Limiting Orientations of Newton's Method in ℝ²

Justin Bush, Wes Cowan, Shaun Harker, Konstantin Mischaikow

For functions from f: ℝ² → ℝ², consider the problem of applying Newton's Method to find roots. In the limit as iterates converge to a root, the angular orientation with respect to that root is governed by a simple self-map of the unit circle, defined by the higher order derivatives of f. We compute a database of the qualitative dynamics of the angular progression under Newton's Method for non-degenerate f.

The following table contains database files from the paper. The database software used to produce these files can be downloaded here. The output files (*.mdb, *.cmdb) from the database can be read using the Database Explorer.

Parameter Space	Parameter Space Depth	Phase Space	Phase Space Min Depth	Phase Space Max Depth	Phase Space Initial	Size Limit	Data
	6	S¹	12	17	11	100	Full_4D Zoo
	10	S¹	12	17	11	100	Full_2D Zoo
	11	S¹	14	19	13	100	2D_Zoom_1 Zoo
	11	S¹	14	19	13	100	2D_Zoom_2 Zoo
	11	S¹	14	19	13	100	Period_28 Zoo