Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen

Summary

For 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 (∂₁ G(x, t)⋮∂₂ G(x, t)) has full rankn along ℭ the method is proven to terminate with a solution (x ^N, 1) of the system fort=1. It works succesfully, too, if the Jacobians ∂₁ G(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.