$10\mathrm{.}6$ Develop your own M$-$file to determine the $\backslash ($ L U $\backslash )$ factorization of a square matrix without partial pivoting. That is$,$ develop a function that is passed the square matrix and returns the triangular matrices $[$L$]$ and $[$U$].$ Test your function by using it to solve the system in Prob. $10\mathrm{.}3.$ Confirm that your function is working properly by verifying that $\backslash ([$L$][$U$]=[$A$]\backslash )$ and by using the built$-$in function lu$.$

Hint: most of the code is listed below.

$```$

function $[$L$,$ U$]=$ LUNaive $($A$)$

$\%$ LUNaive$($A$)$:

$\%$ LU decomposition without pivoting.

$\%$ input:

$\%$ A $=$ coefficient matrix

$\%$ output:

$\%$ L $=$ lower triangular matrix

$\%$ U $=$ upper triangular matrix

m$,$n$]=$size$($A$)$;

if m~$=$n$,$ error$(\text{'}$Matrix A must be square'$)$; end

L $=$ eye$($n$)$;

$```$

$```$

$\%$ forward elimination

for k $=1$:n$-1$

for i $=$ k$+1$:n

L$($i$,$k$)\}=\backslash$textrm$\{$U$\}($i$,$k$)/\backslash$textrm$\{$U$\}($k$,$k$)$

U$($i$,$k$)=0$;

$```$

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