Homework 3
Problem 1
Write a Gaussian elimination program for solving Ax = b. Consider the following test
problems.
1) The matrix A (n × n) is a tri-diagonal matrix with 2 on the main diagonal and −1 on
sub- and super- diagonal (in matlab
A = 2 ∗ diag(ones(n, 1)) + (−1) ∗ diag(ones(n − 1, 1), 1) + (−1) ∗ diag(ones(n − 1, 1), −1)).
In this example, try to take an advantage of the banded structure of the matrix and store
only nonzero elements. Take n = 50, 100 and b = (1, . . . , 1)T .
Problem 2
1) Compute condition numbers using norms kAk2 and kAk∞ :
a+1
a
1 2
α 1
a) A =
, b) A =
, c) A =
,
a
a−1
1 2.01
1 1
2) In solving Ax = b for the matrix A from part b) predict how slight changes in b will
0
affect the solution x, and test your prediction in the concrete case b = (4, 4)T and b = (3, 5)T .
Problem 3
Show that if a square matrix A satisfies kAxk ≥ θkxk for all x, with θ > 0, then A is
nonsingular and kA−1 k ≤ θ −1 . This is valid for any vector norm and its subordinate matrix
norm.
Remark.PNote that if A is diagonally dominant, then kAxk∞ ≥ θkxk∞ , where θ =
mink (|akk | − nj=1,j6=k |akj ).
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