Computing isolated singular solutions of polynomial systems: case of breadth one (Q2882346)

Let us consider an approximate singular solution \(\hat{\eta}\) of a polynomial system \(F=\{f_1, \dots, f_n\}\) satisfying \(\|\hat \eta- \hat \eta_{{e}}\|=\varepsilon\), where \(\hat{\eta}_{{e}}\) denotes the isolated exact singular solution of \(F\), the positive number \(\varepsilon\) is small enough such that there are no other solutions of \(F\) nearby, and assume that the corank of the Jacobian matrix \(F'(\hat{\eta}_{{e}})\) is one. In view to restore the quadratic convergence of the Newton method, the authors apply one regularized Newton iteration to obtain a new approximation \(\hat \eta+\hat \theta\), and then compute the approximate null vector \(\vartheta_1\) of the Jacobian \(F'(\hat \eta+\hat \theta)\) which gives a generalized Newton direction, and the step length \(\delta\) is obtained by solving a linear system. They show that NEWLINE\[NEWLINE\|\hat \eta+\hat \theta+ \delta \vartheta_1- \hat \eta_{{e}}\| = O(\varepsilon^2).NEWLINE\]NEWLINE The size of matrices involved in the algorithm is bounded by \(n\times n\), and the method has been implemented in Maple.