In Matlab, programs may be written and saved in files with a suffix $.$m called M$-$files. There are two

types of M$-$file programs: functions and scripts.

The following function program does $n$ iterations of the bisection method.

function $[xe$ mybisect $(f,a,b,n)$

$\%$ function $[xe]=$ mybisect $(f,a,b,n)$

$\%$ Does $n$ iterations of the bisection method for a function $f$

$\%$ Inputs : $f--$ a function

$\%a,b--$ left and right edges of the interval

$\%n--$ the number of bisections to do$.$

$\%$ Outputs : $x--$ the estimated solution of $f(x)=0$

$\%$ e $--$ an upper bound on the error

$\%$ evaluate at the ends and make sure there is a sign change

$c=f(a)$;

$d=f(b)$;

if $c^{**}d>0\mathrm{.}0$

error $(\text{'}$Function has same sign at both endpoints. $\text{'})$

end

disp$(\text{'}\{$:$x,y^{\text{'}})$

for $i=1$ : $n$

$\%$ find the middle and evaluate there

$x=\frac{a+b}{2}$;

$y=f(x)$;

disp$([x,y])$

$ify=0\mathrm{.}0\%$ solved

$,a=x$;

$b=x$;

break

$\%$ jumps out $of$ the $f$

$\%$ decide which half

$if{c}^{**}y<mn\; id="EditorElement\_447585">0</mn>$

$b=x$;

else

$a=x$;

if $y==0\mathrm{.}0\%$ solved the equation exactly

$a=x$;

$b=x$;

break

$\%$ jumps out of the for loop end

$\%$ decide which half to keep, so that the signs at the ends differ

if $c^{**}y<mn\; id="EditorElement\_447120">0</mn>$

$b=x$;

else

$a=x$;

end

end

$\%$ set the best estimate for $x$ and the error bound

$x=\frac{a+b}{2}$;

$e=\frac{b-a}{2}$;

end

Your turn

Modify mybisect in above to create mybisectwhile to solve until the absolute error is bounded

by a given tolerance. Use a while loop to do this. Run your program on the function

$f(x)=2x^3+3x-1$ with starting interval $0,1$ and a tolerance $of{10}^{-8}.$

How many steps does the program use to achieve this tolerance? $($You can count the

steps by adding $1$ to a counting variable $i$ in the loop of the program.$)$

How big is the final residual $f(x)?$

Turn in your program and a brief summary of the results.

The commands in the shell aremybisectwhile$($f$,$a$,$b$,$tol,n$)$

Here is the sample function with "while" loop.

function $x=$ mynewtontol $,$ tol