# Getting Started

This documentation describes a Python package to compute solutions of polynomial systems using PHCpack.

The 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 and code generated by the CAMPARY library. CAMPARY is the CudA Multiple Precision ARithmetic librarY, by Mioara Joldes, Olivier Marty, Jean-Michel Muller, Valentina Popescu and Warwick Tucker.

## 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 version 3 of the GNU General Public License as published by the Free Software Foundation.

At the poster session EuroSciPy 2013, questions from Max Demenkov led to development of a step-by-step path tracker, a process in which the user can control the pace of the tracker, asking for the next point on a solution path, which is very convenient for plotting. This is one of the features of `phcpy` which makes PHCpack more versatile to the Python programmer.

## installing phcpy

The installation requires the library files `libPHCpack.so` (Linux), `libPHCpack.dll` (windows), and `libPHCpack.dylib` (Mac OS X) to be present in the folder that holds the modules of `phcpy`.

The file `setup.py` defines the instructions to install and is located of the parent folder of `phcpy` modules.

Then the installation happens simply via

    pip install .

executed at the command prompt in the folder where `setup.py` is located. The installation may require superuser priveleges (although virtual environments are recommended); and on older systems with a python2 present, `pip3` may be needed.

To make the library files `libPHCpack`, one can use the alire crate `phcpack`. Alire <https://alire.ada.dev> is the name of the package manager of Ada.

## extending SageMath with phcpy

The SageMath project was one of the motivations for the development of `phcpy`.

To extend SageMath with `phcpy`, locate the python interpreter used by SageMath and then extend that python with `phcpy`.

Importing phcpy apparently changes the configuration of the signal handlers which may lead SageMath 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)

Without the `init_cysignals()`, the statement `pari(1)/pari(0)` crashes SageMath. With the `init_cysignals()`, the `PariError` exception is handled and the user can continue the SageMath session.

## project history

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.

Prior to release 1.1.3, extension modules were applied, which worked on Linux and Mac OS X, but unfortunately not on Windows. Version 1.1.3 relies directly on the object file `libPHCpack` and applies the `ctypes` to interface more directly with the compiled code. The development of version 1.1.3 happened mainly on Windows, in an Anaconda distribution of Python.

## references

1. 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.

2. 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.

3. M. Joldes, J.-M. Muller, V. Popescu, and W. Tucker: CAMPARY: Cuda Multiple Precision Arithmetic Library and Applications. In Mathematical Software - ICMS 2016, pages 232-240, Springer-Verlag 2016.

4. 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.

5. 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.

6. T. Mizutani, A. Takeda, and M. Kojima. Dynamic enumeration of all mixed cells. Discrete Comput. Geom. 37(3):351-367, 2007.

7. 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.

8. K. Piret: Computing Critical Points of Polynomial Systems using PHCpack and Python. PhD Thesis, University of Illinois at Chicago, 2008.

9. 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.

10. 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.

11. 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>.

12. J. Verschelde: Exporting Ada Software to Python and Julia. ACM SIGAda Ada Letters 42(1):76-78, 2022.

13. J. Verschelde and X. Yu: Polynomial Homotopy Continuation on GPUs. ACM Communications in Computer Algebra, 49(4):130-133, 2015.

## acknowledgments

The PhD thesis of Kathy Piret (cited above) described the development of a first Python interface to PHCpack. The 2008 `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.