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: http://fredrikj.net/python-flint/

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

Author: Fredrik Johansson

Installation

Currently python-flint supports CPython versions 3.9-3.12. 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.

See here for instructions on building FLINT:

• http://flintlib.org/

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 .

A script that builds and installs FLINT and python-flint that is tested on Ubuntu can be found in the git repo here:

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

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

• https://fredrikj.net/python-flint/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.)

CHANGELOG

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+.