Orbital mechanics: bisect vs$.$ false position

In orbital mechanics you sometimes need to use root finding to determine trajectories that will get your craft to its destination. The roots of

$f(t)=0\mathrm{.}03-0\mathrm{.}311e^{0\mathrm{.}5t}sin(0\mathrm{.}8t)$

can tell you possible transit times to reach a particular planet from Earth where $t$ is the time in years.

Copy the bisect function and modify it to create a false position function. Compute the possible transit times and the number of iterations required to find these times as described below. Use an error tolerance of $0\mathrm{.}0001\%.$

You will need to modify both functions to include the number of iterations as a return value from the function. Modify the false position function to add a max iteration parameter. If the number of iterations exceeds the

maximum number of allowed iterations, the algorithm should stop and return the current value for the root.

A$.$ Plot the function between $2$ and $13.$

B$.$ Compute the root between $2$ and $13$ for both methods. Limit the false position method to $110$ iterations. Create a table using fprintf with $3$ columns: $1)$ the method, $2)$ the root $($specify what the root represents and its

units in the header$),$ and $3)$ the number of iterations to find the root using that method. Display the root with $10$ decimal places. Just above the table, use an fprintf statement to display a statement such as: 'Using $x$ and $x$

as the bounds'. After the table, explain the difference in the number of iterations between the two methods in text.

C$.$ Create the table again but use $5$ and $10$ as the bounds. Explain the difference in the number of iterations between part $b$ and $c$ for the False Position method as text.

D$.$ Plot all solutions $($in the figure from part a$)$ using:

Bisection part b: red asterisk

False Position part b: red square

Bisection part c: green asterisk

False Position part c: a green square