(This part is worth 15% of the total marks for the course)
There is a signal b collected at M sampling points(b(1), b(2), …b(m), …,b(M)), and you are asked to strategically place N components x (x(1), x(2), …,x(n),…x(N)) into the system to modify the signal profile. The modified signal profile is expressed as
The matrix A and vector b are stored in files: A.mat and b.mat, and in Matlab, you can access the data as follows: load A; load b.
You are asked to implement the following tasks:
Use the Singular Value Decomposition (SVD) technique to find x (consider low-rank approximation r=200).
Based on your SVD results, find the condition number of A.
(Note- SVD can be calculated by Matlab built-in function).
Use the regularisation approach (regularised linear least squares) to find x.
Select different regularisation parameters 𝜆 = 1𝑒−10 , 1𝑒−9, 1𝑒−8,1𝑒−7, 1𝑒−6, 1𝑒−5, 1𝑒−4, 1𝑒−3,1𝑒−2
Plot an L-curve figure that shows the relationship between 2-norm(x) and 2-norm(Axb) with different regularisation parameters.
(Note – refer to lecture notes (LA9, Least square problems).
The signal B will satisfy the following condition: |𝐁 − 𝐁𝟎| ≤ 𝐁𝟎 ∈, where, 𝑩𝟎 is the mean value of B. And ∈ (𝑚) = 1𝑒−4, 𝑚=1,2,…M. In addition, b and B are both positive vectors. The range of components x: 0≤x(n) ≤ 5𝑒−3 , n=1,2,…,N.
Write a Matlab code to minimise the 1-norm of vector x.
Note: please consider the function linprog() in Matlab;
(This part is worth 9% of the total marks for the course)
Assessment Type: Application
The student needs to prepare a presentation (ppt file (at least ten slides) + audio recording file) on real-world applications that use concepts learned in the linear algebra (LA) part.
The presentation should contain the following sections
(3) results and discussion/interpretation;
(4) conclusion and