Creating Our Own Root Finding Function

We will assume, like MATLAB does, that it only makes sense to look for roots in intervals where at least one solution exists. The first part

of our function will throw out anything where the function and given interval does not contain a zero $($that is the endpoints of the function

differ in sign$).$

function x $=$ bisect$($f$,$ a$,$b$,$tol$)$

if nargin $<mtext></mtext><mn>4</mn>$

tol $=1$e$-10$; $\%$ default value

end

$\%$ check input values

assert$($f$($a$)*$f$($bFunction values at the $\text{'}...$

'endpoints must differ in sign'$])$;

assert$($a $$ b$,$ 'a must be less than b$\text{'})$

end

Using this as a starting point, complete bisect. $m$ with the following requirements:

Use a while loop that runs until your interval width is less than $2^{*}$ tol

Calculate the midpoint $m$ of the interval, then detect which interval $a,m$ or $m,b$ contains the sign change

Use an if statement inside the loop to change a to $m$ or $b$ to $m$ depending on which interval $($parts $2$ and $3$ may be done

simultaneously$)$

Once your while loop is finished, report $x=m,$ where $m$ is the midpoint of the final interval $($this guarantees that the reported root is

within tol of the actual root$).$

$($Exceptional Case$)$ When the above works, add an extra elseif which detects if $f(m)==0.$ If this happens, return $m$ as your

answer.