In a class announcement this week, you will be given scripts for the MATLAB functions gs$.$m and mgs$.$m$,$ which implement the codes given in class for classical and,

respectively, modified Gram$-$Schmidt.

Copy them in your MATLAB directory, then go back to the Command Window and

type

A $=$ hilb$(8)$;

This makes A a particularly ill$-$conditioned $8\backslash$times $8$ matrix form MATLAB$$s matrix

library.

$1$

Then, call

$[$Q$1,$ R$1]=$ gs$($A$)$;

$[$Q$2,$ R$2]=$ mgs$($A$)$;

to get the computed factors Q$1$ and R$1,$ respectively, Q$2$ and R$2,$ with the classical,

respectively, modified Gram$-$Schmidt algorithms.

Check how close Q$1$ and Q$2,$ respectively, R$1$ and R$2$ are by typing

checkQ $=$ norm$($Q$1$ Q$2)$

checkR $=$ norm$($R$1$ R$2)$

Finally, let us now test the quality of the Q$1$ and Q$2$ factors. At the prompt, type

orthoQ$1=$ norm$($Q$1$

$$ Q$1$ eye$(8)),$

followed by

orthoQ$2=$ norm$($Q$2$

$$ Q$2$ eye$(8))$

These commands let you know how close to orthogonal the matrices Q$1,$ respectively

Q$2,$ are.

By examining checkQ, checkR, orthoQ$1$ and orthoQ$2,$ explain how classical GramSchmidt and modified Gram$-$Schmidt perform on this ill$-$conditioned matrix. Take

screenshots of the Command Window, making sure the outputs to the last $4$ commands

are visible.