HOM4PS-2.0 is a software package in FORTRAN 90 which implements the polyhedral homotopy continuation method for solving polynomial systems. It updates its original version HOM4PS in three key aspects: (1) new method for finding mixed cells, (2) combining the polyhedral and linear homotopies in one step, (3) new way of dealing with curve jumping. Numerical results show that this revision leads to a spectacular speed-up, ranging up to 1950s, over its original version on all benchmark systems, especially for large ones. It surpasses the existing packages in finding isolated zeros, such as PHCpack (Verschelde in ACM Trans Math Softw 25:251–276, 1999), PHoM (Gunji et al. in Computing 73:57–77, 2004), and Bertini (Bates et al. in Software for numerical algebraic geometry. Available at http://www.nd.edu/~sommese/bertini), in speed by big margins.

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HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method

Though numerical methods to find all the isolated solutions of nonlinear systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomial systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operations of numerical algebraic geometry, although no one package provides all such capabilities. This paper contains an enumeration of the operations that an ideal software package in this field would allow. The current and upcoming capabilities of Bertini, the most recently released package in this field, are also described.

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Software for numerical algebraic geometry: a paradigm and progress towards its implementation

Publisher Summary This chapter presents the numerical solution of polynomial systems by homotopy continuation methods. Solving polynomial systems is an area where numerical computations arise almost naturally. Given the complexity of the problem, e standard machine arithmetic to obtain efficient programs is used.

Numerical Solution of Polynomial Systems by Homotopy Continuation Methods

http://homepages.math.uic.edu/~jan/Students/ailing_thesis.pdf

Newton's method with deflation for isolated singularities of polynomial systems

We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution as a regular root. Using standard bases, a tool for the symbolic computation of multiplicities, we show that the number of deflation stages is bounded by the multiplicity of the isolated root. Our implementation performs well on a large class of applications.

MixedVol is a Cpp software package that computes the mixed volume of n finite subsets of Zn or the support of a system of n polynomials in n variables. The software produces the mixed volume as well as the mixed cells. The mixed cells are crucial for solving polynomial systems by the polyhedral homotopy continuation method. The software leads existing codes for mixed-volume computation in speed by a substantial margin and its memory requirement is very low.

Algorithm 846: MixedVol: a software package for mixed-volume computation

. The Li—Li algorithm produced in [11] for the mixed volume computation of fully mixed polynomial systems is reconstructed in this article for general semi-mixed polynomial systems. Taking the special structure of the semi-mixed supports of the systems into account, the resulting algorithm, illustrated by numerical results, can dramatically speed up the mixed volume computation, especially when the systems are unmixed. Even when applied to fully mixed systems, the new algorithm improves the speed of the Li—Li algorithm by a considerable amount.

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Mixed Volume Computation for Semi-Mixed Systems

A renewed algorithm is presented to calculate the mixed volume of the support ${\cal A}=({\cal A}_1,\dots,{\cal A}_n)$ of a polynomial system $P({\bf x})=(p_1({\bf x}),\dots, p_n({\bf x}))$ in $\Bbb C^n$. The key ingredient is a specially tailored application of LP feasibility tests, which allows us to calculate the {\em mixed cells}, their volumes constituting the mixed volume, in a {\em mixed subdivision} of ${\cal A}$ more efficiently. The problem of finding mixed cells plays a crucial role in polyhedral homotopy methods for finding all isolated zeros of $P({\bf x})$. Our new algorithm advances the speed of mixed volume computation by a considerable margin, illustrated by numerical examples.

https://projecteuclid.org/download/pdf_1/euclid.twjm/1500407294

Tangan Gao

Papers

Algorithm 846: MixedVol: a software package for mixed-volume computation

Mixed Volume Computation for Semi-Mixed Systems

Mixed volume computation via linear programming

Balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methods