91Ó°ÊÓ

Show that \(x=1+\tan ^{-1}(x)\) has a solution \(\alpha .\) Find an interval \([a, b]\) containing \(\alpha\) such that for every \(x_{0} \in[a, b]\), the iteration $$ x_{n+1}=1+\tan ^{-1}\left(x_{n}\right) \quad n \geq 0 $$ will converge to \(\alpha .\) Calculate the first few iterates and estimate the rate of convergence.

Which of the following iterations will converge to the indicated fixed point \(\alpha\) (provided \(x_{0}\) is sufficiently close to \(\alpha\) )? If it does converge, give the order of convergence; for linear convergence, give the rate of -linear convergence. (a) \(x_{n+1}=-16+6 x_{n}+\frac{12}{x_{n}} \quad \alpha=2\) (b) \(x_{n+1}=\frac{2}{3} x_{n}+\frac{1}{x_{n}^{2}} \quad \alpha=3^{1 / 3}\) (c) \(x_{n+1}=\frac{12}{1+x_{n}} \quad \alpha=3\)

Given below is a table of iterates from a linearly convergent iteration \(x_{n+1}=g\left(x_{n}\right) .\) Estimate (a) the rate of linear convergence, (b) the fixed point \(\alpha\), and \((\mathrm{c})\) the error \(\alpha-x_{5}\) \(\begin{array}{lc}\boldsymbol{n} & x_{n} \\ 0 & 1.0949242 \\ 1 & 1.2092751 \\\ 2 & 1.2807917 \\ 3 & 1.3254943 \\ 4 & 1.3534339 \\ 5 & 1.3708962\end{array}\)

Let \(p(x)\) be a polynomial of degree \(n .\) Let its distinct roots be denoted by \(\alpha_{1}, \ldots, \alpha_{p}\), of respective multiplicities \(m_{1}, \ldots, m_{r}\) (a) Show that $$ \frac{p^{\prime}(x)}{p(x)}=\sum_{j=1}^{r} \frac{m_{j}}{x-\alpha_{j}} $$ (b) Let \(c\) be a number for which \(p^{\prime}(c) \neq 0\). Show there exists a root \(\alpha\) of \(p(x)\) satisfying $$ |\alpha-c| \leq n\left|\frac{p(c)}{p^{\prime}(c)}\right| $$