The output argument of the program, orbit, must be a matrix of the format

$$

$$

$$

$$

$$

t$0$ E$0$

u $0$ r$0$ x$0$ y$0$

t$1$ E$1$

u $1$ r$1$ x$1$ y$1$

$..................$

tn En

u n rn xn yn

$$

$$

$$

$$

$$

where ti $=$ iT $/$n$,$ Ei is the solution to Kepler Equation,

u i is the true anomaly corresponding to ti$,$ ri is the

distance between the satellite and the earth at ti and $($xi$,$ yi$)$ are the cartesian coordinates.

Each function must be in a different $.$m file, with the same name as the function.

$3$ Summary

You are required to submit to $$Project $1$ in Canvas the following files$1$:

$1.(40\%)$ The source files $$bis$.$m$,$nwt$.$m$,$bisnwt$.$m$,$tanom$.$m$$ described above$\mathrm{.}2.$

$2.$ A $.$pdf file with a brief report of the project. The report should include the following sections:

$$ Implementation: $(20\%+10\%+10\%+10\%)$

$$ Describe the tests that you have done to validate your code, what you have discovered, the problems

encountered, and their respective solutions. You must include at least the following test

T $=4,$ e $=0\mathrm{.}25,$ n $=100,$

and tolerances tolb$=1$e$-2,$ toln$=1$e$-12$ in $$bisnwt$.$m$.$

$$ Derive a formula for x in terms of E and then decide the value for toln if we want the error in the

x$-$coordinates of the orbit to be smaller than $106.$

$$ Intuitively, root$-$finding solvers find a scalar root x in R and thus, the points in the orbit can be

found one by one, applying Algorithm $1$ and the transformations described in Section $2\mathrm{.}1$ for each

time of the period. A slight change, though, would allow us to work with a vector x in Rn and

find all the points of the orbit at once. Show your modified implementation and discuss about the

differences between the two approaches.

$$ Assume the orbit eccentricity e $=0\mathrm{.}25.$ Reformulate $(1)$ so that fixed$-$point iteration method can be

applied with convergence, and then estimate the number of iterations needed in fixed$-$point iteration

to solve $(1)$ with accuracy $108.$

$$ Solutions: $(10\%)$ Plot the orbits that you have obtained with your selected initial data and parameters.

You also need to submit a Matlab script $$demo$.$m$$ which calls tanom and displays the orbits.