Matlab problem Next, implement the three basic rootfinding algorithms discussed

in class as three separate functions $($bisection$,$ Newton's, and Secant$).$ Each rootfind$-$

ing function should return an array consisting of the iterates $($i$.$e$.,$ the guesses of the

root$).$ That is$,$ return the array $x_1,x_2,x_3,$dots,$x_n,$ where $x_k$ root. The first entry

$x_1$ is the initial root guess, and the last entry $x_n$ is the final $($and most accurate$)$ root

guess.

We will use two checks for the stopping criteria,

$(|x|$ tol

Stop your rootfinding algorithm if either of the above conditions is met. The quan$-$

tity $x$ is given below for each algorithm, tol is some user specified tolerance, and

$lo{n}_m$ is the machine epsilon $($also called machine precision$).$ Matlab has a built$-$in value

eps that equals machine epsilon.

$($a$)$ Bisection: Write a Matlab function that computes the root with the bisection

method. Use the midpoint of the interval $($i$.$e$.,$ bracket$)$ as the guess of the root.

The quantity $x$ equals the length of the current interval $($i$.$e$.,$ bracket$).$

A sample function prototype for bisection would be

function $[$x$\_$arr$]=$ my$\_$bisection$($f$,$ c$,$ d$,$ tol$)$

where $f$ is the function name in question $($technically called a function pointer$),$

$c,d$ is the initial bracket, and tol is the halting tolerance The array x$\_$arr

is the return value, the array $x_1,x_2,x_3,$dots described above.

For instance if the function for rootfinding is called my$\_$fcn$,$ then you would

call bisection with

x$\_$arr$\_$bisection $=$ my$\_$bisection $($my$\_$fcn$,$ c$,$ d$,$ tol$)$;

The function my$\_$fon $($x$)$ will take in an $x$ value and return $f(x).$ It is this

function that you find the zero of$.$

$($b$)$ Newton's: Write a Matlab function that computes the root with Newton's

method. The quantity $x$ equals $x=x_k-x_{k-1},$ which are your two most

recent guesses for the root.

A sample function prototype for Newton's method would be

function $[$x$\_$arr$]=$ my$\_$newton$(f,$ df$,$ x$\_0,$ tol$)$

where $f$ is still the function name in question $($technically called a function

pointer$),df$ is the name of a function that implements the derivative of $f($tech$-$

nically a function pointer$),x_{-}0$ is the initial guess to a root, and tol is the

halting tolerance. The return value array x$\_$arr is the same as for bisection.

The function $df(x)$ will take in an $x$ value and return $f^{\text{'}}(x).$

$($c$)$ Secant: Write a Matlab function that computes the root with the Secant method.

The quantity $x$ equals $x=x_k-x_{k-1}.$

A sample function prototype for the Secant method would be

function $[$x$\_$arr$]=$ my$\_$secant $(f,x_{-}0,x_{-}1,$ tol