Software for Algebraic Geometry pp 15-32 | Cite as

# PHClab: A MATLAB/Octave Interface to PHCpack

## Abstract

PHCpack is a software package for Polynomial Homotopy Continuation, to numerically solve systems of polynomial equations. The executable program “phc” produced by PHCpack has several options (the most popular one “-b” offers a blackbox solver) and is menu driven. PHClab is a collection of scripts which call phc from within a MATLAB or Octave session. It provides an interface to the blackbox solver for finding isolated solutions. We executed the PHClab functions on our cluster computer using the MPI ToolBox (MPITB) for Octave to solve a list of polynomial systems. PHClab also interfaces to the numerical irreducible decomposition, giving access to the tools to represent, factor, and intersect positive dimensional solution sets.

## Keywords

Irreducible Component Embed System Polynomial System Mixed Volume Irreducible Factor## Preview

Unable to display preview. Download preview PDF.

## ^{References}

- [1]R.C. Alperin and R.J. Lang. One and Two-Fold Origami Axioms 2006 4OSME Proceedings, Pasadena, CA, A.K.Peters, 2007.Google Scholar
- [2]J. Fernr1ndez, M. Anguita, E. Ros, and J.L. Bernier. SCE Toolboxes for the development of high-level parallel applications. Proceedings of ICCS 2006, Volume 3992 of Lecture Notes in Computer Science, pages 518-525, Springer-Verlag, 2006Google Scholar
- [3]T. Gao, T.Y. Li, and M. Wu. Algorithm 846: MixedVol: a software package for mixed-volume computation. ACM Trans. Math. Softw., 31(4):555-560, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
- [4]M. Griffis and J. Duffy. Method and apparatus for controlling geometrically simple parallel mechanisms with distinctive connections. US Patent 5,179,525, 1993.Google Scholar
- [5]B. Huber and B. Sturmfels. A polyhedral method for solving sparse polynomial systems. Math. Comp., 64(212):1541-1555, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
- [6]M.L. Husty and A. Karger. Self-motions of Griffis-Duffy type parallel manipu-lators. Proc. 2000 IEEE Int. Conf. Robotics and Automation, CDROM, San Francisco, CA, April 24-28, 2000.Google Scholar
- [7]A. Leykin and J. Verschelde. PHCmaple: A Maple interface to the numerical homotopy algorithms in PHCpack. In Quoc-Nam Tran, editor, Proceedings of the Tenth International Conference on Applications of Computer Algebra (ACA'2004 ), pages 139-147, 2004.Google Scholar
- [8]A. Leykin and J. Verschelde. Interfacing with the numerical homotopy algo-rithms in Phcpack. In Nobuki Takayama and Andres Iglesias, editors, Pro-ceedings of ICMS 2006. Volume 4151 of Lecture Notes in Computer Science, pages 354-360, Springer-Verlag, 2006.Google Scholar
- [9]A. Leykin, J. Verschelde, and A. Zhao. Newton's method with deflation for isolated singularities of polynomial systems. Theoretical Computer Science 359(1-3):111-122,2006.zbMATHCrossRefMathSciNetGoogle Scholar
- [10]T.Y. Li. Numerical solution of polynomial systems by homotopy continuation methods. In F. Cucker, editor, Handbook of Numerical Analysis. Volume XI. Special Volume: Foundations of Computational Mathematics, pages 209-304. North-Holland, 2003.Google Scholar
- [11]M. Maekawa, M. NoR.o, K. Ohara, Y. Okutani, N. Takayama, and Y. Tamura. Openxm - an open system to integrate mathematical softwares. Available at http://www.Openxm.org/.
- [12]M. Maekawa, M. Noao, K. Ohara, N. Takayama, and Y. Tamura. The de-sign and implementation of OpenXM-RFC 100 and 101. In K. Shirayanagi and K. Yokoyama, editors, Computer mathematics. Proceedings of the Fifth Asian Symposium (ASCM 2001) Matsuyama, Japan 26-28, September 2001, Volume 9 of Lecture Notes Series on Computing, pages 102-111. World Sci-entific, 2001.Google Scholar
- [13]J.M. Mccarthy. Geometric Design of Linkages. Volume 11 of Interdisciplinary Applied Mathematics, Springer-Verlag, 2000.Google Scholar
- [14]A.J. Sommese and J. Verschelde. Numerical homotopies to compute generic points on positive dimensional algebraic sets. J. of Complexity,16(3):572-602, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
- [15]A.J. Sommese, J. Verschelde, and C.W. Wampler. Using monodromy to de-compose solution sets of polynomial systems into irreducible components. In C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher, editors, Application of Algebraic Geometry to Coding Theory, Physics and Computation, pages 297-315. Kluwer Academic Publishers, 2001. Proceedings of a NATO Conference, February 25-March 1, 2001, Eilat, Israel.Google Scholar
- [16]A.J. Sommese, J. Verschelde, and C.W. Wampler. Symmetric functions ap-plied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal., 40(6):2026-2046, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
- [17]A.J. Sommese, J. Verschelde, and C.W. Wampler. Advances in polynomial continuation for solving problems in kinematics. ASME Journal of Mechanical Design 126(2):262-268, 2004.CrossRefMathSciNetGoogle Scholar
- [18]A.J. Sommese, J. Verschelde, and C.W. Wampler. Numerical irreducible de-composition using Phcpack. In M. Joswig and N. Takayama, editors, Algebra, Geometry, and Software Systems, pages 109-130. Springer-Verlag, 2003.Google Scholar
- [19]A.J. Sommese, J. Verschelde, and C.W. Wampler. Solving polynomial systems equation by equation. Accepted for publication in the IMA volume on Algorithms for Algebraic Geometry.Google Scholar
- [20]A.J. Sommese and C.W. Wampler. The Numerical solution of systems of poly-nomials arising in engineering and science. World Scientific Press, Singapore, 2005.Google Scholar
- [21]J. Verschelde. Algorithm 795: PHCPACK: A general-purpose solver for poly-nomial systems by homotopy continuation. ACM Trans. Math. Softw., 25 (2):251-276, 1999. Software available at http://www.math.uic.edu/-jan.zbMATHCrossRefGoogle Scholar