Problem $1$: A Hybrid Bracketing Method

A new method tor solving a nonlinear equation $f(x)=0$ is proposed in this problem. The method is a

combination of the bisection and the false position methods. The solution starts by defining an interval

$a,b$ that brackets the solution. Then estimated numerical solutions are determined once with the bisection

method and once with the false position method. $($The first iteration uses the bisection method.$)$ Write a

MATLAB user$-$defined function that solves a nonlinear equation $f(x)=0$ with this new method. Name

the function $x=$ BiFalRoot $($Fun$,$ a$,$ b$,$ Tol$),$ where the output argument x is the solution. The input

argument Fun is a name for the function that calculates $f(x)$ for a given $x($it is a dummy name for the

function that is imported into BiFalRoot$),$ a and b are two initial points that bracket the root, and Tol is

the error tolerance.

Your code should include the following features:

The program should stop when the estimated relative error drops below the specified tolerance Tol.

Check if points a and $b$ are on opposite sides of the solution. If not, the program should halt and

display an error message of your choice.

The number of iterations should be limited to $100($to avoid an infinite loop$).$ If a solution with the

required accuracy is not obtained in $100$ iterations, the program should stop and display an error

message of your choice.

Use the function BiFalRoot to solve the equation $cos(x)-0\mathrm{.}8x^2=0$ with $a=0$ and $b=2$ and observe the

answer. For Tol use $0\mathrm{.}00001.$