## The Singular Value Decomposition for Polynomial Systems (1995)

Citations: | 81 - 9 self |

### BibTeX

@INPROCEEDINGS{Corless95thesingular,

author = {Robert M. Corless and Patrizia M. Gianni and Barry M. Trager and Stephen M. Watt},

title = {The Singular Value Decomposition for Polynomial Systems},

booktitle = {},

year = {1995},

pages = {195--207},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients 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 efficient 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 of polynomial equations to the inexact-coefficient case. We also briefly discuss an application of the modied Lazard's method to the location of singular points on approximately known projections of algebraic curves.

### Citations

4668 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
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83 |
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- 1988
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- 1995
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34 |
T.: Approximate GCD and its applications to ill-conditioned algebraic equations
- Noda, Sasaki
- 1991
(Show Context)
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16 |
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5 |
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- 1992
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3 |
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- Stetter
- 1993
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2 |
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- Seppl, Silhol
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1 |
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- Bunse-Gerstner, Byers, et al.
- 1993
(Show Context)
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1 |
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- Corless, Jeoerey, et al.
- 1989
(Show Context)
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1 |
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1 |
iResolution des systemes d'equations algebriquesj, Theoretical Computer Science 15
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- 1981
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Citation Context ...ok at the solution of possibly overdetermined homogeneous multivariate problems with only nitely many solutions at innity. To do this we provide a constructive reformulation of an algorithm of Lazard =-=[12]-=-, changing his determinantal algorithm to a generalized eigenvalue problem. This work is similar to that in [2, 13, 14, 15, 16, 20, 21], but dioeers in that the method of this paper can handle the ove... |

1 |
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- 1994
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1 |
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1 | iQuasi-GCD - Schnhage - 1981 |

1 |
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- Stetter
- 1993
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1 |
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- Stewart
- 1978
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Citation Context ...envector and x i is a right eigenvector corresponding to the i-th eigenvalue. Similarly s i = y T i M1x i and t i = y T i M2x i . These formulas may also be arrived at by standard perturbation theory =-=[22]-=-, and indeed that is how we rst found them. These formulas can be expressed succinctly as the Rayleigh quotient formula: r ij = y T i M j x i . If all of these quantities are small, then the root is i... |

1 |
iA rst report on the A ] compiler,j
- Watt, Broadbery, et al.
- 1994
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Citation Context ...gorithms can be carried out in purely numerical subroutines, perhaps in FORTRAN, and if the computer algebra system can take suOEcient advantage of connections to good numerical libraries, as A ] can =-=[24]-=-, the actual performance of this approach may not be bad at all. It is possible that some of the ideas presented here may be adapted to improve instead, say, Noda and Sasaki's algorithm (or at least p... |

1 |
iThe Perdious Polynomialj
- Wilkinson
- 1984
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Citation Context ...oblem as a generalized eigenvalue problem. This avoids formation of the determinant as a polynomial in u, v, and w to begin with, which is well-known to induce an instability in the rootnding process =-=[25]-=-. In eoeect, we will be generalizing the companion matrix method for nding roots of univariate polynomials: we will replace the polynomial rootnding problem with a (generalized) eigenvalue problem. Ph... |