Bezout Matrices Research Articles

Computing the topology of an arrangement of parametric or implicit algebraic curves in the Lagrange basis

The study of arrangements of geometric objects is a problem of great interest in combinatorial and computational geometry, and in CAGD, with applications in computer-aided manufacturing, geometric modeling, robotics, NC-machining, molecular modeling, etc. [2, 3] We present a new algebraic method for computing the topology of the arrangement of a finite family of planar algebraic curves. The curves are given either by their implicit equations or by polynomial parametrizations. As far as we know, none of the previous works on this problem include the treatment of parametrized curves. The input of the method is assumed to be given as Lagrange data, i.e., by the values of the polynomials defining the curves, at a suitable set of nodes. The output is a graph representing critical points of the curves, all the intersection points between curves, and segments joining some of these points isotopic to the arcs of the curves. The method follows a sweep-line algorithm. To find critical points, intersections and self-intersections we use Bezout matrices. We take advantage of the existence of a basis of the nullspace of a Bezout matrix that has a nice and useful structure (see [4]). We use algebra by values techniques, which means that all computations (Bezout matrices, polynomial derivatives, etc.) are done in the Lagrange basis [5, 1]. This approach reduces the computational cost of manipulating high degree polynomials, and avoids conversion between different basis, which is known to be a numerically unstable procedure. To compute the roots of the determinant of a Bezout matrix we apply linearization, i.e., we use a straightforward procedure to build (in the Lagrange basis) a companion matrix pencil whose generalized eigenvalues are the roots of the aforementioned determinant. The algorithm has been tested using Maple 17, showing that it is accurate and quite efficient.

The Cyclic Groups via Bezout Matrices

<p>In this paper, we define the Bezout matrices by the aid of the characteristic polynomials of the <em>k</em>-step Fibonacci, the generalized order-<em>k</em> Pell and the generalized order-<em>k</em> Jacobsthal sequences then we consider the multiplicative orders of the Bezout matrices when read modulo <em>m</em>. Consequently, we obtain the rules for the order of the cyclic groups by reducing the Bezout matrices modulo <em>m</em>.</p>

Roots of Difference Quotient Forms of Chebyshev Polynomials

AbstractLet ­
Œ Þ
–s be the Chebyshev polynomials of first kind of degree
Π. In this paper, we show that for
ðI , the polynomial with integer coefficients

˜a
­
Œ Þ
˜sa­
Œ Þ
s has all its roots in c
˜c=
 . Key words : Polynomials, Chebyshev Polynomials, Roots 1. Introduction 1 Let
„Þ
–s and
Þ
—s be polynomials in the single independent variables
– and
— with coefficients in the field
g of complex numbers. Cassels et al . [1,2] studied factorizations of
„Þ
–sa
Þ
—s as the polynomial in the pair of variables
– and
— . They also considered a trivial case when
„ and
are the same polynomial since
„Þ
–sa
„Þ
—s is divisible by
–a
— . In this case, obtaining the factors of the polynomial

–a
—
„Þ
–sa
„Þ
—s (1)is in general rather complicated. The polynomial of the form (1) also arises in Bezout matrices that have appeared in the literature for a long time. Given
“Þ
–sa


ˆa×
Œ
“
ˆ
–
ˆ Þ
“
Œ d×si
”Þ
–sa


ˆa×
Œ
”
ˆ
–
ˆ ,let
FÞ
–i
—sa

–a
—
“Þ
–s
”Þ
—sa
“Þ
—s
”Þ
–sa

A commutativity relation for submatrices of the inverse of a Sylvester matrix

We take the inverse of a Sylvester matrix of two coprime polynomials of degree m and study the family of m×m submatrices formed from consecutive columns of the bottom m rows. We prove that these matrices commute and in the course of the proof we find a similar relation for Bezoutian matrices related to the polynomials.

