Math 411 Numerical Linear Algebra, Professor: Rakhim Aitbayev, April 4, 2003 1 Turn your test papers in on Tuesday, April 6. Summary: Math 411 ­ Numerical Linear Algebra, Professor: Rakhim Aitbayev, April 4, 2003 1 · Turn your test papers in on Tuesday, April 6. · You must work individually on this test. · Attach printouts of your computer code and its output with your comments. Test 2 Problem 1. Describe an algorithm to solve a least squares problem with a full-rank matrix. Problem 2. Write a MATLAB program that solves the least squares problem with a full-rank matrix A Rn×m. Include a reasonable number of comments in your program. The code should first implement a QR decomposition, A = QR, where Q Rn×n is a product of reflectors and R = [ ^R 0]T Rn×m, where ^R is m × m and upper triangular. Make use of algorithms (3.2.37), (3.2.40), and (3.2.45). Then, the code should use the QR decomposition to find x, the solution of the least squares problem. Calculate c = QT b = QmQm-1 . . . Q1b by applying the reflectors subsequently. An additional one-dimensional array is needed for b. This array can also be used for c and intermediate results. The solution x is found by solving ^Rx = ^c by back substitution, where c = [^c d]T . Find the minimum value of Ax - b 2 without computing Ax - b. Problem 3. 1. Use your program to solve the following problems. (a) Find the least squares quadratic polynomial for the data. ti -1 -0.75 -0.5 0 0.25 0.5 0.75 Collections: Mathematics