The singular value decomposition for polynomial systems

@inproceedings{Corless1995TheSV,
  title={The singular value decomposition for polynomial systems},
  author={Robert M Corless and Patrizia M. Gianni and Barry M. Trager and Stephen M. Watt},
  booktitle={ISSAC '95},
  year={1995}
}
This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coecients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD’s which gives exact results for interesting nearby problems, and give ecient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard’s u-resultant algorithm for the solution of overdetermined systems… 
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References

SHOWING 1-10 OF 38 REFERENCES
Computing selected solutions of polynomial equations
  • D. Manocha
  • Computer Science, Mathematics
    ISSAC '94
  • 1994
TLDR
The problem of finding roots is reduced to computing eigenvalues of a generalized companion matrix and the structure of the matrix is used to compute the solutions in the domain of interest only and the resulting algorithm is iterative in nature.
Approximate GCD and its application to ill-conditioned algebraic equations
An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations
A direct numerical method is proposed for the determination of all isolated zeros of a system of multivariate polynomial equations. By “polynomial combination”, the system is reduced to a special…
A matrix pencil based numerical method for the computation of the GCD of polynomials
The authors present a novel numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R(s), P/sub m,d/, of maximal degree d. It is based on a procedure…
Resolution des Systemes d'Equations Algebriques
Effective polynomial computation
  • R. Zippel
  • Mathematics, Computer Science
    The Kluwer international series in engineering and computer science
  • 1993
1. Euclid's Algorithm. 2. Continued Fractions. 3. Diophantine Equations. 4. Lattice Techniques. 5. Arithmetic Functions. 6. Residue Rings. 7. Polynomial Arithmetic. 8. Polynomial GCD's: Classical…
Quasi-GCD computations
A matrix pencil based numerical method for the computation of the GCD of polynomials
TLDR
The method defines the exact degree of GCD, works satisfactorily with any number of polynomials and evaluates successfully approximate solutions.
Algorithms for Intersecting Parametric and Algebraic Curves II: Multiple Intersections
TLDR
Elimination theory is used and the resultant of the equations of intersection are expressed as a matrix determinant and the algorithm for intersection is reduced to computing eigenvalues and eigenvectors of matrices.
Verification in Computer Algebra Systems
TLDR
This paper has attempted to demonstrate that the question of condition, i.e. of the sensitivity of results w.r.t. perturbations of data, may play a role in algebraic algorithms, even if they are carried out in rational arithmetic, and how these ideas may also lead to safe implementations ofgebraic algorithms in floating-point arithmetic.
...
1
2
3
4
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