PYPOLSYS

This package provides a python wrapper to POLSYS_PLP fortran90 package from Layne T. Watson, Steven M. Wise, Andrew J. Sommese, August, 1998.

POLSYS_PLP is a solver for N complex coefficients polynomial systems of equations in N unknowns by a probability-one, globally convergent homotopy method.

For a quick survey on homotopy continuation see the very clear practical presentation available on HomotopyContinuation.jl website or this nice tutorial.

There are several other homotopy softwares (more recent, also with python interface, ...) allowing more advanced applications. See for instance:

• PHCpack
• Bertini
• Hom4PSpy and the wrapper on pypi
• HomotopyContinuation.jl

The advantage of POLSYS_PLP is to be an open-source self-content single f90 file (it requires few blas and lapack functions that are shipped with POLSYS_PLP sources or that may be linked to optimized libraries). The idea of pypolsys project is to create an easy to install and to deploy python package to start with Homotopy method.

This wrapper has been developped to tackle polynomial systems obtained by high order perturbation of multi-parametric eigenvalue problems obtained with EasterEig. This yields to dense polynomial system. In this context, multivariate Horner algorithm has been added to quickly evaluate simultaneously the polynomial and its Jacobian matrix.

POLSYS_PLP

POLSYS_PLP use a probability-one globally convergent homotopy method, to find all finite isolated complex solutions to a system F(X) = 0 of N polynomial equations in N unknowns with complex coefficients. A partitioned linear product (PLP) formulation is used for the start system of the homotopy map.

The whole algorithm and its fortran90 implementation is described in

Wise, Steven M., Andrew J. Sommese, and Layne T. Watson. "Algorithm 801: POLSYS_PLP: A partitioned linear product homotopy code for solving polynomial systems of equations." ACM Transactions on Mathematical Software (TOMS) 26.1 (2000): 176-200.

• The sources are available on netlib.
• The theoretical part of POLSYS_PLP is available here
• POLSYS_PLP is based on HOMPACK

To facilitate the build of this module, a copy of POLSYS_PLP original sources is included in the pypolsys/801 folder.

Installation

You'll need :

• python (tested for v >= 3.5);
• pip (optional);
• fortran compiler (tested with gfortran and with m2w64-toolchain on windows)
• lapack and blas installation (only on linux). The useful lapack/blas routines are also shipped with POLSYS_PLP sources and built by default on Window or on demand on linux (see below). Optimized library are preferred and often already present on linux systems. If not, library like openblas and the required development files can be installed with sudo apt install libopenblas-dev on debian based distribution (including ubuntu).

If needed, please see the steps given in the continuous integration script workflows.

Using pip (preferred)

You can install pypolsys from pip:

pip install pypolsys [--user]

You can also install pypolsys after a download from github:

pip install path/to/pypolsys-version.tar.gz [--user]

or in editable mode if you want to modify the sources

pip install -e path/to/pypolsys

after cloning the repos.

pip will install numpy (mandatory) and sympy and scipy (optional, required only for tests).

Note that on ubuntu, you will need to use pip3 instead of pip and python3 instead of python.

On linux, if you want to force the building with POLSYS_PLP lapack sources, you can tell it to pip with

pip install --no-use-pep517 --install-option="--buildLapack" path/to/pypolsys

The flag --no-use-pep517 avoids side effect of install-option that disable wheels. Using POLSYS_PLP lapack sources is the standard behavior on windows.

Troubleshooting

With old gfortran version present on ubuntu 16.04 (see here) the building may failed. To avoid this, intent(in, out) statement has to be removed from the original pypolsys/801/polsys_plp.f90. It seems to work out of the box with gfortan-8.

Running tests

To execute the full test suite, run :

python -m pypolsys.test

Usage

Note that this projet is a work in progress and the API may change.

Get started

Let us consider the following example

x**2 + y + z - 1 = 0
x + y**2 + z - 1 = 0
x + y + z**2 - 1 = 0

Cox, David, John Little, and Donal O'Shea. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media, 2013, From page 122.

