Hybrid method for computing the nearest singular polynomials (original) (raw)
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The nearest polynomial with a given zero, revisited
ACM SIGSAM Bulletin, 2005
In his 1999 SIGSAM BULLETIN paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter's formula and explicitly extending the results to different polynomial bases.Experiments with our implementation demonstrate that the formula may not after all, fully solve the problem,
The singular value decomposition for approximate polynomial systems
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 modified Lazard's method to the location of singular points on approximately known projections of algebraic curves.
Nearest common root of a set of polynomials: A structured singular value approach
Linear Algebra and its Applications, 2020
The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (µ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an "approximate GCD" of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the "approximate GCD" of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations.
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Advances in Applied Mathematics, 1992
A method to generate accurate approximations to the singular solutions of a system of (complex) polynomial equations is presented. This method is established in a context of polynomial continuation; thus, all solutions are generated, with the singular solutions being approximated more accurately than by standard implementations. The theorem on which the method is based is proven using results from several complex variables and algebraic geometry. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at solutions, are required. A specific implementation is given and the results of numerical experiments in solving four test problems are presented. 0 1992 Academic Press, Inc.
An analytical method for computing the coefficients of the trivariate polynomial approximation
2013
This paper deals with computing the coefficients of the trivariate polynomial approximation (TPA) of large distinct points given on . 3 The TPA is formulated as a matrix equation using Kronecker and Khatri-Rao products of the matrices. The coefficients of the TPA are computed using the generalized inverse of the matrix. It is seen that the trivariate polynomial approximation can be investigated as the matrix equation and the coefficients of the TPA can be computed directly from the solution of the matrix equation.
Some higher-order methods for the simultaneous approximation of multiple polynomial zeros
Computers & Mathematics with Applications, 1986
Applying Newton's and Halley's corrections, some modified methods of higher order for the simultaneous approximation of multiple zeros of a polynomial are derived. Further acceleration of convergence of these methods is performed by approximating all zeros in a serial fashion using the new approximations as they become available. Faster convergence is attained without additional calculations. Lower bounds of the R-order of convergence for the serial (single-step) methods are given.
IMA Journal of Mathematical Control and Information, 2013
In this paper the following problem is considered: Given two co-prime polynomials, find the smallest perturbation in the magnitude of their coefficients, such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed. A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree. Finally, problems involving the calculation of the approximate greatest common divisor (GCD) of univariate polynomials are considered, by proposing a generalization of the definition of the structured singular value involving additional rank constraints.
We present a symbolic-numeric method to refine an approximate isolated singular solutio x of a polynomial system F when the Jacobian matrix of F evaluated a x has corank one. Our new approach is based on the regularized Newton iteration and the computation of Max Noether conditions satisfied at the singular solution. The method has been implemented in Maple and can deal with regular singularities and irregular singularities. For multiplicity being 2 or 3, we prove the quadratical convergence of our algorithm. Numerical experiments show that the new algorithm converges quadratically for arbitrary large multiplicity.
Approximation Methods for Polynomial Optimization
SpringerBriefs in Optimization, 2012
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