We discuss a pathfollowing strategy based on pseudo-arclength
parametrization for the solution of parameter-dependent
boundary value problems for ordinary differential equations.
We formulate criteria which ensure the successful application of this method
for the computation of solution branches with turning points
for problems with an essential singularity. Finally, we demonstrate
that a Matlab implementation of the solution method based on
an adaptive collocation scheme is well suited to solve problems
of practical relevance. As one example, we compute solution branches
for the complex Ginzburg-Landau equation which start from multi-bump
solutions of the nonlinear Schrödinger equation. Following the
branches around turning points, real-valued solutions of the nonlinear
Schrödinger equation can easily be computed.