Algorithm 795

Verschelde, Jan

doi:10.1145/317275.317286

Cited by 462 publications

(318 citation statements)

References 71 publications

Supporting

Mentioning

317

Contrasting

Order By: Relevance

“…Implementation. We have two Maple implementations of our algorithms using the package PHCmaple [17] to interface with PHCpack [33], which performs the numerical polynomial homotopy continuation. PHCmaple produced the graphic of Figure 3.…”

Section: Description Of Softwarementioning

confidence: 99%

See 1 more Smart Citation

Galois groups of Schubert problems via homotopy computation

Leykin

Sottile

2009

Math. Comp.

View full text Add to dashboard Cite

Abstract. Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C 8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S 6006 .

show abstract

Section: Description Of Softwarementioning

confidence: 99%

“…For the continuation, both implementations use PHCpack [33] through its Maple interface PHCmaple [17], and the second implementation may also call Bertini [1]. The advantages of Bertini are that it can use arbitrary precision and it gives an independent verification of our results.…”

Section: Introductionmentioning

confidence: 99%

Galois groups of Schubert problems via homotopy computation

Leykin

Sottile

2009

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…The second one solves the same linkage but assuming that θ 3 is a free variable, yielding a onedimensional continuum of solutions. The same benchmarks have been used previously to show the performance of elimination [15,16], continuation [17,18], and relaxation techniques [2]. We compare our results with those derived by such techniques, and employ the same linkage dimensions used in these papers.…”

Section: Methodsmentioning

confidence: 99%

“…We have checked, however, that our method converges in substantially shorter times than those used by the continuation method in [17,18], using the implementation available at Jan Verschelde's home page, which spent about 3 seconds of CPU time on the same example. We remark, though, that we are comparing our algorithm with a general-purpose solver targeted to arbitrary systems of algebraic equations, and that a better performance of our algorithm was to be expected, given that we exploit the specific structure of the equations involved.…”

Section: A Rigid Butterflymentioning

confidence: 99%

Exact interval propagation for the efficient solution of position analysis problems on planar linkages

Celaya

Creemers

Ros

2012

Mechanism and Machine Theory

View full text Add to dashboard Cite

This paper presents an interval propagation algorithm for variables in planar single-loop linkages. Given intervals of allowed values for all variables, the algorithm provides, for every variable, the whole set of values, without overestimation, for which the linkage can actually be assembled. We show further how this algorithm can be integrated in a branch-and-prune search scheme, in order to solve the position analysis of general planar multi-loop linkages. Experimental results are included, comparing the method's performance with that of previous techniques given for the same task.

show abstract

“…The algorithms outlined above have been implemented with the aid of the path following routines in PHCpack [23]. Recently the package has been upgraded with a module (see [20] for an overview) to numerically decompose solution sets of polynomial systems into irreducible components.…”

Section: A Numerical Experimentsmentioning

confidence: 99%

A method for tracking singular paths with application to the numerical irreducible decomposition

Beltrametti¹,

Catanese²,

Ciliberto³

et al. 2002

Algebraic Geometry

View full text Add to dashboard Cite

Abstract. In the numerical treatment of solution sets of polynomial systems, methods for sampling and tracking a path on a solution component are fundamental. For example, in the numerical irreducible decomposition of a solution set for a polynomial system, one first obtains a "witness point set" containing generic points on all the irreducible components and then these points are grouped via numerical exploration of the components by path tracking from these points. A numerical difficulty arises when a component has multiplicity greater than one, because then all points on the component are singular. This paper overcomes this difficulty using an embedding of the polynomial system in a family of systems such that in the neighborhood of the original system each point on a higher multiplicity solution component is approached by a cluster of nonsingular points. In the case of the numerical irreducible decomposition, this embedding can be the same embedding that one uses to generate the witness point set. In handling the case of higher multiplicities, this paper, in concert with the methods we previously proposed to decompose reduced solution components, provides a complete algorithm for the numerical irreducible decomposition. The method is applicable to tracking singular paths in other contexts as well.

show abstract

Algorithm 795

Cited by 462 publications

References 71 publications

Galois groups of Schubert problems via homotopy computation

Galois groups of Schubert problems via homotopy computation

Exact interval propagation for the efficient solution of position analysis problems on planar linkages

A method for tracking singular paths with application to the numerical irreducible decomposition

Contact Info

Product

Resources

About