On the computation of matrices of traces and radicals of ideals

Let f 1 , … , f s ∈ K [ x 1 , … , x m ] be a system of polynomials generating a zero-dimensional ideal I , where K is an arbitrary algebraically closed field. We study the computation of “matrices of traces” for the factor algebra A ≔ K [ x 1 , … , x m ] / I , i.e. matrices with entries which are trace functions of the roots of I . Such matrices of traces in turn allow us to compute a system of multiplication matrices { M x i ∣ i = 1 , … , m } of the radical I . We first propose a method using Macaulay type resultant matrices of f 1 , … , f s and a polynomial J to compute moment matrices, and in particular matrices of traces for A . Here J is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when I has finitely many projective roots in P K m . We also extend previous results which work only for the case where A is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of A . Here we need the assumption that s = m and f 1 , … , f m define an affine complete intersection. This second method also works if we have higher-dimensional components at infinity. A new explicit description of the generators of I are given in terms of Bezoutians.

Division-free computation of subresultants using Bezout matrices

We present an algorithm to compute the subresultant sequence of two polynomials that completely avoids division in the ground domain, generalizing an algorithm given by Abdeljaoued et al. [J. Abdeljaoued, G. Diaz-Toca, and L. Gonzalez-Vega, Minors of Bezout matrices, subresultants and the parameterization of the degree of the polynomial greatest common divisor, Int. J. Comput. Math. 81 (2004), pp. 1223–1238]. We evaluate determinants of slightly manipulated Bezout matrices using the algorithm of Berkowitz. Although the algorithm gives worse complexity bounds than pseudo-division approaches, our experiments show that our approach is superior for input polynomials with moderate degrees if the ground domain contains indeterminates.

Block LU factorization of Hankel and Bezout matrices and Euclidean algorithm

Given two polynomials, this paper is devoted to describing the natural relation between the Euclidean algorithm and the block LU factorization of the Hankel and Bezout matrices associated to such polynomials.

Intersection and self-intersection of surfaces by means of Bezoutian matrices

The computation of intersection and self-intersection loci of parametrized surfaces is an important task in Computer Aided Geometric Design. We address these problems via four resultants with separated variables; two of them are specializations of general multivariate resultants and the two others are specializations of determinantal resultants. We give a rigorous study in these four cases and provide new formulas in terms of Bezoutian matrix.

The inverse problem for Krein orthogonal matrix functions

In the mid-fifties, in a seminal paper, M. G. Krein introduced continuous analogs of Szegő orthogonal polynomials on the unit circle and established their main properties. In this paper, we generalize these results and subsequent results that he obtained jointly with Langer to the case of matrix-valued functions. Our main theorems are much more involved than their scalar counterparts. They contain new conditions based on Jordan chains and root functions. The proofs require new techniques based on recent results in the theory of continuous analogs of resultant and Bezout matrices and solutions of certain equations in entire matrix functions.

Bernstein–Bezoutian matrices and curve implicitization

A new application of Bernstein–Bezoutian matrices, a type of resultant matrices constructed when the polynomials are given in the Bernstein basis, is presented. In particular, the approach to curve implicitization through Sylvester and Bézout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational parametric equations of the curve are expressed in the Bernstein basis, avoiding the basis conversion from the Bernstein to the power basis. The coefficients of the implicit equation are computed in the bivariate tensor-product Bernstein basis, and their computation involves the bidiagonal factorization of the inverses of certain totally positive matrices.

Computing curve intersection by means of simultaneous iterations

A new algorithm is proposed for computing the intersection of two plane curves given in rational parametric form. It relies on the Ehrlich–Aberth iteration complemented with some computational tools like the properties of Sylvester and Bezout matrices, a stopping criterion based on the concept of pseudo-zero, an inclusion result and the choice of initial approximations based on the Newton polygon. The algorithm is implemented as a Fortran 95 module. From the numerical experiments performed with a wide set of test problems it shows a better robustness and stability with respect to the Manocha–Demmel approach based on eigenvalue computation. In fact, the algorithm provides better approximations in terms of the relative error and performs successfully in many critical cases where the eigenvalue computation fails.

Structured matrix methods for CAGD: an application to computing the resultant of polynomials in the Bernstein basis

AbstractWe devise a fast fraction‐free algorithm for the computation of the triangular factorization of Bernstein–Bezoutian matrices with entries over an integral domain. Our approach uses the Bareiss fraction‐free variant of Gaussian elimination, suitably modified to take into account the structural properties of Bernstein–Bezoutian matrices. The algorithm can be used to solve problems in algebraic geometry that arise in computer aided geometric design and computer graphics. In particular, an example of the application of this algorithm to the numerical computation of the intersection points of two planar rational Bézier curves is presented. Copyright © 2005 John Wiley & Sons, Ltd.

