A quadratically convergent method for computing simple singular roots and its application to determining simple bifurcation points

We present an efficient method for computing roots of mappings on ? ⁿ in the case where the Jacobian has the rankn?1 at the root. For the accurate determination of such a rootx*∈? ⁿ an auxiliary system ofn equations inn+1 variables is constructed which possesses (x ^*, 1) as a turning point. This turning point can be computed by direct methods. We use an adapted method which requires only the solution of (n+1)-dimensional systems of linear equations and the evaluation of one Jacobian and 5 function values per step. This techniques is successfully applied to compute simple bifurcation points by means of a suitable system of nonlinear equations which has the properties mentioned above.