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I am working on an iteration problem for computing inverse of a non singular matrix $A$

I have got following relationship between error matrix defined by $E_k = X_k-A^{-1}$. $\|E_{k+1} \|\leq (1-\beta)E_{k}+\beta E_{k}^3$, $k =0,1,\ldots$, $0<\beta\leq 1$

From here can we conclude that $\|E_{k+1} \|\rightarrow 0$ if $k\rightarrow \infty$ provided $\|E_{0} \|< 1$. What can we say about order of convergence of my iterative method? Could we conclude that order of convergence is 3 for $\beta= 1$ and it has linear convergence for $\beta$ not equal to 1.

Could anybody help me with this?I would be very much thankful.

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    @JitseNiesen Thank you very much sir. I was bit confused. I wanted to know could we draw the same conclusion as I have written above.2012-08-16

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