mod_mpoly, fmpz_series and fmpq_series (about half of the codebase). (OB)
• gh-300, Fix arb.repr which now returns a Python representation that round trips. (OB)
• gh-292, The fmpq constructor now accepts fmpq numerator and denominator as input. (OB)
• gh-289, Add .prec attribute to series types fmpz_series, fmpq_series, arb_series and acb_series. (OB)
• gh-285, Don't use deprecated meson build option. (AK)
• gh-274, Add resultant methods to fmpz_poly, fmpq_poly and nmod_poly. Now all univariate and polynomial types have the resultant method except for fq_default_poly. (RDB)

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+.