Some examples for solving systems of algebraic equations by calculating groebner bases
Journal of Symbolic Computation
Coefficient-parameter polynomial continuation
Applied Mathematics and Computation
A neural network modeled by an adaptive Lotka-Volterra system
SIAM Journal on Applied Mathematics
The cheater's homotopy: an efficient procedure for solving systems of polynomial equations
SIAM Journal on Numerical Analysis
Journal of Symbolic Computation
A polyhedral method for solving sparse polynomial systems
Mathematics of Computation
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics - Proceedings of the international conference on recent advances in computational mathematics
A Rank-Revealing Method with Updating, Downdating, and Applications
SIAM Journal on Matrix Analysis and Applications
Algorithm 846: MixedVol: a software package for mixed-volume computation
ACM Transactions on Mathematical Software (TOMS)
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
Dynamic Enumeration of All Mixed Cells
Discrete & Computational Geometry
Adaptive Multiprecision Path Tracking
SIAM Journal on Numerical Analysis
HOM4PS-2.0para: Parallelization of HOM4PS-2.0 for solving polynomial systems
Parallel Computing
Chern numbers of smooth varieties via homotopy continuation and intersection theory
Journal of Symbolic Computation
Polynomial homotopy continuation with PHCpack
ACM Communications in Computer Algebra
The nearest complex polynomial with a zero in a given complex domain
Theoretical Computer Science
Computing curve intersection by homotopy methods
Journal of Computational and Applied Mathematics
Stability verification for monotone systems using homotopy algorithms
Numerical Algorithms
Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems
ACM Transactions on Mathematical Software (TOMS)
Verified error bounds for real solutions of positive-dimensional polynomial systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Cell decomposition of almost smooth real algebraic surfaces
Numerical Algorithms
On the numerical solution of Kronecker-based infinite level-dependent QBD processes
Performance Evaluation
Leading a continuation method by geometry for solving geometric constraints
Computer-Aided Design
Coordinate-free geometry and decomposition in geometrical constraint solving
Computer-Aided Design
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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.