CS 417/517 Computational Methods and Software
Spring 2004
Assignment 6
Assigned: Thurs March 25, 2004; Due: Thurs April 1, 2004
1. Consider the least-squares problem Ax ≈bdiscussed in class, where
A=
1−1 1
1−0.5 0.25
1 0 0
1 0.5 0.25
1 1 1
, b =
1
0.5
0
0.5
2
.
Solve this problem in Matlab, using the normal equations, and the Cholesky factorization.
Show the matrix and the right-hand side vector in the normal equations, the Cholesky factor,
and the least-squares solution vector x.
2. Consider Problem 3.2from the text book. Instead of using the normal equations approach,
compute the least squares solution using the QR factors of the coefficient matrix via Matlab.
3. Problem 3.5from the book.
4. (a) If Q1and Q2are orthogonal matrices, show that their product Q1Q2is also an orthog-
onal matrix.
(b) If Qis an orthogonal matrix, then for any vector v, the 2-norm of Q v is equal to the
2-norm of v. Use this fact and the definition of the 2-norm of a matrix,
kQk2= max
v
kQvk2
kvk2
,
to show that the 2-norm of an orthogonal matrix is 1. What is the 2-norm condition
number of an orthogonal matrix?
