Understanding Fixed-Point Iteration: Exploring Convergence and Root Determination
The equation $$2x^2 - 7x + 6 = 0$$ has two positive roots. We aim to determine these using the following fixed-point iteration: $$ x_{k+1} = \frac{{8x_k - 2x_k^2}}{{k - 6}} $$
a) It is found that only the larger of the two roots can be determined using this method. Explain why, for example using the figure below.
Solution: Denote the right-hand side of the iteration as $$\phi(x) = 9x - 2x^2 - 6.$$ For convergence, it is required that $$|\phi'(x^*)| < 1$$ where $x^*$ is the root. The figure shows that $\phi$ has a slope of zero at the largest root, and therefore converges towards it. At the smallest root, $\phi$ has a slope greater than $x = y$. Since the latter has a slope of one, the iteration diverges here.
My question the solution is more confusing than the question. I have no idea what they are doing why they changed 8 with a 9. How in the hell the function has a slope 0 for the largest solution which is x=3. Can anyone help me understand what is going on? Thanks beforehand!