Publications (4)0 Total impact
-
[show abstract]
[hide abstract]
ABSTRACT: In this paper, we generalize the algorithm described by Rump and Graillat, as
well as our previous work on certifying breadth-one singular solutions of
polynomial systems, to compute verified and narrow error bounds such that a
slightly perturbed system is guaranteed to possess an isolated singular
solution within the computed bounds. Our new verification method is based on
deflation techniques using smoothing parameters. We demonstrate the performance
of the algorithm for systems with singular solutions of multiplicity up to
hundreds.
12/2012;
-
[show abstract]
[hide abstract]
ABSTRACT: In this paper we describe how to improve the performance of the
symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the
multiplicity structure and refining approximate isolated singular solutions in
the breadth one case. By introducing a parameterized and deflated system with
smoothing parameters, we generalize the algorithm in (Rump and Graillat, 2009)
to compute verified error bounds such that a slightly perturbed polynomial
system is guaranteed to have a breadth-one multiple root within the computed
bounds.
01/2012;
-
[show abstract]
[hide abstract]
ABSTRACT: We present a symbolic-numeric method to refine an approximate isolated
singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, ..., \hat{x}_{n})$ of a
polynomial system $F=\{f_1, ..., f_n\}$ when the Jacobian matrix of $F$
evaluated at $\hat{\mathbf{x}}$ has corank one approximately. Our new approach
is based on the regularized Newton iteration and the computation of approximate
Max Noether conditions satisfied at the approximate singular solution. The size
of matrices involved in our algorithm is bounded by $n \times n$. The algorithm
converges quadratically if $\hat{\xx}$ is close to the isolated exact singular
solution.
07/2010;
-
[show abstract]
[hide abstract]
ABSTRACT: 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.
04/2010; 29:9-24.
