implement the algorithm

$28\mathrm{.}2.$ The preliminary reduction to tridiagonal form would be of little use if

the steps of the QR algorithm did not preserve this structure. Fortunately,

they do$.$

$($a$)$ In the QR factorization A$=$ QR of a tridiagonal matrix A$,$ which entries

of Rare in general nonzero? Which entries of Q$?($In practice we do not form

Q explicitly.$)$

$($b$)$ Show that the tridiagonal structure is recovered when the product RQ is

formed.

$($c$)$ Explain how Givens rotations or $2$ x $2$ Householder reflections can be used

in the computation of the QR factorization of a tridiagonal matrix, reducing

the operation count far below what would be required for a full matrix. Prepare an error table with the successive differences using the previous exercise; for this calculate

ek $=$ log$(||$Pk $$ Pk$1||2).$ Make an example with an array of size $20$ and $15$ iterations.