newton's method: when does it cycle?

Question

or what other scenarios cause non covergence? any help is appreciated.

Answer 1

♠ when you solve an equation F(x)=0 you have to transform it like this:
x=f(x) to use Newton’s method;
♣ if |f’(x)| < 1 in vicinity of supposed root x, the method works and procedure converges;
if |f’(x)| > 1, the procedure diverges;
if some |f’(x)| > 1 while others < 1 the procedure cycles;
♦ example: x^2 = 27, find x; Newton’s method: x=0.5*(27/x +x) =f(x);
f’(x) = 0.5*(-27/x^2 +1); suppose x=5, then f’(5) = 0.5*(-27/25 +1) < 1; thus method is expected to work;
♥ example: x=sqrt(tan(x-0.5)) diverges; try another combination
x=f(x), i.e. x=atan(x^2)+0.5, it converges;

Answer 2

you should draw the graph of the function you are examining to see what could happen

cycling is one
also it can simply diverge