Implement the procedure $1$ Naive Gauss and Gaussin algorithm with sclaed partial pivoting: procedures $2$ Gauss and Solve

Apply the procedures to solve the matrix. Write a single test program for both procedures.

procedure Naive Gauss$($n$,($ai j $),($bi $),($xi $))$ integer i$,$ j$,$ k$,$ n; real sum, xmult

real array $($ai j $)1$:n$\backslash$times $1$:n $,($bi $)1$:n $,($xi $)1$:n

for k $=1$ to n $1$ do

for i $=$ k $+1$ to n do xmult $$ aik$/$akk

aik $$ xmult

for j $=$ k $+1$ to n do

aij $$ aij $($xmult$)$akj end for

bi $$ bi $($xmult$)$bk end for

end for

xn $$bn$/$ann

fori $=$n$1$to$1$ step$1$do

sum $$ bi

for j $=$ i $+1$ to n do

sum $$ sum $$ ai j x j end for

xi $$ sum$/$aii end for

end procedure Naive Gauss

procedure Gauss$($n$,($ai j $),($li $))$

integer i$,$ j$,$ k$,$ n; real r$,$ rmax, smax, xmult

real array $($ai j $)1$:n$\backslash$times $1$:n $,($li $)1$:n ; real array allocate $($si $)1$:n for i $=1$ to n do

li $$i

smax $0$

for j $=1$ to n do

smax $$ max$($smax$,|$ai j $|)$ end for

si $$ smax end for

for k $=1$ to n $1$ do rmax $0$

for i $=$ k to n do

r $|$ali $,$k $/$sli $|$

if $($r $>$ rmax$)$ then rmax $$ r

j$$i end if

end for

lj $$lk

for i $=$ k $+1$ to n do

xmult $$ ali $,$k $/$alk $,$k ali,k $$xmult

for j $=$ k $+1$ to n do

ali,j $$ ali,j $($xmult$)$alk$,$j end for

end for end for

deallocate array $($si $)$ end procedure Gauss

$\left[\begin{array}{cccc}2& 3& -4& 1\\ 1& -1& 0& -2\\ 3& 3& 4& 3\\ 4& 1& 0& 4\end{array}\right]$