By Jacobi C.G.J.
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8) It is clear that this method follows the solution curve if ||xo — C(0)|| and \U — ti-i\ are small enough. Of course, the crucial number for complexity considerations is the number k of Newton steps involved in this embedding method. 2 has to be added. In fact, \\xi — C(*»)|| < «/7 To summarize, we have outlined a program for approaching complexity investigations when Newton steps are the primary tool of path following methods. ), C(s)) < C2I* — s\ with explicit constants C\ and C%. This program was carried out by Shub and Smale (1991) for the case of a homotopy method for calculating all solutions of a system of polynomial equations (Bezout's theorem).
In any case, our opinion is that path following codes always need to be considerably adapted to the special purposes for which they are designed. The path following literature offers various tools for accomplishing such tasks. Although there are some general purpose codes, probably none will slay every dragon. Rheinboldt, Roose and Seydel (1990) present a list of features and options that appear to be necessary or desirable for continuation codes. This should be viewed as a guideline for people who want to create a new code.
However, these algorithms become extremely costly for large N. The piecewiselinear algorithms have been applied to the visualization of body surfaces, see All gower and Gnutzmann (1991), and to the approximation of surface and body integrals, see Allgower, Georg and Widmann (1991d). They can also be used as automatic mesh generators for boundary element methods, see Georg (1991). CONTINUATION AND PATH FOLLOWING 41 6. Complexity In modern complexity investigations of continuationtype methods the so called atheory of Smale (1986) is a convenient tool.