Bernstein–Bezoutian matrices

Several computational and structural properties of Bezoutian matrices expressed with respect to the Bernstein polynomial basis are shown. The exploitation of such properties allows the design of fast algorithms for the solution of Bernstein–Bezoutian linear systems without never making use of potentially ill-conditioned reductions to the monomial basis. In particular, we devise an algorithm for the computation of the greatest common divisor (GCD) of two polynomials in Bernstein form. A series of numerical tests are reported and discussed, which indicate that Bernstein–Bezoutian matrices are much less sensitive to perturbations of the coefficients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the Bernstein and the power basis.

Barnett’s Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices

This article provides a new presentation of Barnett’s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated in a companion matrix as in the original Barnett’s presentation. Moreover, this presentation also allows us to compute the coefficients of the considered greatest common divisor in an easier way than in the original Barnett’s theorems.

Squarefree Decomposition of Univariate Polynomials Depending on a Parameter. Application to the Integration of Parametric Rational Functions

In this paper, we describe the application of a new version of Barnett’s method to the squarefree decomposition of a univariate polynomial with coefficients in K [ x ], x being a parameter andK a characteristic zero field. This new version of Barnett’s method uses Bezoutian matrices instead of matrices obtained from evaluating polynomials in a companion matrix and allows the determination of the squarefree decomposition parametrizing the gcd of the polynomial and its successive derivatives with respect to the main variable. The application of this parametric squarefree decomposition to the integration of parametric rational functions is also presented.

Polynomial Bezoutian matrix with respect to a general basis

We introduce a class of so-called polynomial Bezoutian matrices. The operator representation relative to a pair of dual bases and the generalized Barnett factorization formula for this kind of matrix are derived. An intertwining relation and generalized Bezoutian reduction via polynomial Vandermonde matrix are presented.

Generalized Resultants over Unirational Algebraic Varieties

In this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instance to blowing up varieties and to residual intersection problems. We recall the classical notion of resultant over a variety X. Then we extend it to varieties which are parameterized on a dense open subset and give new conditions for the existence of the resultant over these varieties. We prove that any maximal nonzero minor of the corresponding Bezoutian matrix yields a nontrivial multiple of the resultant. We end with some experiments.

A supplement to "The roles of Sylvester and Bezoutian matrices in the historical study of stability of linear discrete-time systems"

In this supplement, we present the stability criterion of Nour-Eldin, based on the Cauchy index and Hankel matrix arguments, and its reduced form, based on reduced Schur-Cohn-Fujiwara criterion. This simplification of the Nour-Eldin criterion is further elaborated by Anderson and Mansour [1991]. A related result, based on reduced Markov stability criterion, was obtained by Mansour and Anderson [1990].

A hybrid approach to the computation of the inertia of a parametric family of Bezoutians with application to some stability problems for bivariate polynomials

Given two polynomials with coefficients over K [ k], the associated Bezout matrix B( k) with entries over Z [ k] defines a parametric family of Bezout matrices with entries over K . It is intended in this paper to propose a hybrid approach for determining the inertia of B( k) for any value of k in some real interval. This yields an efficient solution to certain root-location problems for bivariate polynomials. We first develop a fast fraction-free method for computing an inverse triangular factorization of the Bezout matrix B( k) over the integral domain Z [ k]. In this way, we may easily compute the sequence { φ i ( k)} of the training principal minors of B( k). For almost any value k of k the associated sign sequence { sign(φ i( k))} specifies the inertia of B( k) . The function sign ( φ i ( k)) is final obtained by numerically computing rational approximations of the real zeros of φ i ( k) ϵ Z [k].

The roles of Sylvester and Bezoutian matrices in the historical study of stability of linear discrete-time systems

In this paper, a historical review of the stability criteria of linear discrete-time systems is presented. In this review the early pioneering works of Hermite, Schur, Cohn, and Fujiwara, and how these paved the way for modern development of other criteria are indicated. It is also mentioned that Sylvester and Bezoutian matrices are the key methods in such developments. Besides this review of early work, some new results and simplifications are brought to light. The paper ends with some important applications of the various stability criteria in diverse fields.