Question $1$

$1\mathrm{.}1$ The table below shows the results of iterations of Bisection method on Manning's formula to find the water height, $h(m),$ in a rectangular open channel. A relative error of less than $3\%$ is used as a stopping criterion.

$\backslash$table$[[$Iteration no$,h_L,h_u,h_i,$Relative error $(\%)],[1,1\mathrm{.}000,2,1\mathrm{.}500,-],[2,1\mathrm{.}000,1\mathrm{.}5,1\mathrm{.}250,20\mathrm{.}000],[3,1\mathrm{.}000,1\mathrm{.}25,1\mathrm{.}125,],[4,1\mathrm{.}125,1\mathrm{.}25,1\mathrm{.}188,],[5,1\mathrm{.}188,1\mathrm{.}25,1\mathrm{.}219,],[6,1\mathrm{.}188,1\mathrm{.}219,1\mathrm{.}204,]]$

$1\mathrm{.}1\mathrm{.}1$ Calculate the relative error for each iteration.

$1\mathrm{.}1\mathrm{.}2$ At which iteration number the analysis terminates?

$(2)$

$1\mathrm{.}1\mathrm{.}3$ What are the two initial guesses used for the root?

$(2)$

$1\mathrm{.}1\mathrm{.}4$ Based on the analysis until iteration $6,$ what the best estimation for the water height $(h)$ in the channel.

$(2)$

$1\mathrm{.}2$ Given the following Matrices A and B $,$ write the Octave commands and provide screenshots to show the following matrix operations: the product of $A.B,A-B$ and the inverse of matrix B$.($Note: command script must include student name, student number, test date and time$).$

$(11)$

$A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 6& -2\\ 0& -2& 8\end{array}\right],B=\left[\begin{array}{ccc}3& -2& 3\\ -2& -1& 0\\ 3& 0& 5\end{array}\right]$

$[25$ marks$]$