1. Compute on paper the QR factorization of the following matrix using both classical
and modified Gram-Schmidt algorithms.
2. Consider the system Ax = b where A is the matrix from problem 1) and b =
[1 1 1 1]t. Solve the system using the QR factorization obtained in 1).
3. Consider the function f: Rm
Rn defined by f fxx = Ax. Show that the Jacobian J
is equal to A. (Hint: Start with a 2 X 3 matrix and generalize.)
4. Consider the regularized least squares problem
min Il A.c -
Show that the regularized problem is equivalent to the least squares problem
min x A b 2
Show that the normal equation of this new least squares problem is
Consider L = I, the identity matrix. Using the SVD of A, show that the solution
of the normal equation is given by
5. Show that computing 2.c by x +
x is backward stable.
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