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MH6502 NUMERICAL LINEAR ALGEBRA AND DIFFERENTIAL EQUATIONS

Group Project on Algorithm, Implementation and Analysis

> Instructions:

It is a group project. Each group will consist of exactly 3 members to be teamed up by yourselves. Pkase email the names of your team by August 22, 2025 (Friday) via [email protected]

(i) Problems for this project will be ackded and updated as the lectures progress. Will stop adding problems after Oct. 1, 2025.

(ii) A written report is recuired, and the submission deadline will be Oct. 29, 2025 (Wednesday). The report should contain the steps of the implementation/algo rithms, presentation of the mumerical results (in figures or tables), and related anal ysis. You may refer to a sample report uploaded!

(iii) Project presentation will be on Oct. 23, 2025 (Thursday), and each group will be 20 minutes including Q&A.

(iv) The project will be graded by group, so each member in a group will be given the same mark.

> Problems and Tasks:

1. Finite Difference & Linear System.

(i) Let a(x) be a given smooth function, and consider the differential operator:

Use the Taylor expansion to show that

(ii) Let xy= jh for some small h, and let u, be the finite difference approximation of u(ry). Denote ajt1/2= a(xy ±h/2). Then we can obtain the second-order finite difference approximation

Follow the steps of the FD disretisation process stated in the lecture slides (Slide 18- Slide 20) of Chapter 1, and construct the second-order FD scheme to solve the differential equation:

               (1)

where a(z) 2 a >0 (bounded away from 0), c(z) 2 0 and f(x) are given functions.

(iii) Test the algorithm and code by the manufactured solution: () = sin(2πr) with a(x)=1+2 and c(x)=e fiom which the boundary conditions u(0)= u(1) = 0 are satisfied and also can evaluate f(æ) by substituting u,a, c into the equation.

(iv) Use the code to compute the mumerical solution with a = 1 + 2,c(z) =e but f(æ) = 1. Justify your solution.

2. Singular Value Decomposition (SVD). Read a gray-scale image (eg, via Matlab) and store it in matrix A (a non-square matrix by dividing 255 to nornalize the entries to [0,1]). Compute its SVD:

and approximate A by its rank-k appxoximation:

Define the error

where Il-lle is the matrix Frobenius nom. Plot the enor E for k = 1,2,...k, and verify the estimate  Depict the images for various k.

You may refer to the sample codes and images in the figure below.