4,755 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
Computing Dynamic Output Feedback Laws
The pole placement problem asks to find laws to feed the output of a plant
governed by a linear system of differential equations back to the input of the
plant so that the resulting closed-loop system has a desired set of
eigenvalues. Converting this problem into a question of enumerative geometry,
efficient numerical homotopy algorithms to solve this problem for general
Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While
dynamic feedback laws offer a wider range of use, the realization of the output
of the numerical homotopies as a machine to control the plant in the time
domain has not been addressed before. In this paper we present symbolic-numeric
algorithms to turn the solution to the question of enumerative geometry into a
useful control feedback machine. We report on numerical experiments with our
publicly available software and illustrate its application on various control
problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly
available via http://www.math.uic.edu/~jan/download.htm
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
On Equivalence of Known Families of APN Functions in Small Dimensions
In this extended abstract, we computationally check and list the
CCZ-inequivalent APN functions from infinite families on for n
from 6 to 11. These functions are selected with simplest coefficients from
CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence
between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT
ASSOCIATION
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