1. (a) (5 points) Use the Gaussian elimination method to find two matrices L,U such that A = LU where L is lower triangular and U is upper triangular.
(b) (5 points) Then solve the linear system Ax = b using the above matrices L and U.
The code should work for any non-singular square matrix A. Use a proper A and b to verify the results.
2. (5 points) Find the conditional number of the following matrix for n = 3,4,3,5,6 using the row-norm, column-norm and Euclidean-norm. Do not use the inbuilt norm function from the Matlab library. What do you observe in the results?
.
3. (5 points) A tridiagonal system with n unknowns is given by,
aixi−1 + bixi + cixi+1 = di; i = 1,2,...,n
where a1 = cn = 0. Solve the above system using the following method which is a special case of Gaussian elimination method:
; ; i = 1
;; i = 2,3,...,n.
xn = d0n xi = d0i − c0ixi+1 ; i = n − 1,n − 2,...,1.
