Abstract
In previous chapters we assumed that the map H : RN+1 → RN was smooth, that zero was a regular value, and that H−1 (0) was a collection of disjoint smooth curves which could be numerically traced using PC-methods. Now we will discuss piecewise linear (PL) methods which can again be viewed as curve tracing methods, but the map H can now be arbitrary. The map H is approximated by a piecewise linear map H T which affinely interpolates H at the nodes of a triangulation T of RN+1. The PL methods trace the piecewise linear 1-manifold H T −1 (0). A connected component of the piecewise linear 1-manifold consists of a polygonal path which is obtained by successively stepping through certain “transverse” (N + 1)-dimensional simplices of the triangulation. Although the PL method works for arbitrary maps, only under some smoothness assumptions on H can one obtain truncation error estimates in terms of the meshsize of the underlying triangulation. In order to be able to discuss these methods it is necessary to introduce a few combinatorial ideas. The first notions we need are those of a simplex and a triangulation.
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© 1990 Springer-Verlag Berlin Heidelberg
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Allower, E.L., Georg, K. (1990). PL Continuation Methods. In: Numerical Continuation Methods. Springer Series in Computational Mathematics, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61257-2_12
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DOI: https://doi.org/10.1007/978-3-642-61257-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64764-2
Online ISBN: 978-3-642-61257-2
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