Find the rate of convergence of given sequence.

Find the rate of convergence of given sequence. $$x_{n+1}=\frac{-9}{x_n-6}$$

Here the what I tried and definition of rate of convergence.

• A sequence is said to be convergent with order $k(> 1)$ if $\exists \space c > 0 $and $N > 0$ such that, $\varepsilon_{n+1} ≤ c\varepsilon^k_n$ $\forall n>N$ where $c$ is the rate of convergence.

Therefore by using definition I tried $$|x_{n+1}-3|=\frac{3|x_{n}-3|}{|x_{n}-6|}=|x_{n}-3|\bigg(\dfrac{3}{|x_{n}-6|} \bigg)$$

I have no idea to handle part $\bigg(\dfrac{3}{|x_{n}-6|} \bigg)$

Can anyone help me to find the rate of convergence?

Solution 1:

If you set $g(x)=\frac{-9}{x-6}$, then $g'(3)=1$. This means that there is no direct claim on stability possible, no linear convergence. One would need less usual methods to explore the convergence.

One can linearize the computation by setting $x_n=\frac{p_n}{q_n}$. This has one degree of freedom, which can be used to separate the recursion equation $$ \frac{p_{n+1}}{q_{n+1}}=\frac{-9q_n}{p_n-6q_n}=\frac{9q_n}{6q_n-p_n} $$ into the linear system $$ \pmatrix{p_{n+1}\\q_{n+1}} = \pmatrix{0&9\\-1&6} \pmatrix{p_n\\q_n} $$ This can be solved via eigen-decomposition, or by reducing to a second order recursion equation $$ q_{n+2}-6q_{n+1}=-p_{n+1}=-9q_n $$

From the explicit solution formula that results one can directly read off the convergence.