Content uploaded by Ronald N. Goldman
Author content
All content in this area was uploaded by Ronald N. Goldman on Oct 03, 2014
Content may be subject to copyright.
Transformations and Transitions from the Sylvester to the Bezout Resultant
Abstract
A simple matrix transformation linking the resultant matrices of Sylvester and Bezout is derived. This transformation matrix is then applied to generate an explicit formula for each entry of the Bezout resultant, and this entry formula is used, in turn, to construct an efficient recursive algorithm for computing all the entries of the Bezout matrix. Hybrid resultant matrices consisting of some columns from the Sylvester matrix and some columns from the Bezout matrix provide natural transitions from the Sylvester to the Bezout resultant, and allow as well the Bezout construction to be generalized to two polynomials of different degrees. Such hybrid resultants are derived here, employing again the transformation matrix from the Sylvester to the Bezout resultant. 1 Introduction Resultants play an important role in algorithmic algebraic geometry [Cox et al 1997], computer aided geometric design [Goldman et al 1984; De Montaudouin and Tiller 1984; Sederberg and Parry 1986; Manocha and Cann...
... In 1994 the first foundation of hybrid resultant was laid in [11], even though the construction was for certain class of polynomials. Another independent work is presented in [12], possibly the first construction of the general type of the system of polynomials. These hybrid constructions consider only the classical resultant formulation, sparse hybrid formulation is constructed due to the frequent appearance of such systems in many engineering applications [13,14]. ...
... Hybrid Resultant Matrix:-The foundation work for hybrid resultant was first introduced in [11], derived for certain class of the multivariate polynomials of multi graded type. Independently, in 1999 [12] had proposed another hybrid construction which possibly is the first construction that can be applied to more general class of system of polynomials. Apart from the classical hybrid resultant matrix, the sparse hybrid formulation was constructed, due to the frequent appearance of such systems in many engineering applications [13]. ...
Availability of computer algebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Groebner basis and Ritt-Wu method due to their high complexity and storage requirement. This paper focuses on the current resultant matrix formulations and investigates their ability or otherwise towards producing optimal resultant matrices. A determinantal formula that gives exact resultant or a formulation that can minimize the presence of extraneous factors in the resultant formulation is often sought for when certain conditions that it exists can be determined. We present some applications of elimination theory via resultant formulations and examples are given to explain each of the presented settings.
... Each term in these three sums is a Cayley expression for two univariate polynomials fg i ; h j g, fh i ; f j g, and ff i ; g j g; hence each term generates a Bezout matrix when written in matrix form De Montaudouin and Tiller 1984; Goldman et al 1984; Chionh et al 1998]. Therefore, each block F +u = B ;u contains three summations of Bezout matrices, where the three matrices interleave row by row. ...
... Explicit formulas for the entries of the Bezout matrix of two univariate polynomials are given in Goldman et al 1984]. More eecient methods for computing the entries of a Bezout resultant or the entries of a sum of Bezout resultants are described in Chionh et al 1998]. Since the blocks F k are Bezoutian, we can adopt these methods here to provide eecient algorithm for computing the entries of F k . ...
Dixon [1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m;n). The first technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn. The second approach uses Cayley's determinant device to form a more compact representation for the resultant as the determinant of a matrix of order 2mn. The third method employs a combination of Cayley's determinant device with Sylvester's dialytic method to build the resultant as the determinant of a matrix of order 3mn. Here relations between these three resultant formulations are derived and the structure of the transformations between these resultant matrices is investigated. In particular, it is shown that these transformation matrices all have similar, simple, upper triangular, block symmetric structures and the blocks themselves have elegant symmetry properties. Elementary entry formulas for the transformation matrices are...
... Recently a new class of determinant formulations for the resultant of two univariate polynomials has been discovered, consisting of matrices of all orders between n (Bezout) and 2n (Sylvester). These new representations are hybrid matrices, containing k columns from the Bezout matrix and 2n?2k truncated columns from the Sylvester matrix Weyman and Zelevinsky 1994; Sederberg et al 1997; Chionh et al 1998b]. When k = 0, these hybrids reduce to the usual Sylvester matrix, and when k = n, they reduce to the standard Bezout matrix. ...
A. L. Dixon [Proc. Lond. Math. Soc. 6, 49-69 (1908; JFM 40.0207.01)] describes three distinct homogeneous determinant representations for the resultant of three bivariate polynomials of bidegree (m,n). These Dixon resultants are the determinants of matrices of orders 6mn, 3mn and 2mn, and the entries of these matrices are respectively homogeneous of degrees 1, 2, and 3 in the coefficients of the original three polynomial equations. Here we mix and match columns from these three Dixon matrices to construct a large assortment of new hybrid determinant representations of orders ranging from 2mn to 6mn for the resultant of three bivariate polynomials of bidegree (m,n).
