Newton-Raphson sequence?

Question

Prove that the Newton-Raphson sequence for the equation f(x)=(x-1)^2=0 converges for every real initial guess. Does the result still hold if the initial guess is a complex number?

Answer 1

given equation is(x-1)^2=0
f(x)= (x-1)^2

dy/dx = 2(x-1)

as dy/dx becomes zero at x=1

Newton Raphson formula is

Yn+1 = Yn + F(Xn)/(dF/dx) at Xn

therefore thenewton Raphson method is converges for every real initial guess except x=1( Bcoz at x=1 denominator becomes zero)

The result holds for all complex real guesses as for no complex no. the dy/dx becomes zero

Answer 2

1) Each iteration halves the distance to the point x=1, so the sequence always converges.

2) Yes. See Part 1.

Answer 3

a(n+1) - a(n) = - f[a(n)]/f'[a(n)]

Therefore |a(n+1) - a(n)| = [1/2](^n) |A - 1| if A is the first guess.

The sequence converges.

This seems to work for complex A.