Is the iterative method suitable?? We have the equation $g(x)=x^{2}-3x-4=0$ that has the roots $-1$ and $4$ and we are looking for a suitable  iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges to the root $4  \forall  x_{0} \in [3,5] $.Is this one: $\varphi(x)=\frac{x^{3}-3x^{2}}{4}$ suitable?How can I check this??
 A: @evinda;
One obvious way to get the idea that you form will not work is to try it. Using initial conditions of 4.1, 3.9, -1.1 and -.9 the iteration either blows up or heads to a false root.
A more analytical idea is to use the rule that for 
$x = g(x)$
convergence will occur if 
$\left | g'(x) \right |<1$
so
$g'(x)=\frac{1}{4} \left(3 x^2-6 x\right)$
plugging in x = 4 and taking the absolute value yields 6 which is greater than 1 and therefore convergence will not occur. Plugging in -1 yields 9 / 4 so again no convergence.
A: Recall, you need $x = g(x)$. 
The function you are trying does not meet the criteria $x = g(x)$, so is not a candidate.
Possible choices are:


*

*$x = \dfrac{x^2-4}{3}$, (fails to converge for any $x_0$), 

*$x = (3x + 4)^{1/2}$, (converges to $4$ for $x_0 \in [3,~5]$)

*Other choices are possible, try some to see if you can find another example that does and does not work.


You can test iterates to see if they pass the criteria for this method and can even write the general error term.
A: To check that an iterative method is stable, you need to relate the error of $x_n$ to the error of $x_{n+1}$. In particular, you need to show that the errors do not grow.
As an example, let $x_n = 4 + e_n$; here $e_n$ is the error. Then $x_{n+1} = \varphi(x_n) = \varphi(4 + e_n) = 4 + 6 e_n + (9/4) e_n^2 -(3/4)e_n^3$, so then the new error is
$$ e_{n+1} = 6 e_n + (9/4) e_n^2 -(3/4)e_n^3$$
It is clear that for $e_n$ very small, $e_{n+1}$ is in fact larger than $e_n$, so the method is not stable.
