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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 Johanssonfredrik.johansson@gmail.com

Currently python-flint supports CPython versions 3.11-3.14 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 withpip

pip install python-flint

Alternatively python-flint can be installed usingconda

conda install -c conda-forge python-flint

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 Flintsuch as 2.9 are not supported any more. Note that as of Flint 3 Arb no longerneeds 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) canbe 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 ofFLINT 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 installedusing:

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

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

pip install .

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

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

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]

• 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 likeZZ(1) etc rather thanfmpz_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 Pythonexceptions 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.)

Generally each release of python-flint will be compatible with a range ofPython versions. Since python-flint 0.5.0 the minimum supported Flint versionis3.0 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 ofpython-flint.

As of python-flint 0.7.0, CPython 3.13PEP703 free-threaded (no-GIL) builds ofpython-flint are provided. In the the free-threaded build, mutating matrices orpolynomials from multiple threads can lead to memory corruption. There are someother known issues with the free-threaded build so it should be consideredexperimental.

Contributors (0.9.0):

• Rémy Oudompheng (RO)
• Agriya Khetarpal (AK)
• Oscar Benjamin (OB)
• Daniel Simmons-Marengo (DSM)

Changes (0.9.0):

• gh-322,Addmul_low andpow_trunc methods tofmpz_poly,fmpq_poly andnmod_poly. (RO)
• gh-318,Add emscripten build in CI. Polynomial factors and roots arenow sorted into a consistent order fornmod_poly andfq_default_poly. Some tests are fixed so that they pass on32-bit systems. (AK, OB)
• gh-312,Adddiscriminant method tofmpz_poly,fmpq_poly andnmod_poly. (RO)
• gh-336,Fixed a bug inarb.neg() which caused it to return its inputwithout negating it. (DSM)
• gh-339,Addfmpq.__float__ method so thatfloat(fmpq) andcomplex(fmpq)work. (OB)

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 runnerspython-flint 0.8.0 is likely the last release to provideprebuilt wheels for MacOS x86-64 (MacOS arm64 remainsfully supported). It is likely that future versions willstill work fine on MacOS x86-64 but would require buildingfrom 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 nowsupported for building (and required for the freethreadedbuild). Wheels are provided for CPython 3.14 and 3.14t(free-threaded) and PyPy 3.11. (OB)
• gh-310,Addtruncate,left_shift andright_shift methods tofmpz_poly,fmpq_poly,nmod_poly,acb_poly,arb_polyto match other univariate polynomial types. (RO)
• gh-287,gh-293,gh-305,gh-307,gh-309,Add type annotations forfmpz,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 andfmpq_series(about half of the codebase). (OB)
• gh-300,Fixarb.repr which now returns a Python representation thatround trips. (OB)
• gh-292,Thefmpq constructor now acceptsfmpq numerator and denominatoras input. (OB)
• gh-289,Add.prec attribute to series typesfmpz_series,fmpq_series,arb_series andacb_series. (OB)
• gh-285,Don't use deprecated meson build option. (AK)
• gh-274,Add resultant methods tofmpz_poly,fmpq_poly andnmod_poly. Now all univariate and polynomial types have theresultant method except forfq_default_poly. (RDB)

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 (previously3.0.1 was used). All versions from FLINT 3.0.0 to FLINT 3.2.0are compatible with python-flint but some features requirenewer FLINT versions and the PyPI packages now use FLINT 3.2.0.
• gh-97,gh-182:Addfq_default andfq_default_poly for finite fields andunivariate polynomials over finite fields. This exposes allof the different implementations of finite fields (fq_zech,fq_nmod etc) via thefq_default interface. (GP)
• gh-132,gh-164,gh-190,gh-191:gh-192:gh-216:gh-225:gh-228:Addfmpz_mpoly,fmpq_mpoly,nmod_poly andfmpz_mod_polytypes for multivariate polynomials with integer, rational orintegers modn coefficients. (JM)
• gh-142Addacb_theta module for the numerical evaluation ofthetafunctions (onlyavailable forFlint >= 3.1). (EC)
• gh-218gh-254gh-255An experimental interface for FLINT's generic rings has beenadded. This provides access to many of FLINT's types thatare not yet wrapped by python-flint such as Gaussian integer,number fields, qqbar, calcium, as well as both univariate andmultivariate polynomials and series over these rings (nomatrices yet though). (OB and TH)
• gh-129gh-208Use meson/meson-python instead of setuptools as the build systemfor parallel builds and better detection of build and dependencyrequirements. (OB)
• gh-201gh-202The documentation has been updated and is now atreadthedocs.(OB)gh-235Nightly 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-259Add PyPI wheels for Linux aarch64 (Linux on ARM CPU). (OB)

