A methodology for solving chemical equilibrium systems
Applied Mathematics and Computation
Numerical solution of a class of deficient polynomial systems
SIAM Journal on Numerical Analysis
A homotopy for solving general polynomial systems that respects m-homogenous structures
Applied Mathematics and Computation
Computing all solutions to polynomial systems using homotopy continuation
Applied Mathematics and Computation
Coefficient-parameter polynomial continuation
Applied Mathematics and Computation
Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms
ACM Transactions on Mathematical Software (TOMS)
Algorithm 555: Chow-Yorke Algorithm for Fixed Points or Zeros of C2 Maps [C5]
ACM Transactions on Mathematical Software (TOMS)
A Method for Computing All Solutions to Systems of Polynomials Equations
ACM Transactions on Mathematical Software (TOMS)
Algorithm 596: a program for a locally parameterized
ACM Transactions on Mathematical Software (TOMS)
Number of Solutions for Motion and Structure from Multiple FrameCorrespondence
International Journal of Computer Vision
Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms
ACM Transactions on Mathematical Software (TOMS)
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Finding all real zeros of polynomial systems using multi-resultant
Journal of Computational and Applied Mathematics
ACM Transactions on Mathematical Software (TOMS)
Incomplete Gröbner basis as a preconditioner for polynomial systems
Journal of Computational and Applied Mathematics
Motion of nonrigid objects from multiframe correspondences
Journal of Visual Communication and Image Representation
Mathematical and Computer Modelling: An International Journal
Lipschitz condition for finding real roots of a vector function
Journal of Computational Methods in Sciences and Engineering
Hi-index | 0.00 |
Although the theory of polynomial continuation has been established for over a decade (following the work of Garcia, Zangwill, and Drexler), it is difficult to solve polynomial systems using continuation in practice. Divergent paths (solutions at infinity), singular solutions, and extreme scaling of coefficients can create catastrophic numerical problems. Further, the large number of paths that typically arise can be discouraging. In this paper we summarize polynomial-solving homotopy continuation and report on the performance of three standard path-tracking algorithms (as implemented in HOMPACK) in solving three physical problems of varying degrees of difficulty. Our purpose is to provide useful information on solving polynomial systems, including specific guidelines for homotopy construction and parameter settings. The m-homogeneous strategy for constructing polynomial homotopies is outlined, along with more traditional approaches. Computational comparisons are included to illustrate and contrast the major HOMPACK options. The conclusions summarize our numerical experience and discuss areas for future research.