Computation of singular points

A solution point (x*, lambda*, alpha*) of F(x, lambda, alpha) = 0 where F : R-n X R-p X R-k --> R-m : (x, lambda, alpha) F(x, lambda, alpha) is called a singular point if rank partial derivative(x)F(x*, lambda*, alpha*) = m - q with max{1, m - n + 1} less than or equal to q less than or equal to 5 m. For many types of singular points we can find an extended system F(x, lambda, alpha) + D mu = 0, f (x, lambda, alpha) = 0 with D epsilon R-mxl, mu epsilon R-l and l = m - rank partial derivativeF(x*, lambda*, alpha*) greater than or equal to 0 such that (x*, lambda*, alpha*, 0) is a regular solution of this system. The function f depends on the type of the singular point and contains derivatives of implicitly defined auxiliary functions. The singular points can be computed solving the extended system using Newton-type methods. All derivatives needed can be computed by computational differentiation and solving some linear systems with the same coefficient matrix in each step.