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 Λ such that T=QΛQT 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 Λ 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 2×2 matrix is an interesting question... (You may use the matlab function eig for this.)