Verified error bounds for isolated singular solutions of polynomial systems (Q2927825)

The article deals with the numerical approximation of isolated singular solutions of polynomial systems over the complex numbers.NEWLINENEWLINELet \(f_1,\dots,f_n\) be polynomials of \(\mathbb{C}[\mathbf{x}]:=\mathbb{C}[x_1,\dots,x_n]\), let \(F:=\{f_1,\dots,f_n\}\) and consider the system \(F(\mathbf{x})=\mathbf{0}\). Usually, isolated nonsingular solutions to the system \(F(\mathbf{x})=\mathbf{0}\) are approximated by means of the Newton method. On the other hand, there is a number of articles devoted to describe variants of the Newton method which restore quadratic convergence for the approximation of an isolated singular solution \(\hat{\mathbf{x}}\) of the system \(F(\mathbf{x})=\mathbf{0}\) (see, e.g., [\textit{A. Griewank}, SIAM Rev. 27, 537--563 (1985; Zbl 0598.65026)]). Algorithms which use certain regularized Newton iterations, based on the computation of differentials at a given approximation of an isolated singular solution, allow one to approximate the solution under consideration to the full machine precision (see, e.g., [\textit{B. H. Dayton} et al., Math. Comput. 80, No. 276, 2143--2168 (2011; Zbl 1242.65102)]). The general idea of adding differentials at the given approximation is called a deflation method.NEWLINENEWLINESince isolated singular solutions may be transformed into clusters of regular solutions by means of arbitrarily small perturbations, it is more difficult to verify that a polynomial system has a multiple root. In [Theor. Comput. Sci. 479, 163--173 (2013; Zbl 1291.65161)], the authors developed a verification method which permits to compute guaranteed error bounds for certain isolated singular solutions.NEWLINENEWLINEIn the paper under review, a further deflation method is proposed for the numerical approximation of isolated singular solutions of a system \(F(\mathbf{x})=\mathbf{0}\) as above. This variant is a modification of a method due to \textit{N. Yamamoto} [J. Inf. Process. 7, 16--21 (1984; Zbl 0564.65036)]. The new deflation method returns a regular and square augmented system in a number of steps which is bounded in terms of the local structure of the ideal generated by \(f_1,\dots, f_n\) at the isolated singular solution \(\hat{\mathbf{x}}\) under consideration. Then an algorithm for computing verified error bounds for a given isolated solution is described. If successful, the algorithm produces a ``slightly perturbed'' polynomial system having an isolated singular solution within the computed bounds.NEWLINENEWLINEThe paper ends by presenting examples which illustrate the performance of the proposed methods to deal with singular solutions of a few polynomial systems previously considered in the literature.