Computing multiple turning points by using simple extended systems and computational differentiation

A point (x *,λ*) is called a turning point of multiplicity p ≥1 of the nonlinear system if and if the Ljapunov–Schmidt reduced function has the normal form . A minimally extended system F(x, λ)=0 F(x, λ)=0 is proposed for defining turning points of multiplicity p, where is a scalar function which is related to the pth order partial derivatives of g with respect to ξ. When Fdepends on m≤p-1 additional parameters the system F(x, λ α)=0 can be inflated by m + 1 scalar equations f 1(x, λ α)=0,…f m+1(x, λ α)=0 The functions depend on certain partial derivatives of gwith respect to ξ where f m+1 corresponds to f The regular solution (x *, λ *,α *) of the extended system of n+m+1 equations delivers the desired turning point (x *, λ*). For numerically solving these systems, two-stage New tonype methods are proposed, where only one LU decomposition of an (n+1) ×(++1) matrix and some back substitutions have to be preformed per iteration step if Gaussian elimination is used for solving the linear systems. Moreover, ...