A point (x*, lambda*) is called a turning point of multiplicity p greater than or equal to 1 of the nonlinear system F(x, lambda) = 0, F:R-n x R-1 --> R-n, if rank partial derivative(x)F(x*, lambda*) = n - 1, rank[partial derivative(x)F(x*, lambda*) \ partial derivative(lambda)F(x*, lambda*)] = n, and if the Ljapunov-Schmidt reduced function has the normal form g(xi, mu) = +/- xi(p+1) +/- mu, g:R-1 x R-1 --> R-1. A minimally extended system F(x, lambda) = 0, f(x, lambda) = 0 is proposed for defining turning points of multiplicity p, where f:R-n x R-1 --> R-1 is a scalar function which is related to the pth order partial derivatives of g with respect to xi. When F depends on m less than or equal to p - 1 additional parameters alpha is an element of R-m, the system F(x, lambda, alpha) = 0 can be inflated by m + 1 scalar equations f(1)(x, lambda, alpha) = 0,...,f(m+1)(x, lambda, alpha) = 0. The functions f(i):R-n x R-1 x R-m --> R-1 depend on certain partial derivatives ofg with respect to xi where f(m+1) corresponds to f. The regular solution (x*, lambda*, alpha*) of the extended system of n + m + 1 equations delivers the desired turning paint (x*, lambda*). For numerically solving these systems, two-stage Newton-type methods are proposed, where only one LU decomposition of an (n f 1) x (n + 1) matrix and some backsubstitutions have to be preformed per iteration step if Gaussian elimination is used for solving the linear systems. Moreover, the methods require the computation of certain higher order partial derivatives of F with respect to low dimensional subspaces as well as derivatives of implicitly defined related functions. Both tasks are realized via computational differentiation, often also called automatic differentiation. For doing this the special structure of the higher order derivatives is exploited, and the problem is reformulated so that the Taylor coefficients technique of computational differentiation can efficiently be integrated into the algorithm. Some numerical tests show the behavior of the algorithms in case of turning points of multiplicity p = 2, 3.
