Computing the SVD of a square matrix

Question #1: In this problem, you will implement, in Matlab, a number of functions for computing the SVD of a square matrix. There are three matlab files:

• test_svd.m
  • Implicit_bidiag_QR.m
  • Implicit_bidiag_QR_SVD.m

Here is your assignment:

• Implement the reduction to bidiagonal form, BiRed discussed in “Ponder This 11.2.3.3” (https://www.cs.utexas.edu/users/flame/laff/alaff/chapter11-reduction-to-bidirectional-form.html). All you need to implement and test this function is discussed in that exercise. What to turn in: m

• Once you have this, you can also test it with the file test_SVD.m, which also will help you test other components of this first part of the exam. In particular, in test_SVD.m, find the section where you are asked to compute U_A and V_A such that A = U_ABV_A^T , where B is the bidiagonal matrix computed by BiRed. In the script, enter the lines needed to compute U_A and V_A. (You will eventually upload test_SVD.m, but only after you complete other parts of it as well.)

• Next, you are to implement a function Bidiag_Francis_Step that implements the introduction of the bulge into the bidiagonal matrix and the chasing out of that bulge. You will again use test_SVD.m to test your implementation. In test_SVD.m, there are hints as to how to do the implementation so that you slowly make it more sophisticated. Once completed, you upload m. (Note: Here, you are allowed to change the entire matrix and hence you don’t have to extract parts of the matrix, make them symmetric, compute, and then reinsert.)

• Next, you copy Bidiag_Francis_Step.m into Bidiag_Francis_Step_U_V.m and modify this copy so that you also update UA and VA by applying the Givens’ rotations, thus accumulating the U and V such that A = UΣV^T. Look at how Bidiag_Francis_Step_Update_U_V is called in Implicit_bidiag_QR_SVD to see how your function should its inputs and outputs There are additional instructions in test_SVD.m. When finished, submit m. Make sure you understand how Givens’ rotation have to be applied to what columns of U_A and V_A. This is where in the past a lot of people got confused and spent a lot of time.

• In the end, you will also need to upload your final m.

While initially you will want to just get it running, in the end you need to make sure that operating on the bidiagonal matrix is only O(n) and updating U_A and V_A is only O(n²) per call to the routine that implements the Francis step.