This documentation describes a collection of Python modules to compute solutions of polynomial systems using PHCpack.
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.
The computation of the mixed volume in phcpy calls MixedVol (ACM TOMS Algorithm 846 of T. Gao, T.Y. Li, M. Wu) as it is integrated in PHCpack. DEMiCs (Dynamic Enumeration of all Mixed Cells, by T. Mizutani, A. Takeda, and M. Kojima) is faster than MixedVol for larger systems with many different supports. A function to compute mixed volumes with DEMiCs is available in phcpy.
For double double and quad double arithmetic, PHCpack incorporates the QD library of Y. Hida, X.S. Li, and D.H. Bailey. See the References section for pointers to the literature.
While PHCpack has been under development for over twenty years, phcpy is still working through its proof-of-concept stage. In its present state, working with phcpy will require persistence and plenty of patience.
what is phcpy?¶
The main executable phc (polynomial homotopy continuation) defined by the source code in PHCpack is a menu driven and file oriented program. The Python interface defined by phcpy replaces the files with persistent objects allowing the user to work with scripts or in interactive sessions. The computationally intensive tasks such as path tracking and mixed volume computations are executed as compiled code so there will not be a loss of efficiency.
Both phcpy and PHCpack are free and open source software packages; 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 2 of the License, or (at your option) any later version.
One application of phcpy is to run regression tests. The Python interface phcpy to PHCpack is a programmer’s interface. The long-term goal is to make PHCpack more versatile, at least for those programmers familiar with the Python scripting language.
The source for PHCpack can be downloaded from <http://www.math.uic.edu/~jan/download.html> and is also available at <https://github.com/janverschelde/PHCpack>. For the installation from source, the gnu-ada compiler (available for free at <http://libre.adacore.com/>) is needed. Also your Python interpreter must most likely have been built with gcc. In addition to the Python interpreter, the file Python.h of of the developer’s libraries will need to exist on your system. Otherwise, PHCpack and phcpy are self contained and do not require the installation of other software.
Up to version 0.3.7, phcpy was written with versions 2.6 and 2.7 of Python. Version 0.3.8 of phcpy was ported to Python 3.5, using a modified C interface phcpy2c3.c and the corresponding shared object phcpy2c3.so.
The code runs on Red Hat Linux 64-bit, Ubuntu Linux, and on Mac OS X. There is no support for Windows. Below is a step-by-step installation procedure.
The gnu-ada compiler must be installed to compile the shared object file. Although several Linux distributions come with gcc that have Ada enabled, check whether
gnatmakeis in your execution path. In a terminal window, at the command prompt, type
which gnatmake. If the system answers with the location of
gnatmake, then the gnu-ada compiler is installed on your system.
If you have multiple versions of gcc installed on your system, then the binaries of the gnu-ada compiler should appear first in your execution path. Typing
gcc -vshould show
for GNAT GPLin the reply, or most recently (in 2018):
GNAT Community 2018.
gcc -vgave satisfactory replies, then you can proceed to step 2 and skip the installation of the gnu-ada compiler.
By default one needs to have superuser privileges to install the gnu-ada compiler at standard locations, but otherwise it is not hard to install the gnu-ada compiler in your own directory.
Following the instructions of the gnu-ada compiler, the location with the binaries must be added in front of the execution path. You may have to edit
.bashrc(for the Bourne shell) or
.cshrc(for the C shell).
On Linux systems, make sure to compile and install the library with the option -fPIC. When configuring, run configure as
./configure CXX=/usr/bin/g++ CXXFLAGS='-fPIC -O3'to set the flags of the c++ compiler.
If you rather would not (or cannot) install QDlib, then it is possible to compile the library in your home directory. All you need is to be aware of the location of the header files for the include statement and you need the
qdlib.afile for linking. For an example, consider the makefile for Windows computers. The
phc.exewith the QD library compiled in a home directory, not installed on the system.
The source code directory of PHCpack contains the directory
Objects, a directory with file
makefilein it. Depending on whether you are on a Unix-like system or on a mac, you will have to edit the
MAKEFILEvariable either refers to
makefile_mac. Once the makefile is adjusted you could type, just as a test,
make phcto compile the main executable program. Note that for phcpy, you will not need this executable.
To make the shared object file, your python system needs to have been installed with the development version, that is: the file
Python.hmust be available on your disk. Often, following binary installations of the Python interpreter, this
Python.hmight be absent.
