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1. Compute on paper the QR factorization of the following matrix using both classical and modified Gram-Schmidt algorithms. -8 3 2 -8 2 7 A= 3 2 - 1 2 7 - 1 4 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 2 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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