462 research outputs found
Generalized companion matrix for approximate GCD
We study a variant of the univariate approximate GCD problem, where the
coefficients of one polynomial f(x)are known exactly, whereas the coefficients
of the second polynomial g(x)may be perturbed. Our approach relies on the
properties of the matrix which describes the operator of multiplication by gin
the quotient ring C[x]=(f). In particular, the structure of the null space of
the multiplication matrix contains all the essential information about GCD(f;
g). Moreover, the multiplication matrix exhibits a displacement structure that
allows us to design a fast algorithm for approximate GCD computation with
quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201
Recommended from our members
Numerical and Symbolical Methods for the GCD of Several Polynomials
The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccuracies in the input data and produce meaningful approximate results. Therefore, we describe and compare numerically and symbolically methods such as the ERES, the Matrix Pencil and other resultant type methods, with respect to their complexity and effectiveness. The combination of numerical and symbolic operations suggests a new approach in software mathematical computations denoted as hybrid computations. This combination offers great advantages, especially when we are interested in finding approximate solutions. Finally the notion of approximate GCD is discussed and a useful criterion estimating the strength of a given approximate GCD is also developed
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving
Univariate polynomial root-finding is both classical and important for modern
computing. Frequently one seeks just the real roots of a polynomial with real
coefficients. They can be approximated at a low computational cost if the
polynomial has no nonreal roots, but typically nonreal roots are much more
numerous than the real ones. We dramatically accelerate the known algorithms in
this case by exploiting the correlation between the computations with matrices
and polynomials, extending the techniques of the matrix sign iteration, and
exploiting the structure of the companion matrix of the input polynomial. We
extend some of the proposed techniques to the approximation of the real
eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm
Selecting Algorithms for Black Box Matrices: Checking for Matrix Properties That Can Simplify Computations
Processes to automate the selection of appropriate algorithms for various
matrix computations are described. In particular, processes to check for, and
certify, various matrix properties of black box matrices are presented. These
include sparsity patterns and structural properties that allow "superfast"
algorithms to be used in place of black-box algorithms. Matrix properties that
hold generically, and allow the use of matrix preconditioning to be reduced or
eliminated, can also be checked for and certified - notably including in the
small-field case, where this presently has the greatest impact on the
efficiency of the computation.Comment: Department of Computer Science Technical Report 2016-1085-0
- …