If packaged distributions for the development version of Python fail, then you may have to download the source code from <http://www.python.org>, configure, compile, and install the Python system. An additional benefit of such a Python installation is that then the Python interpreter could be built with the gnu-ada compiler, so both the scripting environment as the compiled code are using the same version of gcc.
Once you have located the file
Python.hon your system, you most likely will have to adjust the definitions in the files
makefile_mac. Assign to the variables
PYTHON3the directories where
In the directory
Objectsof the PHCpack source distribution, type
make phcpy2c2.soto make the shared object file for python2, or type
make phcpy2c3.sofor the python3 version of phcpy. If all goes well, the shared object
phcpy2c2.socan then be found in
phcpy2c3.sois then in
If you run the Python interpreter in a terminal window in the directory
phcpyof the source code distribution, then the
import phcpyshould already work.
To extend your python2 installation with phcpy, go to the
Python/PHCpy2directory and run
python2 setup.py installas superuser or as sudoer. For python3, go to
python3 setup.py installas superuser or as sudoer.
The documentation is typeset with Sphinx. Sphinx depends on the default version of Python on your system. If phcpy is installed with a different version of Python than the version Sphinx depends on, then this may cause problems to typeset the documentation.
extending Sage with phcpy¶
The current version of Sage uses python2. So the instructions to extend Sage with phcpy work only with the Python2 version of phcpy.
If you have installed Sage from source on your computer, then this installation comes with its own python libraries and interpreter. Then it is not too much work any more (in comparison to the steps in last section) to extend the python interpreter of Sage with phcpy.
On Linux systems, locate the python interpreter of Sage.
Most likely this python is
/local/bin of in the downloaded directory.
Use the absolute path name for the location of the Sage python
interpreter and navigate to the
Python/PHCpy2 directory which
setup.py for phcpy.
python setup.py install at
the command prompt. This does not require superuser access,
but you must execute this setup with the same account you used
to install Sage with.
We check the installation at the command prompt, as shown in Fig. 1.
On Mac OS X, extending Sage with phcpy requires a bit more work as
phcpy2c2.so must be compiled with the Python library that
comes with the Sage installation. For this, the
must be modified with the correct definition for the location of
the Python library of Sage, as defined by
With this definition, do
make sage_phcpy2c2.so and then move
this file under the name
phcpy2c2.so to the directory
/Python/PHCpy2/phcpy. The installation is then similar as
for Linux, type
python setup.py install at the command prompt
in the directory where
setup.py exists and for
using the absolute file name of the executable, e.g., type
/Users/jan/Downloads/sage-7.2/local/bin/python setup.py install.
Importing phcpy apparently changes the configuration of the signal handlers which may lead Sage to crash when exceptions occur. Thanks to Marc Culler for reporting this problem and for suggesting a work around:
sage: import phcpy sage: from cysignals import init_cysignals sage: init_cysignals() sage: pari(1)/pari(0)
pari(1)/pari(0) crashes Sage.
PariError exception is handled
and the user can continue the Sage session.
This section describes some milestones in the development history.
The Python interface to PHCpack got to a first start when Kathy Piret met William Stein at the software for algebraic geometry workshop at the IMA in the Fall of 2006. The first version of this interface is described in the 2008 PhD Thesis of Kathy Piret.
The implementation of the Python bindings depend on the C interface to PHCpack, developed for use with message passing on distributed memory computers.
Version 0.0.1 originated at lecture 40 of MCS 507 in the Fall of 2012, as an illustration of Sphinx. In Spring of 2013, version 0.0.5 was presented at a graduate computational algebraic geometry seminar. Version 0.1.0 was prepared for presentation at EuroSciPy 2013 (August 2013). Improvements using pylint led to version 0.1.1 and the module maps was added in version 0.1.2. Version 0.1.4 added path trackers with a generator so all solutions along a path are returned to the user. Multicore path tracking was added in version 0.1.7.
The paper Modernizing PHCpack through phcpy written for the EuroSciPy 2013 proceedings and available at <http://arxiv.org/abs/1310.0056> describes the design of phcpy.
