Newton's Method

You will create two zero$-$finding functions newton.m$$and secant.m$.$ These functions will be similar, but the first will use an analytical derivative and the second will compute the derivative numerically.

Base your functions around the main Newton's Method iteration formula

to find the root, stopping when $,$ where $$is the specified tolerance.

Your function declaration for newton.m$$should look like

function $[$x$,$n$]=$ newton$($f$,$fp$,$x$0,$tol$)$

where f$$is a the function you're finding the root for, fp$$is the derivative of f$($also a handle$),$ x$0$is the starting guess, and tol$$is an optional argument for the error tolerance $($default $1$e$-10).$ Have your program count the number of iterations for comparisons to bisect.m$.$

For secant.m$,$ instead of passing an analytical derivative, you will make a subfunction that computes forward difference$$derivative approximation

$.$

Your subfunction will take f$,$ x$0,$ and tol$$as its inputs and return a numerical approximation to the derivative to use in the above formula for Newton's Method. The declaration of the main function secant.m$$will look like

function $[$x$,$n$]=$ secant$($f$,$x$0,$tol$)$

where the variables and the their meanings remain the same as in newton.m$.$

What to Turn In:

Make a PDF of you testing these functions on our example from class $$and compare it to bisect.m$.$ Start with bisect.m$$and find the root in $[0\mathrm{.}5,1],$ having the number of iterations required display in your results. Then use newton.m$$and secant.m$$on the same function starting at $0\mathrm{.}5$ and showing the iteration count.