Question
1-LU decomposition
given a lower triangular matrix L, write down a program to solve linear system Lx = b. Count the number of multiply-divide (MD) operations involved. Run your program on L = [1, 0, 0; 1, 2, 0; 1, 2, 3] and b = [1; 5; 14]
2- Tridiagonal System
Solve the following tridiagonal system Ax = b by hand calculations:
A = [2, 1, 0, 0; 1, 2, 1, 0; 0, 1, 2, 1; 0, 0, 1, 2] and b = [1; 0; 0;-1]. compute all the 's, 's, g's and x's to show intermediate steps.
3- Stability and conditioning
Let A = [19, 20; 20, 21] and b = [39; 41]. Find the condition number cond(A). Solve for the system Ax = b. Now relate small perturbation to b. Let bp = [39.01; 40.99]. Now find xp. What is the relative error in ||b|| and in ||x||. How are they associated?