$5\mathrm{.}9[$A$]$ We now build on problem $5\mathrm{.}8,$ which recast Newton$$s method as a version of

fixed$-$point iteration. Assume x

$$

is a root of multiplicity m$,$ that is $($for q$($x

$$

$),0)$:

f$($x$)=($x $$ x

$$

$)$

m

q$($x$)(5\mathrm{.}141)$

$($a$)$ Analytically derive a formula for the g$($x$)$ of Eq$.(5\mathrm{.}140)$ where you$$ve employed

Eq$.(5\mathrm{.}141).$

$($b$)$ Use that result to show that g

$$

$($x

$$

$)=($m$1)/$m$.$ From there, since m $=2$ or higher,

draw the conclusion that Newton$$s method converges linearly for multiple roots.

$($c$)$ It should now be straightforward to see that if you had used:

g$($x$)=$ x $$ m

f$($x$)$

f

$$

$($x$)$

$(5\mathrm{.}142)$

instead of Eq$.(5\mathrm{.}140),$ then you would have found g

$$

$($x

$$

$)=0,$ in which case

Newton$$s method would converge quadratically $($again$).$ by python