HELP ME PLEASE!!! $$T$\text{'}$S NUMER$$CAL ANALYS$$S$!!!!$

$($Bisection Method$).$ All numerical answers should be rounded to $6-$digit floating$-$point numbers.

$($i$)$ Consider the polynomial

$f(x)=x^3+2x^2-45x-116.$

Please accept as a given that the polynomial $f(x)$ has three real roots in $-10,10.$

$($a$)$ Let then $r_1,r_2,r_3$ be the roots of $f(x)$ written in increasing order. For each of the roots $r_i,$ find a pair of integer

numbers $m,m+1$ that bracket the root $r_i$ :

$r_1$ is between

and

$r_2$ is between

and

$r_3$ is between

and

$($b$)$ Now, according to $($a$),$ is it true that the polynomial $f(x)$ has a unique root in the closed interval $-7,-6?$

Yes

No

$($ii$)$ Use the Bisection Method to find an approximation $p_N$ of the unique root of the function $f(x)$ in $-7,-6$ satisfying

$RE(tilde(p{)}_N$~~tilde

where tilde$(z)$ denotes the result $fl(z)$ of rounding of a real number $z$ to a $6-$digit floating$-$point number.

$($iii$)$ Show then your work by filling in the table that follows. In each input field in the column labelled by

$f(a_n)f(p_n),($iii$)$ Show then your work by filling in the table that follows. In each input field in the column labelled by

$f(a_n)f(p_n),$

please enter either a plus sign $+($if $f(a_n)f(p_n)>0),$ or a minus sign $-($if If a particular row of the

table is not necessary, enter an asterisk $**$ in each input field in the row. In order to calculate the relative error

$RE(tilde(p{)}_1$~~tilde$(p{)}_0)$

in the first row, assume formally that $p_0=-7.$If you are going to use a scientific calculator, use the program we have discussed in class. Also, you can create an OpenOffice $($or Excel$)$ worksheet with

a copy of the table given above in order to smooth up the calculations. Make sure that all numbers in your table are shown to be rounded to $6-$digit floating$-$

point numbers $($as it was discussed in class, in OpenOffice use the command Format$/$Cells$/$Numbers$/$Scientific from the main menu, and

replace the pattern in the Format code field to $0\mathrm{.}00000E+000).$

In the process of calculations, enter in the rows of your table in the worksheet the terms

$a_n,p_n,b_n,$

followed by the sign of the number $f(a_n)f(p_n)$ as described above.

If you'll feel that the terms $p_n$ may have become close enough to satisfy the stopping criterion, fill in the last column of the table in your worksheet by

calculating the required relative errors

$RE(\frac{?}{\text{b}\text{a}\text{r}}(p{)}_n$~~tilde$(p{)}_{n-1}).$

Note the step $N$ at which the stopping criterion became true; if all your relative errors are still greater than the tolerance, continue generating the terms

$a_n,p_n,b_n,$ and so on$.$

Once the table in your worksheet is ready, check your answers, redo the table if necessary, and then copy$-$paste your answers to the table in this page.

$5$a$)$ To stress, the relative errors of the form $(2\mathrm{.}1)$ are in general different from the relative errors of the form

$RE(p_n$~~$p_{n-1}).$

To give an example, let

$x=$ and $x^{**}=\frac{22}{7.}$

Then

$RE(x$~~$x^{**})0\mathrm{.}402499{10}^{-3}$

and

$RE((\stackrel{}{x})$~~${\stackrel{}{x}}^{**})=RE(fl(x)$~~$fl(x^{**}))0\mathrm{.}404254{10}^{-3}$

$($verify both results to ensure a better understanding!$).$

$5$b$)$ The relative errors of the form $(2\mathrm{.}1)$ are normally used when the numerical root$-$finding methods are implemented with scientific calculators, for

evaluating the relative errors of the form $(2\mathrm{.}2)$ on the fly would make the corresponding programs for scientific calculators harder to execute. So a nun

of the terms $p_n$ is generated first, and only then, as discussed above, the relative errors of the form $(2\mathrm{.}1)$ are evaluated. Understandably, the users take the

values $\frac{?}{\text{b}\text{a}\text{r}}(p{)}_n$ from the tables they have created.$($iii$)$ Accordingly, by $($i$)$ and $($ii$),$

$p_N$