44,928 research outputs found
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Stable normal forms for polynomial system solving
This paper describes and analyzes a method for computing border bases of a
zero-dimensional ideal . The criterion used in the computation involves
specific commutation polynomials and leads to an algorithm and an
implementation extending the one provided in [MT'05]. This general border basis
algorithm weakens the monomial ordering requirement for \grob bases
computations. It is up to date the most general setting for representing
quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay
bases and new representation that do not fit into the previous categories. With
this formalism we show how the syzygies of the border basis are generated by
commutation relations. We also show that our construction of normal form is
stable under small perturbations of the ideal, if the number of solutions
remains constant. This new feature for a symbolic algorithm has a huge impact
on the practical efficiency as it is illustrated by the experiments on
classical benchmark polynomial systems, at the end of the paper
On post-Lie algebras, Lie--Butcher series and moving frames
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on
differential manifolds. They have been studied extensively in recent years,
both from algebraic operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid
On the Lie enveloping algebra of a post-Lie algebra
We consider pairs of Lie algebras and , defined over a common
vector space, where the Lie brackets of and are related via a
post-Lie algebra structure. The latter can be extended to the Lie enveloping
algebra . This permits us to define another associative product on
, which gives rise to a Hopf algebra isomorphism between and
a new Hopf algebra assembled from with the new product.
For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
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