$(50$ points$)$

$(1)(5$ points$)$ Consider a normalized floating point number system with $\backslash$beta $=10($base$),$ p $=2,$ L $=4$

and U $=4.$ Compute the underflow and overflow levels of the system, as well as its machine

precision. If you use Matlab, report your results up to $4$ decimal digits.

$(2)$ Consider the following matrix

A $=$

$0\mathrm{.}0011$

$11$

$.$

$($a$)(10$ points$)$ Compute the LU factorization of A $($without partial pivoting$)$ by hand: A $=$ L$1$U$1.$

$($b$)(5$ points$)$ Assume that the floating point system approximates numbers by rounding. Compute

the floating point approximation of L$1$ and U$1$ in the floating point system given in question $(1)$

and denote Le $1=$ fl$($L$1),$ Ue $1=$ fl$($U$1).$ Hint: the floating point approximation of an n $\backslash$times n matrix

B$,$ denoted Be $,$ is defined such that Be $($i$,$ j$)=$ fl$($B$($i$,$ j$)),1<=$ i$,$ j $<=$ n$.$

$($c$)(5$ points$)$ Compute Ae $1=$ Le $1$Ue $1$ and $||$Ae $1$ A$||1/||$A$||1.$ If you use Matlab, report your results

up to $4$ decimal digits.

$($d$)(10$ points$)$ Compute the LU factorization with partial pivoting of A by hand: P A $=$ L$2$U$2.$

$($e$)(5$ points$)$ Assume that the floating point system approximates numbers by rounding. Compute

the floating point approximation of P $,$ L$2,$ U$2$ in the floating point system given in question $(1)$

and denote Pe $=$ fl$($P $),$ Le $2=$ fl$($L$2),$ Ue $2=$ fl$($U$2).$

$($f$)(5$ points$)$ Compute Ae $2=$ Pe Le $2$Ue $2$ and $||$Ae $2$A$||1/||$A$||1.$ If you use Matlab, report your results

up to $4$ decimal digits.

$($g$)(5$ points$)$ Is LU factorization backward stable? Is LU factorization with partial pivoting backward stable?