Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen

SummaryFor solving the nonlinear systemG(x, t)=0,G|ℝn × ℝ1→ℝn, which is assumed to have a smooth curve ℭ of solutions a continuation method with self-choosing stepsize is proposed. It is based on a PC-principle using an Euler-Cauchy-predictor and Newton's iteration as corrector. Under the assumption thatG is sufficiently smooth and the total derivative (∂1G(x, t)⋮∂2G(x, t)) has full rankn along ℭ the method is proven to terminate with a solution (xN, 1) of the system fort=1. It works succesfully, too, if the Jacobians ∂1G(x, t) become singular at some points of ℭ, e.g., if ℭ has turning points. The method is especially able to give a point-wise approximation of the curve implicitly defined as solution of the system mentioned above.