Newton’s iteration as map – Part 1

Newton Iterations is a well known methodology to compute the solution of the problem f(x) = 0. It is very interesting to look at it as a map. The iteration step is as follows for x(n+1).

Output

Let the function be Sin(x) then it results in the following map

Output

The plot of the function and the line x=x is shown below. The points of the intersection of the line with x-Tan(x) function forms the solution of the newton iteration.

As you can see at the points where the sin(x) function change direction the map becomes very unstable. But in other areas it can converge to one solution depending on the initial condition.

Output
Blue: x=x line , Yellow:The map . Solution of the newtons iteration is the solution of f(x) = sin(x)=0 obtained by looking at the intersection of the Blue line with yellow curves. Sin(x) is plotted in green.

To be continued . . .

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