$10\mathrm{.}5\mathrm{.}1$ Additional homework $$

$$Homework $10\mathrm{.}5\mathrm{.}1\mathrm{.}1.$ You may want to do a new "git pull" to update directory Assignments $.$

In Assignments$/$Week$10/$matlab you will find the files

Givens$\_$rotation.m: A function that computes a Givens' rotation from a

$2$

$\backslash$times

$1$

vector x$.$

Francis$\_$Step.m: A function that performs a Francis Implicit QR Step with a tridiagonal matrix

T

$($stored as the diagonal and subdiagonal of T$).$

Test$\_$Francis$\_$Step.m: A very rudimentary script that performs a few calls to the function Francis$\_$Step. Notice that our criteria for the routine being correct is that the matrix retains the correct eigenvalues.

With this,

Investigate the convergence of the

$($

m

$,$

m

$$

$1$

$)$

element of matrix T$1.$

Write a function

function T $=$ Spectral$\_$Decomposition$\_$Lambda$($ T $)$

That returns

$\backslash$Lambda

such that

T

$=$

Q

$\backslash$Lambda

Q

T

is the Spectral Decomposition of

T

$.$

The input matrix

T

is a tridiagonal matrix where only the lower triangular part of the matrix is stored in the diagonal and first subdiagonal of array T$.$ The diagonal matrix

$\backslash$Lambda

is returned in T $.$ The upper triangular part of the array should not change values. You are encouraged to call the function Francis$\_$Step from the function Spectral$\_$Decomposition$\_$Lambda. Obviously, you need to incorporate deflation in your implementation. How to handle the final

$2$

$\backslash$times

$2$

matrix is an interesting question... $($You may use the matlab function eig for this.$)$ PLEASE WRITE THE MATLAB CODE EXPLICITLY