Another case: the method fails

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Another case: the method fails

Change the MATLAB function g.m to solve the equation $\bgroup\color{black}$x=g(x)= \frac{1}{2} \tan x $\egroup$ , where $\bgroup\color{black}$x \in [ -0.5, 1.5]$\egroup$ . There are two solutions in this domain, one at $\bgroup\color{black}$x=0$\egroup$ . You should find it easy to get the iteration.m to find the $\bgroup\color{black}$x=0$\egroup$ solution, but it turns out that you cannot find the solution for $\bgroup\color{black}$x \approx 1.2$\egroup$ using functional iteration.

$\bgroup\color{red}\framebox{\em HAND CALCULATION}\egroup$ $\bgroup\color{black}$\phantom{0}$\egroup$ Demonstrate theoretically that the computational results found above are to be expected. Show that $\bgroup\color{black}$\vert g'(x)\vert<1$\egroup$ at $\bgroup\color{black}$x=0$\egroup$ and that $\bgroup\color{black}$\vert g'(x)\vert>1$\egroup$ near $\bgroup\color{black}$x=1.2$\egroup$ . Now rewrite $\bgroup\color{black}$x=g(x)$\egroup$ in such a way that the magnitude of the derivative is less than one near $\bgroup\color{black}$x=1.2$\egroup$ .

With your new $\bgroup\color{black}$g(x)$\egroup$ find the root near $\bgroup\color{black}$x=1.2$\egroup$ using iteration.m.

$\bgroup\color{blue}\framebox{\em CHECKPOINT: submit solution}\egroup$ $\bgroup\color{black}$\phantom{0}$\egroup$ $\bgroup\color{red}\framebox{ 4.}\egroup$ Record the value of the variable x(21).

Charlie Macaskill 2004-07-26