... Now recall that Bezout matrices are symmetric Goldman et al 1984; Chionh et al 1998b], so M f v is symmetric. Similarly, M g v , M h v are symmetric. ...
Resultants for bivariate polynomials are often represented by the determinants of very big matrices. Properly grouping the entries of these matrices into blocks is a very effective tool for studying the properties of these resultants. Here we derive a set of convolution identities relating the blocks of two Dixon bivariate resultant representations. 1 Introduction For three bivariate polynomials of bidegree (m; n), [Dixon 1908] describes three distinct homogeneous determinant representations for the resultant. The first formulation, generated by Sylvester's dialytic method, is a 6mn Theta 6mn determinant. All the entries of 1 this determinant are of degree 1 in the coefficients of the original three polynomials. The second formulation, generated by Cayley's determinant device, is a 2mn Theta 2mn determinant whose entries are of degree 3 in the coefficients of the three original polynomials. The third formulation, obtained from a combination of Cayley's determinant device and Sylve...
Objective: To evaluate and compare different hybrid resultant formulations in relation to computational complexity, performance and optimality condition. Methods/Statistical Analysis: Hybrid matrices are evaluated using computer algebra system. Findings: we have shown that, none of the hybrid formulation works well with the exception of. However, after deleting the zero rows and columns the resulting matrix may not be a square matrix, on the other hand, we studied and established that none of the hybrid formulation produces a square matrix in general and likewise predicted that, the density or sparseness of the polynomial equations does not influence the performance of these hybrid matrix formulations. Applications/Improvements: This comparison reveals that, the existing hybrid methods for computing resultant are not efficient and therefore there is need for another formulations with will focus on the current limitations described in this paper.
Timing drive systems of recent passenger car engines are usually equipped with chain or synchronous belt drives. The comparatively low stiffness of chains and belts, combined with the large moments of inertia, resulting from moving parts of camshafts and valvetrains, lead to natural frequencies in the frequency range between 80 and 250 Hz for the entire timing drive [1]. Obviously, engine orders dominating the noise excitation are in the same frequency range, and resonance effects cannot be avoided. Furthermore, a guarantee is requested that occurring vibration amplitudes do not exceed acceptable limits. Hence, the predicted results of the primary dynamics of the timing drive are key data for the design analysis of a combustion engine. Besides the low frequency range excitation determined by main engine orders, higher-order structure-borne noise excitation is gaining more importance. This is particularly true for chain-driven systems where specific effects lead to considerable excitation of the engine structure in a higher frequency range (typical whine noise). The excitations are caused by the polygon effect, meshing impacts between chain links and sprockets and impacts in the engagement and disengagement between chain, sprockets and guides. This paper presents a comprehensive multi-body dynamics (MBD) model and the relevant simulation environment, which takes into account the entire timing drive as a fully coupled system. While simple MBD models (e.g. longitudinal span representation) are sufficient for the prediction of the primary dynamics of the timing drive, usually the analysis of higher-frequency dynamics and structural excitation requires a high degree of discretization of drive components to ensure a high-quality representation of the physical behaviour. This leads to a discrete model of chains and belts. Each link or tooth is taken into account separately by its mass properties, viscoelastic characteristics and by its contacts to sprockets, pulleys and guides. This paper describes the mathematical modelling of relevant engine components and the relevant interaction of different force elements. In particular, the consideration of the elastic contact between bodies represented by their real surface shapes is outlined. Finally, simulation results and their comparison with experimental data are presented for a chain-driven system, with discussion of low-frequency primary dynamics and higher-frequency structural excitation. Discrete models have also been developed for timing belts, but their verification with measurements is not completely finished. Hence, this paper only gives an overview of modelling issues.
Computational methods for manipulating sets of polynomial equations are becoming of greater importance due to the use of polynomial in various applications. Dixon resultant algorithm provides one of the most efficient methods for solving the system of polynomial equations or eliminating variables. When computing Dixon resultant, we first construct the Dixon polynomial for input polynomial system. However, as the entries in the determinant are symbolic, large intermediate symbolic expression is generated and leads to computer algebra systems run for a long time or crash. To avoid the intermediate expression swell, we propose using Zippel's multivariate probabilistic interpolation to compute Dixon polynomial. At first, the method of truncated formal power series, which converts the operation of division to multiplication, is used to preprocess the expression of Dixon polynomial. Secondly, Dixon polynomial is interpolated heuristically by Zippel's method. In order to solve the linear equation effectively in Zippel's method, Kaltofen, E.'s efficient algorithm[4] is introduced to solve the equations of transposed Vandermonde system in interpolation. Besides these, we combine Zippel's multivariate method with Lagrange interpolation so as to take advantage of the sparsity of polynomial system sufficiently. Intermediate symbolic expressions can be reduced because the algorithm converts the symbolic computation to the numerical.
