$$

$\%$ Complete this program to perform the modified Gram$-$Schmidt procedure on $\%$ the columns of a matrix A$.$ This entails doing a forward sweep of Gram

$\%$ Schmidt followed by a reverse sweep. The advantage of this over the

$\%$ standard Gram$-$Schmidt procedure is much greater numerical stability,

$\%$ since reusing old columns that are only approximately orthogonal results

$\%$ in a "cascading error" effect. You wil$1$ need to implement each sweep.

$\%$ When completed, the program should output a before$-$error of $8\mathrm{.}0**{10}^{-3}$

: $($Matlab$)$ or $1\mathrm{.}5**{10}^{-2}($Octave$)$ and an after$-$error of ${4\mathrm{.}4}^{***}{10}^{-16},$ the

$\%$ latter of which is around machine precision $($i$.$e: as good as we can hope for

$\%$ in finite$-$precision arithmetic$).$ Please read the entire code before making

$\%$ changes.

clear all : This command erases past variables & other data

format compact : This suppresses extra Iines in the output

$A=$zeros$(24,17)$; A is a matrix of size $24$ by $17,$ initially a zero matrix

$\%$ now we will fill in the entries in A in a somewhat random way

Efor $i=1$:$24$

for $j=1$:$17$

$A(i,j)=\frac{1}{i^{{?}^{?}}2+j^2}$

end

end

: forward cycle of Gram$-$Schmidt

for $i=1$:$17$

for $j=1$:$(i-1)$

this$)$

end

$A($:$,i)=A($:$,i)$;$$ normalizing the ith col of A $($modify this$)$

end

residual$\_$error$\_$before $=$max$(max(\frac{?}{abs}(A^{\text{'}}**A-$eye$(17))))$; don't modify this!

$\%$ reverse cycle of Gram$-$Schmidt

for $i=17$:$-1$:$1$

for $j=17$:$-1$:$(i+1)$

$A($:$,i)=A($;$,i)$; $($modify this$)$

end

$A($:$,i)=A($:$,i)$;$\%$ renormalizing the ith col of $A($modify this$)$

end

$7$

$8$ residual$\_$error$\_$after $=$max$(max(\frac{?}{abs}(A^{\text{'}}**A-$eye$(17))))$;

$9\%$ By seeing how close $A^{\text{'}}**A$ is to the $1717$ identity matrix, we can get a

$0$ sense of "how" orthogonal the columns of A are after the procedure.