$($Continued exponentiation$)$ Let $y$ be defined by

$y=x^{x^{x^{x^2}}}$

where $x$ is a real number in the interval $(e^{-e},e^{\frac{1}{e}}).$

$($a$)$ Noting that $y=x^y$ and using the fact provided below, write down an analytical

formula for $y.$ This part is to be done by hand, and your answer must be written in

terms of $x$ and a special function $W$ given below.

Fact. Let $a0.$ Then the solution of the equation $e^z=az$ is given by the

formula

$z=-W(-\frac{1}{a}),$

where $W$ is so$-$called the Lambert's $W$ function.

${?}^1$ References:

FNC: Fundamentals of Numerical Computation $($Driscoll and Braun$)$

LM: Learning MATLAB, Problem Solving, and Numerical Analysis Through Examples $($Overman$)$

The notation LM $2\mathrm{.}2-5$ indicates Problem $5$ at the end of section $2\mathrm{.}2$ of the textbook by Overman.

${?}^2$ To learn about the difference between numerical and symbolic computations, please read the prologue of FNC$.$

$($b$)$ Using the formula found in $($a$)$ and the MATLAB function lambertW provided,

calculate the value of $y$ numerically for $x=\frac{1}{2},\frac{\sqrt[\phantom{2}]{2}}{2},\frac{\sqrt[\phantom{2}]{3}}{2}.$

Using lambertW: First, make sure that the file lambertW.m is saved in the

same directory as your homework mlx file. To evaluate $W(x)$ using the provided

MATLAB function, just type

$>>$ lambertW$($x$)$

$($c$)$ Even without knowing the analytical formula, one may still compute $y$ as follows.

Let $y_0=x$ and

$y_1=x^x=x^{y_0}$

$y_2=x^{x^x}=x^{y_1}$

$y_3=x^{x^{x^x}}=x^{y_2}$

vdots

So$,$ for any ninN,$y_n=x^{y_{n-1}}$ and $y=\underset{n}{lim}{y}_n.$ This limit is known to exist provided

that xin$(e^{-e},e^{\frac{1}{e}}).$ Thus $y$ can be approximated by $y_n$ for sufficiently large $n.$ Now,

for each of $x=\frac{1}{2},\frac{\sqrt[\phantom{2}]{2}}{2},\frac{\sqrt[\phantom{2}]{3}}{2},$ determine the smallest integer $n$ such that $|y-y_n|{10}^{-10}.$

For the exact values of $y,$ use the results from $($b$).$

lambertW function :

y $=$ lambertW$($x$)$

y $=$ zeros$($size$($x$))$;

for i $=1$:numel$($x$)$

y$($i$)=$ fzero$($@$($y$)$ x$($i$)-$y$.*$exp$($y$),-1)$;

end

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