ALAFF Additional homework

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Subsection 10.5.1 Additional homework

Homework 10.5.1.1.

You may want to do a new "git pull" to update directory Assignments .

In Assignments/Week10/matlab you will find the files

Givens_rotation.m: A function that computes a Givens' rotation from a \(2 \times 1 \) 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 T1.
Write a function
```
function T = Spectral_Decomposition_Lambda( T )
```
That returns \(\Lambda \) such that \(T = Q \Lambda Q^T \) is the Spectral Decomposition of \(T \text{.}\) 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 \(\Lambda \) 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 \(2 \times 2 \) matrix is an interesting question... (You may use the matlab function eig for this.)