Newton's Method in ℝ²

For a function with a zero at the origin, the question of along which direction Newton's method converges to 0 is a question about the dynamics of a map on the circle, S¹.

If we restrict our attention to generic functions with Jacobians that do not vanish at the origin and with nondegenerate nonlinearities, we can parameterize such functions with the four-dimensional parameter space:



The dynamics of the map with these parameters is then given by the following:

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	Newton4D Zoo
	7	S¹	7	12	6	100	Newton3D-1 Zoo
	7	S¹	7	12	6	100	Newton3D-2 Zoo
	7	S¹	7	12	6	100	Newton3D-3 Zoo
	7	S¹	7	12	6	100	Newton3D-4 Zoo
	7	S¹	7	12	6	100	Newton3D-5 Zoo
	10	S¹	12	17	11	100	Newton2D-1 Zoo
	10	S¹	12	17	11	100	Newton2D-2 Zoo
	10	S¹	12	17	11	100	Newton2D-3 Zoo
	10	S¹	12	17	11	100	Newton2D-4 Zoo
	10	S¹	12	17	11	100	Newton2D-5 Zoo
	10	S¹	12	17	11	100	Newton2D-6 Zoo
	11	S¹	14	19	13	100	Newton2D-7 Zoo
	11	S¹	14	19	13	100	Newton2D-8 Zoo
	11	S¹	14	19	13	100	Newton2D-9 Zoo
	11	S¹	14	19	13	100	Newton2D-10 Zoo
	11	S¹	14	19	13	100	Newton2D-11 Zoo
	11	S¹	14	19	13	100	Newton2D-12 Zoo
	10	S¹	12	17	11	100	newton-2-10-12
	8	S¹	12	17	NA	100	newton-2-8-12