Approximate square-free decomposition and root-finding of lll-conditioned algebraic equations
Journal of Information Processing
Computing singular solutions to polynomial systems
Advances in Applied Mathematics
Gröbner duality and multiplicities in polynomial system solving
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Approximate polynomial greatest common divisors and nearest singular polynomials
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Analysis of zero clusters in multivariate polynomial systems
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Detection and validation of clusters of polynomial zeros
Journal of Symbolic Computation - Special issue: validated numerical methods and computer algebra
Efficient algorithms for computing the nearest polynomial with constrained roots
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
On approximate GCDs of univariate polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Efficient algorithms for computing the nearest polynomial with a real root and related problems
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Newton's method for overdetermined systems of equations
Mathematics of Computation
Finding a cluster of zeros of univariate polynomials
Journal of Complexity
A multivariate Weierstrass iterative rootfinder
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Relations between roots and coefficients, interpolation and application to system solving
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
On approximate irreducibility of polynomials in several variables
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
The subresultant and clusters of close roots
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Computer algebra handbook
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
On Location and Approximation of Clusters of Zeros of Analytic Functions
Foundations of Computational Mathematics
Approximate radical of ideals with clusters of roots
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One
Foundations of Computational Mathematics
Computing the multiplicity structure from geometric involutive form
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Deflation and certified isolation of singular zeros of polynomial systems
Proceedings of the 36th international symposium on Symbolic and algebraic computation
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
Computing the multiplicity structure of an isolated singular solution: Case of breadth one
Journal of Symbolic Computation
Computing the nearest singular univariate polynomials with given root multiplicities
Theoretical Computer Science
Hi-index | 0.00 |
We present a symbolic-numeric technique to find the closest multivariate polynomial system to a given one which has roots with prescribed multiplicity structure. Our method generalizes the ''Weierstrass iteration'', defined by Ruatta, to the case when the input system is not exact, i.e. when it is near to a system with multiple roots, but itself might not have multiple roots. First, using interpolation techniques, we define the ''generalized Weierstrass map'', a map from the set of possible roots to the set of systems which have these roots with the given multiplicity structure. Minimizing the 2-norm of this map formulates the problem as an optimization problem over all possible roots. We use Gauss-Newton iteration to compute the closest system to the input with given root multiplicity together with its roots. We give explicitly an iteration function which computes this minimum. These results extends previous results of Zhi and Wu and results of Zeng from the univariate case to the multivariate case. Finally, we give a simplified version of the iteration function analogously to the classical Weierstrass iteration, which allows a component-wise expression, and thus reduces the computational cost of each iteration. We provide numerical experiments that demonstrate the effectiveness of our method.