Compatibility break (0.7.0):

• gh-189As of python-flint 0.7.0fmpq_poly.factor() now returnsprimitive rather than monic factors i.e.2*x + 1 rather thanx + 1/2. This ensures consistency between all poly typesincluding betweenfmpq_poly andfmpq_mpoly. (OB)

Other changes (0.7.0):

• gh-269All univariate and multivariate polynomial types haveis_zero,is_one andis_constant methods. All polynomialtypes now consistently handle negative powers where possible.
• gh-261Addfmpz_mat.fflu for fraction-free LU decomposition ofan integer matrix.
• gh-251Add mpmath-style precision context managers for arbextraprec,extradps,workprec andworkdps. (TH)
• gh-250Addfmpq.gcd() method.
• gh-215gh-219The FLINT binding declarations are now fully generatedautomatically from the FLINT docs. (OB)
• gh-203gh-204gh-205gh-206gh-207gh-211gh-212gh-271Various linting fixes and codebase improvements (FC and GP).
• gh-189All scalar and poly types now havesqrt. All poly types nowhavefactor_squarefree andleading_coefficient methods.Exception types raised in a number of places were changed toDomainError for better consistency. (OB)
• gh-196Supported Python versions are 3.10-3.13 (3.9 dropped). CITesting added for 3.13 free-threaded CPython.
• gh-194Add version checking for build requirements. (OB)
• gh-180Addequal_trunc,add_trunc,sub_trunc,mul_low,mul_mod andpow_trunc methods tofmpz_mod_poly. (GP)
• gh-177Remove old Py2 code for compatibility with Cython 3.1. (OB)
• gh-176Fix the error messages fromfmpq constructor. (OB)
• gh-174Addpow_mod andcompose_mod methods tonmod_poly andfmpz_mod_poly. Also add some missing methods tonmod_polythat other poly types already have. (GP)
• gh-172Addfmpz_is_square. (JR)
• gh-168Make comparisons consistent between different types. Addis_one andis_zero for all poly types. (OB)
• gh-161Addacb.lerch_phi to compute the Lerch transcendent. (OB)
• gh-160Addbits toarb andacb, addlog_base toarb. (JR)
• gh-148Remove debug symbols to make smaller Linux binaries. (OB)
• gh-144Addrel_one_accuracy_bits toarb andacb. (EC)
• gh-137Adderfinv anderfcinv forarb. (JR)
• gh-119Add compatibility with Flint 3.1. (OB)

• gh-112,gh-111,gh-110,gh-108:Add pyproject.toml and build dependencies. This means thatpython-flint can be built from source without--no-build-isolation.
• gh-109:Use exact division for non-field domains. Nowfmpz(6)/fmpz(3)returns an exact resultfmpz(2) or raises an error if an exactresult is not possible. Similar changes forfmpz_poly/fmpz,fmpz_mat/fmpz, and for polynomial division withfmpz_poly,fmpq_poly,nmod_poly andfmpz_mod_poly.
• gh-106:Addfmpz_mod_mat for matrices of integers modn wheren islarger than word sized.
• gh-104:Bump Flint from 3.0.0 to 3.0.1

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 ofnumpy.distutils.

New features:

• gh-87:Addsfmpz_mod_poly type for polynomials overfmpz_mod.
• gh-85:Adds discrete logarithms tofmpz_mod.
• gh-83:Introduces thefmpz_mod type for multi-precision integer mods.

Bug fixes:

• gh-93:Fixes a bug withpow(int, int, fmpz) which previously gaveincorrect results.
• gh-78,gh-79:minor fixes for thenmod type.

• gh-75,gh-77:finish bulk of the work in refactoringpython-flint intosubmodules
• gh-72:The roots method ofarb_poly is not supported. Use either thecomplex_roots method oracb_roots(p).roots() to get the oldbehaviour of returning the complex roots. Theroots method onfmpz_poly andfmpq_poly now return integer and rationalroots respectively. To access complex roots on these types, usethecomplex_roots method. Foracb_poly, bothroots andcomplex_roots behave the same
• gh-71:Include files in sdist and fix issuegh-70
• gh-67:Continue refactoring job to introduce submodules intopython-flint

• gh-63:Theroots method ofarb_poly, andnmod_poly is no longersupported. Useacb_roots(p).roots() to get the old behaviourof returning the roots asacb. Note that theroots method offmpz_poly andfmpq_poly currently returns the complex rootsof the polynomial.
• gh-61:Start refactoring job to introduce submodules intopython-flint

• gh-57:Adds manylinux wheels

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

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

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