where R$=0\mathrm{.}7$ i need help solving the following i$.$ Using an initial guess of $x_0=y_0=5,$ apply the Steepest Descent

method to the above system. Using a fixed value of $t=0\mathrm{.}1,$ calculate

the first iterate $($that is$,x_1,y_1)$ by hand, then verify these values and

complete the iteration using the MATLAB steep command. Let's call

the final values given by MATLAB $x^{**}$ and $y^{**}.$ Why does MATLAB stop

here? $($To answer this, calculate the next iterate, $(x^{****},y^{****}),$ by hand and

consider the values of $S(x^{**},y^{**})$ and

Here $x$ represents the population of species $1($in $1000$s$),y$ represents the

population of species $2($in $1000$s$),A=33+7r$ and $B=17+3r,$ where $r$ is

your random number.

i$.$ Using an initial guess of $x_0=y_0=5,$ apply the Steepest Descent

method to the above system. Using a fixed value of $t=0\mathrm{.}1,$ calculate

the first iterate $($that is$,x_1,y_1)$ by hand, then verify these values and

complete the iteration using the MATLAB steep command. Let's call

the final values given by MATLAB $x^{**}$ and $y^{**}.$ Why does MATLAB stop

here? $($To answer this, calculate the next iterate, $(x^{****},y^{****}),$ by hand and

consider the values of $S(x^{**},y^{**})$ and $S(x^{****},y^{****}).)$

ii$.$ Using an initial guess of $(x^{**},y^{**})$ and showing all working, apply New$-$

ton's method to find an answer correct to four decimal places. $($Hint:

this should not take many steps!$)$

iii. Returning to the Steepest Descent method and using an initial guess of

$x_0=y_0=5$ in both cases: calculate $(x_1,y_1)$ using $t=0\mathrm{.}8$; calculate

$(x_1^{\text{'}},y_1^{\text{'}})$ using $t=0\mathrm{.}4.$ Also calculate $S(x_1,y_1)$ and $S(x_1^{\text{'}},y_1^{\text{'}}).$ You now

have the following data:

$t=0,S(x_0,y_0)$

$t=0\mathrm{.}4,S(x_1^{\text{'}},y_1^{\text{'}})$

$t=0\mathrm{.}8,S(x_1,y_1)$

Using MATLAB, fit a quadratic through these data. Find $($by hand$)$

which value of $t$ minimises the quadratic and hence give 'improved'

values of $(x_1,y_1).$ Do these 'improved' values result in a reduced value

of $S?****.)$