Part $1-$ Gauss Elimination Function

Develop a MATLAB function called GaussElimination that solves a given system of linear equations of the form $Ax=b$ using the Gauss Elimination method. The function should take three input arguments, which must be listed in this order: the coefficient matrix $A,$ the right$-$side vector $b,$ and a true$/$false value. If the third value is true, the function should perform partial pivoting while solving the linear system. If the third value is false, the function should not perform partial pivoting. The function should return the vector $x.$ You must create your own implementation of the Gauss Elimination method; you may not use MATLAB's built$-$in matrix solvers $($e$.$g$.,$ you may not use the left division operator$).$ However, while developing your code, you may want to use the built$-$in matrix solvers to check the answers your code produces.

Refer to the pseudo code form lecture which shows where partial pivoting is performing during Gauss Elimination. You should place a conditional statement containing the partial pivoting code there. If a value of true is received in the input, the partial pivoting code should be executed, otherwise it should not.

$1$

gauss$($A$,$b$)$ : Performs Gauss elimination to solve for $x$ in $Ax=b$

Input:

$A=$ coefficient matrix $(nn)$

$b=$ right hand side vector

Output:

$x=$ solution vector

Confirm that the coefficient matrix $A$ is square

if $A$ is not square

stop execution and return an error value: NaN

end

setup augmented matrix $Ab=[A,b]$

Number of columns $nc=n+1$

$2$ back substitution

$3$ forward elimination

initialize solution vector $x($set all to zero$)$

$x(n)=AB\frac{n,nc}{A}b(n,n)$

for $k=n-1$ down to $1$

$x(k)=Ab(k,nc)$

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

for $p=1$ to $n-1,\frac{p}{0}$ is current pivot row

$x(k)=x(k)-Ab(k,j)**x(j)$

end

$x(k)=x\frac{k}{A}b(k,k)$

end

$\%$ if it is used, partial pivoting goes here

for $i=p+1$ to $n,\frac{0}{\text{@}}$ operate on all rows below pivot

factor $=Ab\frac{i,p}{A}b(p,p)$

for $c=p$ to $nc$

$Ab(i,c)=Ab(i,c)-$factor $***Ab(p,c)$;

end

end

end