94,472 research outputs found
New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less superfluous factors. In this thesis, three different techniques for computing the resultant matrix are presented, namely the Jouanolou-Jacobian method, the Dixon-Jouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The Dixon-Jouanolou method is proposed based on the existing Jouanolou matrix method which is subjected to bivariate systems. To further extend this method to multivariate systems, the entry formula for computing the Dixon resultant matrix is first generalized. This extended application of the loose entry formula leads to the possibility of generalizing the Dixon-Jouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the Dixon-Jouanolou method on systems of polynomials over the affine and projective space, respectively, the concept of pseudohomogenization is introduced. Each space is subjected to its respective conditions; thus, pseudo-homogenization serves as a bridge between them by introducing an artificial variable. From the computing time analysis of the generalized loose entry formula used in the computation of the Dixon matrix entries, it is shown that the method of computing the Dixon matrix using this approach is efficient even without the application of parallel computations. These results show that the cost of computing the Dixon matrix can be reduced based on the number of additions and multiplications involved when applying the loose entry formula. These improvements can be more pronounced when parallel computations are applied. Further analyzing the results of the hybrid Dixon-Jouanolou construction and implementation, it is found that the Dixon-Jouanolou method had performed with less computational cost with cubic running time in comparison with the running time of the standard Dixon method which is quartic. Another independent construction produced in this thesis is the Jouanolou- Jacobian method which is an improvement of the existing Jacobian method since it avoids multi-polynomial divisions. The Jouanolou-Jacobian method is also able to produce a considerably smaller resultant matrix compared to the existing Jacobian method and is therefore less computationally expensive. Lastly all the proposed methods have considered a systematic way of detecting and removing extraneous factors during the computation of the resultant matrix whose determinant gives the polynomial resultant
New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less superfluous factors. In this thesis, three different techniques for computing the resultant matrix are presented, namely the Jouanolou-Jacobian method, the Dixon-Jouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The Dixon-Jouanolou method is proposed based on the existing Jouanolou matrix method which is subjected to bivariate systems. To further extend this method to multivariate systems, the entry formula for computing the Dixon resultant matrix is first generalized. This extended application of the loose entry formula leads to the possibility of generalizing the Dixon-Jouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the Dixon-Jouanolou method on systems of polynomials over the affine and projective space, respectively, the concept of pseudohomogenization is introduced. Each space is subjected to its respective conditions; thus, pseudo-homogenization serves as a bridge between them by introducing an artificial variable. From the computing time analysis of the generalized loose entry formula used in the computation of the Dixon matrix entries, it is shown that the method of computing the Dixon matrix using this approach is efficient even without the application of parallel computations. These results show that the cost of computing the Dixon matrix can be reduced based on the number of additions and multiplications involved when applying the loose entry formula. These improvements can be more pronounced when parallel computations are applied. Further analyzing the results of the hybrid Dixon-Jouanolou construction and implementation, it is found that the Dixon-Jouanolou method had performed with less computational cost with cubic running time in comparison with the running time of the standard Dixon method which is quartic. Another independent construction produced in this thesis is the Jouanolou- Jacobian method which is an improvement of the existing Jacobian method since it avoids multi-polynomial divisions. The Jouanolou-Jacobian method is also able to produce a considerably smaller resultant matrix compared to the existing Jacobian method and is therefore less computationally expensive. Lastly all the proposed methods have considered a systematic way of detecting and removing extraneous factors during the computation of the resultant matrix whose determinant gives the polynomial resultant
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
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
An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes
We develop an incremental algorithm to compute the Newton polytope
of the resultant, aka resultant polytope, or its
projection along a given direction.
The resultant is fundamental in algebraic elimination and
in implicitization of parametric hypersurfaces.
Our algorithm exactly computes vertex- and halfspace-representations
of the desired polytope using an oracle producing resultant vertices in a
given direction.
It is output-sensitive as it uses one oracle call per vertex.
We overcome the bottleneck of determinantal predicates
by hashing, thus accelerating execution from to times.
We implement our algorithm using the experimental CGAL package {\tt
triangulation}.
A variant of the algorithm computes successively tighter inner and outer
approximations: when these polytopes have, respectively,
90\% and 105\% of the true volume, runtime is reduced up to times.
Our method computes instances of -, - or -dimensional polytopes
with K, K or vertices, resp., within hr.
Compared to tropical geometry software, ours is faster up to
dimension or , and competitive in higher dimensions
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