Write a function usolve, analogous to function lsolve in section $7\mathrm{.}2\mathrm{.}2,$

to solve an upper triangular system $Ux=y.[$Hint: Loops can be run

backwards, say, from $n$ down to $1,$ by typing in MATLAB: for $i=n$ :$-$

$1$:$1.$ Remember also that the diagonal entries in $U$ are not necessarily $1.$ Use this starting code in MATLAB:

$\%$ Gaussian elimination without pivoting

$\%$ for solving linear systems Ax$=$b

$\%$

$\%$A $=[123$; $456$; $780]$; b $=[102]\text{'}$;

$\%$A $=[2110$; $4331$; $8795$; $6798]$; b $=[2345]\text{'}$; $\%$ another example

n $=$ size$($A$,1)$;

$\%$

$\%---------------------------------------------\%$

$\%$ This is Step $1$ of Gaussian Elimination

$\%----------------------------------------------\%$

$\%$

for i$=2$:n $\%$ Loop over rows below row $1$

mult $=$ A$($i$,1)/$A$(1,1)$; $\%$ Subtract this multiple of row $1$ from

$\%$ row i to make A$($i$,1)=0.$

A$($i$,$:$)=$ A$($i$,$:$)-$mult$*$A$(1,$:$)$; $\%($this line is equivalent to the "for loop:

$\%$ for k$=1$:n$,$ A$($i$,$k$)=$ A$($i$,$k$)-$mult$*$A$(1,$k$)$; end; $")$

b$($i$)=$ b$($i$)-$ mult$*$b$(1)$;

end

U $=$ A $\%$ display U

$\%$

$\%$

$\%----------------------------------------------\%$

$\%$ All steps of Gaussian elimination

$\%----------------------------------------------\%$

$\%$

A $=[123$; $456$; $780]$; b $=[102]\text{'}$;

$\%$A $=[2110$; $4331$; $8795$; $6798]$; b $=[2345]\text{'}$; $\%$ another example

n $=$ size$($A$,1)$;

for j$=1$:n$-1\%$ Loop over columns.

for i$=$j$+1$:n $\%$ Loop over rows below j$.$

mult $=$ A$($i$,$j$)/$A$($j$,$j$)$; $\%$ Subtract this multiple of row j from

$\%$ row i to make A$($i$,$j$)=0.$

A$($i$,$:$)=$ A$($i$,$:$)-$ mult$*$A$($j$,$:$)$; $\%$ This does more work than necessary! WHY?

b$($i$)=$ b$($i$)-$ mult$*$b$($j$)$;

end

end $\%$ the resulting A is an upper triangular matrix

U $=$ A $\%$ display U

$\%$

$\%$

$\%----------------------------------------------\%$

$\%$ Modified Gaussian elimination

$\%($to avoid recomputing zeros$)$

$\%----------------------------------------------\%$

$\%$

A $=[10^-161$; $11]$; b $=[2$; $3]$;

$\%$A $=[123$; $456$; $780]$; b $=[102]\text{'}$;

$\%$A $=[2110$; $4331$; $8795$; $6798]$; b $=[2345]\text{'}$; $\%$ another example

n $=$ size$($A$,1)$;

for j$=1$:n$-1\%$ Loop over columns.

for i$=$j$+1$:n $\%$ Loop over rows below j$.$

mult $=$ A$($i$,$j$)/$A$($j$,$j$)$; $\%$ Subtract this multiple of row j from

$\%$ row i to make A$($i$,$j$)=0.$

A$($i$,$j:n$)=$ A$($i$,$j:n$)-$ mult$*$A$($j$,$j:n$)$; $\%$ modified! no more recomputing of

$\%$ of zeros in columns $1$ to j$-1$ and rows j to n$.$

b$($i$)=$ b$($i$)-$ mult$*$b$($j$)$;

end

end $\%$ the resulting A is an upper triangular matrix

U $=$ A $\%$ display U