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A global analysis of Newton iterations for determining turning points. (English) Zbl 0806.65052

One way to understand the global convergence properties of the Newton method goes back to the results of Julia and Fatou. These indicate that boundaries of the basins of attraction may reproduce themselves. Another concept is to apply singularity theory; that’s the way the authors analyse the global convergence of Newton iterations for determining turning points of a parameter-dependent mapping.
It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is a bifurcation singularity. Now the theory of imperfect bifurcation offers a particular scenario for the split of the singular root into a fininte number of regular roots (turning points) due to a given parameter imperfection. A lot of theoretical and experimental arguments are presented in this interesting note. For details the reader is referred to the paper.
Reviewer: H.Ade (Mainz)

MSC:

65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
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References:

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