48,756 research outputs found
GPGCD: An iterative method for calculating approximate GCD of univariate polynomials
We present an iterative algorithm for calculating approximate greatest common
divisor (GCD) of univariate polynomials with the real or the complex
coefficients. For a given pair of polynomials and a degree, our algorithm finds
a pair of polynomials which has a GCD of the given degree and whose
coefficients are perturbed from those in the original inputs, making the
perturbations as small as possible, along with the GCD. The problem of
approximate GCD is transfered to a constrained minimization problem, then
solved with the so-called modified Newton method, which is a generalization of
the gradient-projection method, by searching the solution iteratively. We
demonstrate that, in some test cases, our algorithm calculates approximate GCD
with perturbations as small as those calculated by a method based on the
structured total least norm (STLN) method and the UVGCD method, while our
method runs significantly faster than theirs by approximately up to 30 or 10
times, respectively, compared with their implementation. We also show that our
algorithm properly handles some ill-conditioned polynomials which have a GCD
with small or large leading coefficient.Comment: Preliminary versions have been presented as
doi:10.1145/1576702.1576750 and arXiv:1007.183
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
The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials
We present a general, asymptotical solution for the discretised harmonic
oscillator. The corresponding Schr\"odinger equation is canonically conjugate
to the Mathieu differential equation, the Schr\"odinger equation of the quantum
pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian
of an isolated Josephon junction or a superconducting single-electron
transistor (SSET), we obtain an asymptotical representation of Mathieu
functions. We solve the discretised harmonic oscillator by transforming the
infinite-dimensional matrix-eigenvalue problem into an infinite set of
algebraic equations which are later shown to be satisfied by the obtained
solution. The proposed ansatz defines a new class of generalised Hermite
polynomials which are explicit functions of the coupling parameter and tend to
ordinary Hermite polynomials in the limit of vanishing coupling constant. The
polynomials become orthogonal as parts of the eigenvectors of a Hermitian
matrix and, consequently, the exponential part of the solution can not be
excluded. We have conjectured the general structure of the solution, both with
respect to the quantum number and the order of the expansion. An explicit proof
is given for the three leading orders of the asymptotical solution and we
sketch a proof for the asymptotical convergence of eigenvectors with respect to
norm. From a more practical point of view, we can estimate the required effort
for improving the known solution and the accuracy of the eigenvectors. The
applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression
Counting Solutions of a Polynomial System Locally and Exactly
We propose a symbolic-numeric algorithm to count the number of solutions of a
polynomial system within a local region. More specifically, given a
zero-dimensional system , with
, and a polydisc
, our method aims to certify the existence
of solutions (counted with multiplicity) within the polydisc.
In case of success, it yields the correct result under guarantee. Otherwise,
no information is given. However, we show that our algorithm always succeeds if
is sufficiently small and well-isolating for a -fold
solution of the system.
Our analysis of the algorithm further yields a bound on the size of the
polydisc for which our algorithm succeeds under guarantee. This bound depends
on local parameters such as the size and multiplicity of as well
as the distances between and all other solutions. Efficiency of
our method stems from the fact that we reduce the problem of counting the roots
in of the original system to the problem of solving a
truncated system of degree . In particular, if the multiplicity of
is small compared to the total degrees of the polynomials ,
our method considerably improves upon known complete and certified methods.
For the special case of a bivariate system, we report on an implementation of
our algorithm, and show experimentally that our algorithm leads to a
significant improvement, when integrated as inclusion predicate into an
elimination method
Computing the common zeros of two bivariate functions via Bezout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (� 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions
Orthogonal polynomials and Riesz bases applied to the solution of Love's equation
In this paper we reinvestigate the structure of the solution of a well-known Love’s problem, related to the electrostatic field generated by two circular coaxial conducting disks, in terms of orthogonal polynomial expansions, enlightening the role of the recently introduced class of the Lucas–Lehmer polynomials. Moreover we show that the solution can be expanded more conveniently with respect to a Riesz basis obtained starting from Chebyshev polynomials
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
Improved Complexity Bounds for Counting Points on Hyperelliptic Curves
We present a probabilistic Las Vegas algorithm for computing the local zeta
function of a hyperelliptic curve of genus defined over . It
is based on the approaches by Schoof and Pila combined with a modeling of the
-torsion by structured polynomial systems. Our main result improves on
previously known complexity bounds by showing that there exists a constant
such that, for any fixed , this algorithm has expected time and space
complexity as grows and the characteristic is large
enough.Comment: To appear in Foundations of Computational Mathematic
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