2,504 research outputs found
On isolation of singular zeros of multivariate analytic systems
We give a separation bound for an isolated multiple root of a square
multivariate analytic system satisfying that an operator deduced by adding
and a projection of in a direction of the kernel of
is invertible. We prove that the deflation process applied on and this kind
of roots terminates after only one iteration. When is only given
approximately, we give a numerical criterion for isolating a cluster of zeros
of near . We also propose a lower bound of the number of roots in the
cluster.Comment: 17 page
Clustering Complex Zeros of Triangular Systems of Polynomials
This paper gives the first algorithm for finding a set of natural
-clusters of complex zeros of a triangular system of polynomials
within a given polybox in , for any given . Our
algorithm is based on a recent near-optimal algorithm of Becker et al (2016)
for clustering the complex roots of a univariate polynomial where the
coefficients are represented by number oracles.
Our algorithm is numeric, certified and based on subdivision. We implemented
it and compared it with two well-known homotopy solvers on various triangular
systems. Our solver always gives correct answers, is often faster than the
homotopy solver that often gives correct answers, and sometimes faster than the
one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
Certifying isolated singular points and their multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construc-tion uses a single linear differential form
defined from the Jacobian matrix of the input, and defines the deflated system
by applying this differential form to the original system. The advantages of
this new deflation is that it does not introduce new variables and the increase
in the number of equations is linear instead of the quadratic increase of
previous methods. The second construction gives the coefficients of the
so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new vari-ables. We show
that the roots of this new system include the original singular root but now
with multiplicity one, and the new variables uniquely determine the
multiplicity structure. Both constructions are "exact", meaning that they
permit one to treat all conjugate roots simultaneously and can be used in
certification procedures for singular roots and their multiplicity structure
with respect to an exact rational polynomial system
Punctual Hilbert Schemes and Certified Approximate Singularities
In this paper we provide a new method to certify that a nearby polynomial
system has a singular isolated root with a prescribed multiplicity structure.
More precisely, given a polynomial system f , we present a Newton iteration on an extended deflated system
that locally converges, under regularity conditions, to a small deformation of
such that this deformed system has an exact singular root. The iteration
simultaneously converges to the coordinates of the singular root and the
coefficients of the so called inverse system that describes the multiplicity
structure at the root. We use -theory test to certify the quadratic
convergence, and togive bounds on the size of the deformation and on the
approximation error. The approach relies on an analysis of the punctual Hilbert
scheme, for which we provide a new description. We show in particular that some
of its strata can be rationally parametrized and exploit these parametrizations
in the certification. We show in numerical experimentation how the approximate
inverse system can be computed as a starting point of the Newton iterations and
the fast numerical convergence to the singular root with its multiplicity
structure, certified by our criteria.Comment: International Symposium on Symbolic and Algebraic Computation, Jul
2020, Kalamata, Franc
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