Let A be the 10 × n matrix formed by the first n columns of the 10 × 10 Hilbert matrix. Let c be the n-vector [1,…,1], and set b = Ac. Use the normal equations to solve the least squares problem Ax = b for (a) n = 6 (b) n = 8, and compare with the correct least squares solution  = c. How many correct decimal places can be computed? Use condition number to explain the results. (This least squares problem is revisited in Computer Problem 4.3.7.)

Computer Problem 4.3.7

Let A be the 10 × n matrix formed by the first n columns of the 10 × 10 Hilbert matrix. Let c be the n-vector [1,…,1], and set b = Ac. Use the QR factorization to solve the least squares problem Ax = b for (a) n = 6 (b) n = 8, and compare with the correct least squares solution  = c. How many correct decimal places can be computed? See Computer Problem 4.1.8, where the normal equations are used