Problem 1

Let A∈Rn×n�∈��×� be any matrix, and I� be the n×n�×� identity matrix. Show that the matrix Lk=I+mkeTk��=�+����� is the inverse of the matrix Mk=I−mkeTk��=�−�����, where Mk�� is the elimination matrix for the kth��ℎ column ak�� of A�.

Problem 2

Write a program to compute the LU decomposition of a matrix A� using the concept of elimination matrices. Use it to solve the following system:

Problem 3

Suppose that ∥⋅∥a‖⋅‖� and ∥⋅∥b‖⋅‖� are two equivalent vector norms in Rn��. Thus, by definition, there exist two constants c,d>0�,�>0 such that for any vector x∈Rn�∈��, we have

Prove that their induced matrix norms are also equivalent, i.e., there exist two constants c′,d′>0�′,�′>0 such that for any matrix A∈Rn×n�∈��×�

Problem 4

1. Use a single-precision routine for Gaussian elimination to solve the system Ax=b��=� defined below

1. Compute the residual r=b−Ax�=�−�� using double-precision arithmetic, but storing the final result in a single-precision vector r�. (Note that the solution routine may destroy the matrix A�, so you may need to save a separate copy for computing the residual.)
2. Solve the linear system Az=r��=� to obtain the “improved” solution x+z�+�. (Note that the matrix A� need not be refactored.)
3. Repeat steps b)�) and c)�) until no further improvement is observed.

Problem 5

Use Gaussian elimination without pivoting to solve the linear system

for ε=10−2k�=10−2�, where k=1,2,…,10�=1,2,…,10. The exact solution is x=(1,1)T�=(1,1)�, independent of the value of ε�. How does the acccuracy of the computed solution behave as the value of ε� decreases?