8. Computational Methods
Society for Industrial and Applied Mathematics eBooks · 2000
Currently there appear to be no reliable general-purpose algorithms available on which production-level software may be based for solving the DAEs of the form (6.1) or, more generally, (7.1), arising as models of problems involving kinematic or mixed kinematic and geometric constraints. In this chapter we show that a class of algebraically explicit DAEs considered by Rheinboldt [23], based on local parametrizations, includes a method that can be adapted to the construction of an algorithm for solving the DAEs covered by Theorem 6.1. This algorithm is meant to show that it is indeed possible to construct an effective method for solving this class of DAEs. In section 8.3 we indicate some possible further directions of work that may lead to the development of other methods for the DAEs discussed in the previous chapters. 8.1 Computations on Manifolds The DAE methods developed in [23] were based on a package of supporting algorithms, called MANPACK, for computing local parametrizations on submanifolds of that are implicitly defined by local submersions (see Theorem A.1). This section provides a brief overview of some of the relevant MANPACK algorithms, as given in [22]. The presentation is independent of the earlier results. It should be noted that the computational tasks associated with implicitly defined manifolds differ considerably from those arising in connection with manifolds defined in explicit, parametric form as in, e.g., computational graphics. In fact, unlike in the latter case, for implicitly defined manifolds the algorithms for determining local parametrizations and their derivatives still need to be made available. Throughout this section it is assumed that , , is a smooth mapping and a submersion on its nonempty zero set Σ≔ Γ−1 (0) = {x∈ Rn : Γ (x)=0 } . 8.1 Thus Σ is a -dimensional submanifold of . Definition A.2 recalls the concept of a local parametrization of such submanifolds.