This article is devoted to presenting new expressions for Subresultant Polynomials, written in terms of some minors of matrices different from the Sylvester matrix. Moreover, via these expressions, we provide new proofs for formulas which associate the Subresultant polynomials and the roots of the two polynomials. By one hand, we present a new proof for the formula introduced by J. J. Sylvester in 1839, formula written in terms of a single sum over the roots. By other hand, we introduce a new expression in terms of the roots by considering the Newton basis.
We present an algorithm to compute the subresultant sequence of two polynomials that completely avoids division in the ground domain, generalizing an algorithm \revised{given by} Abdeljaoued et al.\ (see Abdeljaoed et al.: Minors of Bezout Matrices\ldots, Int.\ J.\ of Comp.\ Math.\ 81, 2004). 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.
Dixon [1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m;n). The first technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn. The second approach uses Cayley's determinant device to form a more compact representation for the resultant as the determinant of a matrix of order 2mn. The third method employs a combination of Cayley's determinant device with Sylvester's dialytic method to build the resultant as the determinant of a matrix of order 3mn. Here relations between these three resultant formulations are derived and the structure of the transformations between these resultant matrices is investigated. In particular, it is shown that these transformation matrices all have similar, simple, upper triangular, block symmetric structures and the blocks themselves have elegant symmetry properties. Elementary entry formulas for the transformation matrices are...
A functionF(x,y,t)that assigns to each parametertan algebraic curveF(x,y,t)=0is called a moving curve. A moving curveF(x,y,t)is said to follow a rational curvex=x(t)/w(t),y=y(t)/w(t)ifF(x(t)/w(t), y(t)/w(t),t)is identically zero. A new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2nwith no base points the method of moving conics generates the implicit equation as the determinant of ann×nmatrix, where each entry is a quadratic polynomial inxandy, whereas standard resultant methods generate the implicit equation as the determinant of a 2n× 2nmatrix where each entry is a linear polynomial inxandy. Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.
This paper shows that the ideal of any degree n planar rational curve can be generated by two polynomials that are each linear in x, y and degree n1 and n2 (⩾ n1) in t, n1 + n2 = n. The value of n1 is fixed for a given rational curve, and serves to split all degree n curves into equivalence classes. These classes bear on the determinantal form of the implicit equation of the rational curve.
Many current geometric modeling systems use the rational parametric form to represent surfaces. Although the parametric representation is useful for tracing, rendering and surface fitting, many operations like surface intersection desire one of the surfaces to be represented implicitly. Moreover, the implicit representation can be used for testing whether a point lies on the surface boundary and to represent an object as a semi-algebraic set. Previously resultants and Gröbner bases have been used to implicitize parametric surfaces. In particular, different formulations of resultants have been used to implicitize tensor product surfaces and triangular patches and in many cases the resulting expression contains and extraneous factor. The separation of these extraneous factors can be a time consuming task involving multivariate factorization. Furthermore, these algorithms fail altogether if the given parametrization has base points.In this paper we present an algorithm to implicitize parametric surfaces. One of the advantages of the algorithm is that we do not need multivariate factorization. If a parametrization has no base points, the implicit representation can be represented as a determinant of a matrix, otherwise we use perturbation techniques. In the latter case we make use of GCD operation to compute the implicit representation. We also describe an efficient implementation of the algorithm based on interpolation techniques.
This paper treats the problem of how one can best compute the points of intersection of two planar rational curves. Three different algorithms are compared: the well known Bezier subdivision algorithm, a subdivision algorithm based on interval arithmetic, and the implicitization approach. Implementation considerations are discussed, with particular focus on how to make the implicitization method robust and fast. Report is made on a test in which the algorithms solved hundreds of randomly generated problems to eight digits of accuracy. The implicitization algorithm was faster than the others by a factor of five for degree two curves; by a factor of four for cubic curves; by a factor of three for quartic curves; and the interval method was faster for quintic curves by a factor of two.
A thorough treatment of the Cayley method of elimination is given, with particular emphasis on its implementation to solve problems in computational geometry. An easily programmable formula for the Cayley matrix is derived, it is shown how to extract the birational correspondence from the Cayley matrix, numerous examples are worked, and the connection between the Cayley method and the Euclidean Algorithm is explained.
In this paper vector techniques and elimination methods are combined to help resolve some classical problems in computer aided geometric design. Vector techniques are applied to derive the Bezout resultant for two polynomials in one variable. This resultant is then used to solve the following two geometric problems: Given a planar parametric rational polynomial curve, (a) find the implicit polynomial equation of the curve (implicitization); (b) find the parameter value(s) corresponding to the coordinates of a point known to lie on the curve (inversion). The solutions to these two problems are closed form and, in general, require only the arithmetic operations of addition, subtraction, multiplication, and division. These closed form solutions lead to a simple, non-iterative, analytic algorithm for computing the intersection points of two planar parametric rational polynomial curves. Extensions of these techniques to planar rational Bezier curves are also discussed.