Version 0.2.9 coincides with version 2.4 of PHCpack and gives access to the first version of the GPU accelerated path trackers. Sweep homotopies to explore the parameter space with detection and location of singularities along the solution paths were exported in the module sweepers.py in version 0.3.3 of phcpy. With the addition of a homotopy membership test in verion 0.3.7, the sets.py module provides the key ingredients for a numerical irreducible decomposition. Version 0.5.0 introduced Newton’s method on power series. Use cases were added to the documentation in versions 0.5.2, 0.5.3, and 0.5.4. With static linking, the dependencies on the gnat runtime libraries are removed and the Sage python interpreter could be extended with version 0.6.2. Better support of Laurent polynomial systems was added in version 0.6.8. In version 0.6.9, the large module sets.py was divided up, leading to the new modules cascades.py, factor.py, and diagonal.py. Code snippets for jupyter notebook menu extensions were defined in version 0.7.4. Version 0.8.3 gave access to DEMiCs to compute mixed volumes by dynamic enumeration of all mixed cells.
Version 0.9.2 was presented in a talk at the Python devroom at FOSDEM 2019. Another important milestone was the poster presentation at the 18th Python in Science Conference, SciPy 2019, with a paper which appeared in the proceedings (see the references below).
As the original goal of phcpy was on exporting the functionality of PHCpack, its design is functional and phcpy is a collection of modules. Since version 1.0.0, two class definitions were added, one to represent systems of polynomials and another class to represent solutions of polynomial systems. The 1.x.y releases series will continue to develop phcpy with an object-oriented interface to the functionality of PHCpack.
T. Gao, T. Y. Li, M. Wu: Algorithm 846: MixedVol: a software package for mixed-volume computation. ACM Transactions on Mathematical Software, 31(4):555-560, 2005.
Y. Hida, X.S. Li, and D.H. Bailey: Algorithms for quad-double precision floating point arithmetic. In 15th IEEE Symposium on Computer Arithmetic (Arith-15 2001), 11-17 June 2001, Vail, CO, USA, pages 155-162. IEEE Computer Society, 2001. Shortened version of Technical Report LBNL-46996.
A. Leykin and J. Verschelde. Interfacing with the numerical homotopy algorithms in PHCpack. In N. Takayama and A. Iglesias, editors, Proceedings of ICMS 2006, volume 4151 of Lecture Notes in Computer Science, pages 354–360. Springer-Verlag, 2006.
T. Mizutani and A. Takeda. DEMiCs: A software package for computing the mixed volume via dynamic enumeration of all mixed cells. In M. E. Stillman, N. Takayama, and J. Verschelde, editors, Software for Algebraic Geometry, volume 148 of The IMA Volumes in Mathematics and its Applications, pages 59-79. Springer-Verlag, 2008.
T. Mizutani, A. Takeda, and M. Kojima. Dynamic enumeration of all mixed cells. Discrete Comput. Geom. 37(3):351-367, 2007.
J. Otto, A. Forbes, and J. Verschelde. Solving Polynomial Systems with phcpy. In the Proceedings of the 18th Python in Science Conference (SciPy 2019), edited by Chris Calloway, David Lippa, Dillon Niederhut and David Shupe, pages 58-64, 2019.
K. Piret: Computing Critical Points of Polynomial Systems using PHCpack and Python. PhD Thesis, University of Illinois at Chicago, 2008.
A. J. Sommese, J. Verschelde, and C. W. Wampler. Numerical irreducible decomposition using PHCpack. In Algebra, Geometry, and Software Systems, edited by M. Joswig and N. Takayama, pages 109-130. Springer-Verlag, 2003.
J. Verschelde: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software, 25(2):251–276, 1999.
J. Verschelde: Modernizing PHCpack through phcpy. In Proceedings of the 6th European Conference on Python in Science (EuroSciPy 2013), edited by Pierre de Buyl and Nelle Varoquaux, pages 71-76, 2014, available at <http://arxiv.org/abs/1310.0056>.
J. Verschelde and X. Yu: Polynomial Homotopy Continuation on GPUs. ACM Communications in Computer Algebra, 49(4):130-133, 2015.
The PhD thesis of Kathy Piret (cited above) described the
development of a first Python interface to PHCpack.
phcpy.py provided access to the blackbox solver,
the path trackers, and the mixed volume computation.
In the summer of 2017, Jasmine Otto helped with the setup of
jupyterhub and the definition of a SageMath kernel.
Code snippets with example uses of
phcpy in a Jupyter notebook
were introduced during that summer. The code snippets,
listed in a chapter of this document, provide another good way
to explore the capabilities of the software.
This material is based upon work supported by the National Science Foundation under Grants 1115777, 1440534, and 1854513. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
about this document¶
This document arose as an exercise in exploring restructured text and Sphinx. All good software documents contain the following four items: an installation guide, a getting started, a tutorial, and a reference manual. This document combines all four.