I have an exam on Tuesday 27th. I need to know well the bisection method, secant method, newton method and a lot more.

I have a sheet of around 25 exercises. I could do some pretty easy, but others are too hard for me.

I have a lot of questions in relation with these methods. (That I will post in other threads if I feel the need to).

Here comes an exercise I couldn't do.

1)Demonstrate that if f is a function that has a double root in a, then the Newton method applied to f has an order of convergence of 1.

Demonstrate also that the modified Newton has a convergence order of 2.

2)Study what happens if $\displaystyle p>2$.

As p is not defined, I guess it's the number of multiple roots.

End of the problem.

My approach was to write $\displaystyle f$ as $\displaystyle f(x)=(x-a)^2$. $\displaystyle f'(x)=2x-2a

$ and $\displaystyle f^{(2)}(x)=2$. Therefore the Newton method will converges regardless of the starting point.

From it, I supposed $\displaystyle x_0=0$. I calculated $\displaystyle x_1$, which is worth $\displaystyle \frac{a}{2}$. Also $\displaystyle x_2=\frac{3a}{4}$ and lastly $\displaystyle x_3=\frac{7a}{8}$.

So our intuition tells us that the method converges to a. But how can I determine the order? And how can I prove that it effectively converges to a?

I'd be glad if you could start me up for the common Newton method, this way I hope to finish the rest alone.