Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
Numerische Mathematik
Matrix computations (3rd ed.)
A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
SIAM Journal on Optimization
Numerical Polynomial Algebra
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
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
Numerical local rings and local solution of nonlinear systems
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Computing the multiplicity structure from geometric involutive form
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Journal of Symbolic Computation
Computing the multiplicity structure of an isolated singular solution: Case of breadth one
Journal of Symbolic Computation
An improved method for evaluating Max Noether conditions: case of breadth one
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Verified error bounds for isolated singular solutions of polynomial systems: Case of breadth one
Theoretical Computer Science
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We present a symbolic-numeric method to refine an approximate isolated singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, \ldots, \hat{x}_{n})$ of a polynomial system $F=\{f_1, \ldots, f_n\}$, 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 differential 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 $\hat{\mathbf{x}}$ is close to the isolated exact singular solution.