On the computation of multi-dimensional solution manifolds of parametrized equations (Q1122945)

The author considers the numerical treatment of a system \(F(x)=0\) where F is a nonlinear mapping from \({\mathbb{R}}^ n\) to \({\mathbb{R}}^ m\) with \(p:=n- m>1\). The p-dimensional manifold of regular solutions is being computed under mild natural assumptions. The basis of the continuation algorithm is the construction of a local coordinate system in the tangent space which is smoothly changed along the manifold (moving frame algorithm). This coordinate system defines a local net of gridpoints in the tangent space which comes as a projection from a master grid in \({\mathbb{R}}^ p\). A Gauss-Newton process projects the grid points in the tangent space onto the manifold. A basic operation of the algorithm is a QR-factorization of \(DF(x)^ T\). The paper contains many details of the actual numerical processes involved. It ends with interesting numerical experiments.