With 5 distinct solutions in C : (1,0,0) (0,1,0), (0,0,1), (-1+√2, -1+√2, -1+√2), (-1 -√2, -1-√2, -1-√2). The main steps of pypolsys's workflow are:

import numpy as np
import pypolsys
# Declare the number of equation of the polynomial system
N = 3
# and the number of monoms of each equation (here 4 monoms for each equation).
n_coef_per_eq = np.array([4, 4, 4], dtype=np.int32)
# Provide the coefficients as 1D array
all_coef = np.array([1, 1,  1, -1,
                     1, 1,  1, -1,
                     1, 1,  1, -1], dtype=np.complex)
# then the degree of each monom
all_deg = np.zeros((np.sum(n_coef_per_eq), N), dtype=np.int32)
all_deg[0, 0] = 2
all_deg[1, 1] = 1
all_deg[2, 2] = 1
all_deg[4, 0] = 1
all_deg[5, 1] = 2
all_deg[6, 2] = 1
all_deg[8, 0] = 1
all_deg[9, 1] = 1
all_deg[10, 2] = 2
# Pass it to POLSYS_PLP
pypolsys.polsys.init_poly(N, n_coef_per_eq, all_coef, all_deg)
# Create homogeneous partition
# (N.B. Partitions are important to limit the number of paths
# to track associated to solutions at infinity)
part = pypolsys.utils.make_h_part(3)
# Pass it to POLSYS_PLP
pypolsys.polsys.init_partition(*part)
# Show coef
pypolsys.polsys.show_coef()
# Found 8 solutions, and track 8 paths; some solutions appear twice
bplp = pypolsys.polsys.solve(1e-8, 1e-14, 0.0)
# Get the roots, array of size (N+1) x bplp
r = pypolsys.polsys.myroots
# Get status of the solving process
pypolsys.polsys.report()

Note that due to projective transformation, the r[N, :] is the homogeneous variable. All results are returned ready to used (unscaled and untransformed).

Another way, sometimes more convenient, is to setup the polynomial with function pypolsys.utils.fromSympy,

import numpy as np
import pypolsys
import sympy as sym

x, y, z = sym.symbols('x, y, z')
pol = pypolsys.utils.fromSympy([sym.poly(x**2 + y + z - 1, (x, y, z)),
                                sym.poly(x + y**2 + z - 1, (x, y, z)),
                                sym.poly(x + y + z**2 - 1, (x, y, z))])
# Pass it to POLSYS_PLP
pypolsys.polsys.init_poly(*pol)
# Create homogeneous partition
part = pypolsys.utils.make_h_part(3)
# Pass it to POLSYS_PLP
pypolsys.polsys.init_partition(*part)
# Solve
bplp = pypolsys.polsys.solve(1e-8, 1e-14, 0.0)
# Get the roots, array of size (N+1) x bplp
r = pypolsys.polsys.myroots
# Get status of the solving process
pypolsys.polsys.report()

More examples are available in the pypolsys/test.py file. For dense polynomial system, dense flag can be passed to the solve method to speed up evaluation.

Organisation

This python package is divided into two parts

• pypolsys.polsys corresponds to the interface to POLSYS_PLP and maps the following fortran subroutines (more doc is emmbedded in the f90 files) :

  • init_poly, initialize the polynomials monomials,
  • init_partition, initialize the variables partition,
  • solve, solve the polynomial system,
  • refine, refine the given paths,
  • bezout, compute the Bezout PLP number corresponding to the variable partition, usefull to test partition without solving,
  • report, show a report on the homotopy solving process
  • show_partition and show_coef, show the variables partitions and the monomials respectively.

  some attributs are also available :

  • myroots, containing the roots,
  • path_status and solve_status, containing the status of each tracked path and of the global solving process,
  • num_jac_eval, containing the number of evaluation of the Jacobian matrix.

• pypolsys.utils contains utilities to facilitate the creation of the partition, polynomial or the estimation of the number of path (full python).

View documentation

The doctrings are compatible with several Auto-generate API documentation, like pdoc3. Once the package has been installed or the at pypolsys location run,

pdoc3 --html --force --config latex_math=True pypolsys

The html files are generated in place. Then open the pypolsys/index.html file. This interactive doc is particularly useful to see latex includes.

Limitation

Due to shared variables in modules, only one polynomial instance can be handled. Concurrent calls to the .so file may create unexpected behavior. To solve several polynomials, you can use concurrent.futures.

How to contribute

This project started because we need some easy to deploy Homotopy solver. We are users but not experts on these methods.

If you want to contribute to pypolsys, your are welcomed! Don't hesitate to

• report bugs, installation problems or ask questions on issues;
• propose some enhancements in the code or in documentation through pull requests (PR);
• add or enhance support to other plateforms

To ensure code homogeneity among contributors, we use a source-code analyzer (eg. pylint). Before submitting a PR, run the tests suite ;-)

Developpement

If you need to modify wrapper.f90 fortran source file. You will need to re-build the Fortran extension. This should be done into 2 steps

1. The pyf file that avoid trouble with user_defined fortran type should be updated with f2py utilities :

```
f2py -m polsys -h polsys.pyf wrapper.f90 --overwrite-signature
```

2. Re run the install

License

This file is part of pypolsys, a simple python wrapper to fortran package polsys_plp. pypolsys is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. pypolsys is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with amc2moodle. If not, see https://www.gnu.org/licenses/.